Use Pythagoras' theorem to work out the length of the hypotenuse in the triangle on the right, below.
Give your answer in centimetres (cm) and give any decimal answers to 1 d.p.
Answers
This is an exercise of the Pythagorean Theorem, which establishes that in every right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides, called legs.
This can be expressed mathematically as:
c = √(a² + b²) ⇔ a² + b² = c²Where "a" and "b" are the lengths of the legs and "c" is the length of the hypotenuse. This theorem is one of the fundamental bases of geometry and has many applications in physics, engineering, and other areas of science.
The Pythagorean Theorem formula is used to calculate the length of an unknown side of a right triangle, as long as the lengths of the other two sides are known. It can also be used to determine if a triangle is right if the lengths of its sides are known.
To calculate the hypotenuse, we will apply the formula:
c = √(a² + b²)
Knowing that:
a = 8cm
b = 15cm
Now we just substitute the data in the formula, and calculate the hypotenuse, then
c = √(a² + b²)c = √((8 cm)² + (15 cm)²)c = √(64 cm² + 225 cm²c = √(289 cm²)c = 17 cmThe hypotenuse C of the triangle is 17 cm.
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Related Questions
Find the value of x.
If necessary, you may learn what the markings on a figure indicate.
to
73°
X =
Answers
The value of the angle is 34 degrees
How to determine the valueTo determine the value of the variable, we need to the following;
The sum of triangle theorem states that the sum of the angles in a triangle is 180 degreesAlternate angles are know to be equalAn isosceles triangle has two of its sides equalTwo of its angles are equalFrom the information given, we have that the angles are;
73 degrees
73 degrees
x degrees
Equate the angles
73 + 73 +x = 180
collect the like terms
x = 180 - 146
subtract the values
x = 34 degrees
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The mode is ...
A) the middle number in a numerical data set when the values have been arranged in
numerical order.
B) the number or numbers occurring most frequently in a data set.
C) a measure of dispersion.
D) The difference of the highest value and lowest value in the data set.
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Complete the inductive step, identifying where you use the inductive hypothesis. (You must provide an answer before moving to the next part.) Multiple Choice O Replacing the quantity in brackets on the left-hand side of part (c) by what it equals by virtue of the Inductive hypothesis, we have (kok+") + (x + 1)2 = (x + 1)2 +68+4)* (+1)(x+2) as desired. O Replacing the quantity in brackets on the left-hand side of part (c) by what It equals by virtue of the inductive hypothesis, we have (k++) + (x + 1)2 = (x + 1)2 *****!) -(+1Xk+2) * as desired. O Replacing the quantity in brackets on the left-hand side of part (c) by what it equals by virtue of the inductive hypothesis, we have ( kk+) + (k+ 1)2 = (x + 1)2( 344x+2) = (x+1}+2) as desired. O Replacing the quantity in brackets on the left-hand side of part (c) by what it equals by virtue of the inductive hypothesis, we have (64 + (k+ 1)2 = (k+ 1)2 (+4x+1) = (+1}x+2) as desired.
Answers
Completing the inductive step and identifying where the inductive hypothesis is used, the correct multiple choice answer is: Replacing the quantity in brackets on the left-hand side of part (c) by what it equals by virtue of the Inductive hypothesis, we have (kok+") + (x + 1)² = (x + 1)² +68+4)* (+1)(x+2) as desired.
In an induction proof, the inductive step involves assuming a statement is true for some arbitrary value, say k, and then proving it's true for the next value, k+1.
Here, the inductive hypothesis corresponds to the term (kok+"). By replacing this term on the left-hand side of part (c) with its equivalent based on the inductive hypothesis, we can show that the equation holds for the (k+1) case as well. This is crucial for proving the statement using induction, as it establishes the necessary pattern for all cases.
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38% adults favor the use of unmanned drones by police agencies. Twelve U.S. adults are randomly selected. Find the probability that the number of U.S. adults who favor the use of unmanned drones by police agencies is:
(a). exactly three: P(3) =
(b). at least four: P(x\geq4)=
(c). less than eight: P(x<8)=
Answers
The probability that the number of U.S. adults who favor the use of unmanned drones by police agencies is:
(a) P(3) = 0.2636
(b) P(x≥4) = 0.1814
(c) P(x<8) = 0.9997
(a) To find the probability that exactly three out of twelve U.S. adults favor the use of unmanned drones by police agencies, we can use the binomial probability formula:
P(3) = (12 choose 3) * (0.38)^3 * (1-0.38)^(12-3) = 0.2636
where (12 choose 3) = 12! / (3! * 9!) represents the number of ways to choose 3 out of 12 adults.
(b) To find the probability that at least four out of twelve U.S. adults favor the use of unmanned drones by police agencies, we can use the complement rule and subtract the probability of having three or fewer adults who favor the use of drones from 1:
P(x≥4) = 1 - P(x≤3) = 1 - [(12 choose 0) * (0.38)^0 * (1-0.38)^(12-0) + (12 choose 1) * (0.38)^1 * (1-0.38)^(12-1) + (12 choose 2) * (0.38)^2 * (1-0.38)^(12-2) + (12 choose 3) * (0.38)^3 * (1-0.38)^(12-3)] = 0.1814
(c) To find the probability that less than eight out of twelve U.S. adults favor the use of unmanned drones by police agencies, we can sum up the probabilities of having zero to seven adults who favor the use of drones:
P(x<8) = P(x=0) + P(x=1) + ... + P(x=7) = (12 choose 0) * (0.38)^0 * (1-0.38)^(12-0) + (12 choose 1) * (0.38)^1 * (1-0.38)^(12-1) + ... + (12 choose 7) * (0.38)^7 * (1-0.38)^(12-7) = 0.9997
Note that the probability of having eight or more adults who favor the use of drones is negligible.
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Suppose you have a regression model with an interaction term and a dummy variable. In this case, we can have a only one slope and only one intercept b.only one slope, but more than one intercept. c. more than one slope, but only one intercept d. more than one slope and more than one intercept.
Answers
When a regression model has an interaction term and a dummy variable in statistics and probability, there will be more than one slope and more than one intercept (D)
When there is an interaction term and a dummy variable in a regression model, we can have more than one slope and more than one intercept. The interaction term allows for different slopes for different levels of the dummy variable, while the intercepts represent the expected value of the dependent variable when the dummy variable is equal to zero for each level of the interaction term.
When a regression model has an interaction term and a dummy variable, it means that the effect of one independent variable on the dependent variable varies depending on the value of the other independent variable. In other words, the slope and intercept of the regression line will change depending on the value of the dummy variable.
More specifically, the model will have one intercept and two slopes: one for the dummy variable and one for the interaction term. As a result, the relationship between the dependent variable and the independent variables will vary depending on the value of the dummy variable, which will result in different slopes and intercepts.
Therefore, the correct answer is (d): more than one slope and more than one intercept.
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Three factories produce the same tool and supply it to the market. Factory A produces 30% of the tools for the market and the remaining 70% of the tools are produced in factories B and C. 98% of the tools produced in factory A, 95% of the tools produced in factory B and 97% of the tools produced in factory C are not defective. What percent of tools should be produced by factories B and C so that a tool picked at random in the market will have a probability of being non defective equal to 96%?
Answers
The percent of tools should be produced by factories B and C so that a tool picked at random in the market will have a probability of being non defective equal to 96% = 0.96 are 5% and 95% respectively.
We have three factories produce the same tool and supply it to the market. Let's consider three events defined as, A = event for tools produced by factory A
B = event for tools produced by factory B
C = event for tools produced by factory C and N be the count that tools produced by all factories is not defective.
The probability that the tools produced by factory A for the market, P( A) = 30%
= 0.30
The probability that the tools produced by factories B and C for the market, P( B and C) = 70% = 0.70
The Probability that tools are non- defective and that are produced in factory A, P( N/A) = 98%
= 0.98
The Probability that tools are non- defective and that are produced in factory B, P( N/B) = 95%
= 0.95
The Probability that tools are non- defective and that are produced in factory C, P( N/C) = 97% = 0.97
Now, since only three factories supply to the whole market, then by probability law, P(A) + P(B) + P( C) = 1
=> 0.3 + P(B) + P( C) = 1
=> P(B) = 0.7 - P(C) --(1)
We have to determine percent of tools should be produced by factories B and C that is P(C) and P(B) when probability of non defective, P(N) is 96% = 0.96. From the law of total probability law, P(N) is written by, P( N) = P( N/A) P(A) + P( N/B) P(B) + P( N/C) P( C)
=> 0.96 = 0.98 × 0.3 + 0.95 × ( 0.7 - P(C) ) + 0.97 × P(C)
=> 0.96 = 0.98 × 0.3 + 0.95 × 0.7 - 0.95 P(C) + 0.97 × P(C)
=> 0.96 = 0.98 × 0.3 + 0.95 × 0.7 - 0.95 P(C) + 0.97 × P(C)
=> 0.96 = 0.294 + 0.665 + 0.02 × P(C)
=> 0.96 = 0.959 + 0.02 × P(C)
=> 0.02 × P(C) = 0.96 - 0.959
=> 0.02 × P(C) = 0.001
=> P(C) = 0.05 = 5%
from equation (1), P(B) = 1 - P(C)
=> P( B) = 1 - 0.05 = 0.95
Hence, required percentage is 95%.
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21.5 ÷ 5 + (80.6 - 12.5 ÷ 2)
PEMDAS
Answers
Answer:
78.65
Step-by-step explanation:
the binomial theorem states that for any real numbers a and b (a b)^n
Answers
The binomial theorem cannot be used to expand expressions of the form (a - b)^n, where n is an even integer and a and b are real numbers.
The binomial theorem states that for any real numbers a and b, and a non-negative integer n, the expression (a+b)^n can be expanded as the sum of the terms in the form:
(a+b)^n = C(n,0) * a^n * b^0 + C(n,1) * a^(n-1) * b^1 + ... + C(n,n) * a^0 * b^n
where C(n,k) denotes the binomial coefficient, which can be calculated using the formula:
C(n,k) = n! / (k! * (n-k)!)
In this expansion, each term represents a product of the powers of a and b, with the exponents summing up to n. The binomial coefficients, C(n,k), indicate the number of ways to choose k items from a set of n items.
So, the binomial theorem allows us to expand expressions involving the sum of two real numbers raised to a power, using the binomial coefficients and the powers of the real numbers.
The binomial theorem is a powerful formula that allows us to expand expressions of the form (a + b)^n, where n is a non-negative integer. Specifically, the theorem states that (a + b)^n = sum from k=0 to n of (n choose k) * a^(n-k) * b^k, where (n choose k) denotes the binomial coefficient, which is equal to n! / (k! * (n-k)!). However, if we let b = -a, then (a + b)^n becomes (a - a)^n = 0^n = 0. Therefore, the binomial theorem cannot be used to expand expressions of the form (a - b)^n, where n is an even integer and a and b are real numbers. In such cases, we need to use alternative methods such as the difference of squares formula or the factor theorem.
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Test the series for convergence or divergence. [infinity]
Σ (-1)^n+1/3n^4 . n=1 - converges
- diverges
Answers
The answer is: Test the series for convergence or divergence. [infinity] Σ (-1)n+1/3n² - converges.
To test the series Σ (-1)n+1/3n² for convergence or divergence, we can use the alternating series test. This test states that if a series alternates in sign and the absolute value of its terms decreases monotonically to zero, then the series converges.
In this case, the series Σ (-1)n+1/3n² alternates in sign and the absolute value of its terms is given by 1/3n², which decreases monotonically to zero as n increases. Therefore, we can apply the alternating series test and conclude that the series converges.
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16 Write a decimal number on each answer line to make each statement correct.
8.43
843 hundredths =
84 tenths and 3 thousandths
8 ones 4 hundredths and 3 thousandths
8+0.4+ 0.03
Answers
The required decimal numbers are 8.43, 8.403, 8.403, and 8.43.
Place value and decimal notation:In mathematics, place value is the value of a digit in a number based on its position. For example, in the number 123, the digit 3 is in the one's place, representing the value of 3 ones.
Decimal notation is a system of writing numbers using a base value of 10 and the digits 0-9. In decimal notation, each digit in a number represents a multiple of a power of 10. For example, in the number 123.45, The digit 4 is in the tenth place, representing the value of 4 tenths.
Here we have 8.43
The number can be expressed as follows
8.43 = 843 hundredths = 8.43
8.43 = 84 tenths and 3 thousandths = 8.403
8 ones 4 hundredths and 3 thousandths = 8.403
8.43 = 8 + 0.4 + 0.03 = 8.43
Therefore,
The required decimal numbers are 8.43, 8.403, 8.403, and 8.43.
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Colin, Dave and Emma share some money.
Colin gets 3⁄10 of the money.
Emma and Dave share the rest of the money in the ratio 3 : 2 What is Dave's share of the money
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Make the amount of money they have £100 because this makes the question easier.
Colin gets 3/10 of the money, so Colin will get £30.
After Colin has taken his share £70 will be left over.
The ratio give is 3 : 2. So 3 + 2 is equal to 5.
The amount of money left over is then divided by the ratio added in this case its 70/5.
70/5 gives us an answer of 14 .
This means that each share is equal to £14.
Emma gets the ratio of 3 so we do 3 x 14 which gives us he answer of £42.
And if we do 3 x 2 we get the answer of £28.
We then know Dave gets £26 pounds from the £100 at the start.
26/100 converted to a percentage is 26%.
Ashlee purchased a house for $875 000. She made a down payment of 15% of the purchase price and took out a mortgage for the rest. The mortgage has an interest rate of 6.95% compounded monthly, and amortization period of 20 years, and a 5 year term. Calculate Ashley’s monthly payment.
Answers
$5744 is Ashley’s monthly payment.
The amount of the down payment made by Ashlee is 15% of $875,000, which is:
Down payment = 0.15 x $875,000 = $131,250
The amount that Ashlee took out on a mortgage is:
Mortgage amount = Purchase price - Down payment
= $875,000 - $131,250
= $743,750
The monthly payment on a mortgage:
[tex]M = P [ i(1 + i)^n ] / [ (1 + i)^n - 1 ][/tex]
where:
M = monthly payment
P = principal amount (mortgage amount)
i = monthly interest rate (annual interest rate / 12)
n = total number of monthly payments (amortization period x 12)
In this case, the annual interest rate is 6.95% and the term is for 5 years, so we need to first calculate the monthly interest rate and the total number of monthly payments.
Monthly interest rate = 6.95% / 12 = 0.57917%
Total number of monthly payments = 20 years x 12 = 240
Substituting these values into the formula, we get:
M = $743,750 [ 0.0057917 (1 + 0.0057917)^240 ] / [ (1 + 0.0057917)^240 - 1 ]
= $5744.002
Therefore, Ashley's monthly payment on the mortgage is $5744.
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S
1
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2
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Which translation maps the graph of the function f(x) = x² onto the function g(x) = x² − 6x + 6?
Oleft 3 units, down 3 units
Oright 3 units, down 3 units
Oleft 6 units, down 1 unit
Oright 6 units, down 1 unit
Answers
Answer:
right 3 units, down 3 unitsStep-by-step explanation:
You want the translation that maps f(x) = x² to g(x) = x² -6x +6.
GraphA graph of the two functions shows g(x) is right 3 units and down 3 units from f(x).
Vertex formWe know the vertex of f(x) = x² is the origin (0, 0). The vertex of g(x) will tell us the translation. Putting that function in vertex form, we have ...
g(x) = x² -6x +6
g(x) = (x² -6x) +6
g(x) = (x² -6x +9) +6 -9 . . . . . add and subtract 9 to complete the square
g(x) = (x -3)² -3
Compare this to ...
y = (x -h)² +k . . . . . . has vertex (h, k)
We see that (h, k) = (3, -3).
g(x) is translated right 3 units and down 3 units.
(c) immediately after the switch is open (after being closed a long time)... ...the current through the inductor is = 20.4 correct: your answer is correct. ma ...the current through r2
Answers
The current through R2 will depend on the values of the components in the circuit and the initial current through the inductor. Without more information, it is not possible to determine the current through R2.
After the switch is open, the current through the inductor will continue to flow in the same direction but will gradually decrease over time. The current through R2 will depend on the values of the components in the circuit and the initial current through the inductor. Without more information, it is not possible to determine the current through R2.
We want to know the current through resistor R2 immediately after the switch is opened, given that the current through the inductor is 20.4 mA. To provide an accurate answer, I would need more information about the circuit, such as the values of the resistors, inductor, and any voltage sources. However, I will explain the concept behind the problem.
When the switch is opened after being closed for a long time, the inductor behaves like a current source due to its stored energy. Since the current through the inductor is given as 20.4 mA, the current flowing through R2 will be the same (20.4 mA) immediately after the switch is opened, assuming there are no other current paths in the circuit.
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Help please, i don't get it i need it done asap
Answers
The number of boxes that can fit into the crate is 7 boxes.
What is the shape of a cuboid?A cuboid has a hexahedron six-faced solid shape and the volume is determined by multiplying the length by width by height. Here; the volume of the crate is determined by finding the volume of the cuboid.
Volume of the cuboid is: 2.4 m × 1.8 m × 1.1 m
Volume of the cuboid = 4.752 m³
To cm, volume of the cuboid = 475.2 cm³
Now, since the cube has a length of 60 cm, then the number of boxes that will fit into the crate can be estimated by dividing the volume of the cuboid shape by the length of the cube.
Thus, the number of boxes that can fit into the crate is:
= 475.2 cm/ 60 cm
= 7. 92
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The general solution of y" - 8y' + 16y = 4e^4x + e^4x/x is: (a) y = C_1e^-4x + C_2 xe^-4x + 4x^2 e^4x + e^4x ln x (b) y = C_1 e^4x + C_2 xe^4x + 2e^4x + xe^4x ln x (c) y = C_1 e^4x + C_2 xe^4x + 2x^2 e^4x + xe^4x ln x (d) y = C_1 e^4x + C_2xe^4x + 2xe^4x + xe^4x ln x (e) None of the above.
Answers
The general solution of differential equation y" - 8y' + 16y = 4e^4x + e^4x/x is y = C_1e^-4x + C_2 xe^-4x + 4x^2 e^4x + e^4x ln x. So, the correct answer is A).
The given differential equation is
y" - 8y' + 16y = 4e^(4x) + e^(4x)/x
The characteristic equation is
r^2 - 8r + 16 = 0
Solving this equation, we get
r = 4 (repeated root)
So, the homogeneous solution of the differential equation is
y_h = (C_1 + C_2x) e^(4x)
To find the particular solution, we will use the method of undetermined coefficients.
For the first term 4e^(4x), we can take the particular solution as
y_p1 = A e^(4x)
Differentiating and substituting in the differential equation, we get
16A e^(4x) - 32A e^(4x) + 16A e^(4x) = 4e^(4x)
Simplifying, we get
A = 1/4
So, the particular solution for 4e^(4x) is
y_p1 = (1/4) e^(4x)
For the second term e^(4x)/x, we can take the particular solution as
y_p2 = B e^(4x) ln x
Differentiating and substituting in the differential equation, we get
16B ln x e^(4x) - 8B e^(4x) + 16B e^(4x) ln x = e^(4x)/x
Simplifying, we get
B = 1/8
So, the particular solution for e^(4x)/x is
y_p2 = (1/8) e^(4x) ln x
Therefore, the general solution of the given differential equation is
y = y_h + y_p1 + y_p2
y = (C_1 + C_2x) e^(4x) + (1/4) e^(4x) + (1/8) e^(4x) ln x
Hence, the correct option is (a) y = C_1e^-4x + C_2 xe^-4x + 4x^2 e^4x + e^4x ln x.
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what are the values of these sums? a) ∑ 5 k =1 (k 1) b) ∑4 j=0 (−2)j c) ∑ 10 i=1 3 d) ∑ 8 j=0 (2j 1 − 2j )
Answers
The values for the sums are: a) 20, b) 11, c) 30, and d) -430.
Here are the values for each:
a) ∑_(k=1)^5 (k+1) = (1+1) + (2+1) + (3+1) + (4+1) + (5+1) = 2 + 3 + 4 + 5 + 6 = 20
b) ∑_(j=0)^4 (-2)^j = (-2)^0 + (-2)^1 + (-2)^2 + (-2)^3 + (-2)^4 = 1 - 2 + 4 - 8 + 16 = 11
c) ∑_(i=1)^10 3 = 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 = 30 (since there are 10 terms, each with a value of 3)
d) ∑_(j=0)^8 (2j+1 - 2^j) = ∑_(j=0)^8 (2j+1) - ∑_(j=0)^8 (2^j)
First, find the two separate sums:
∑_(j=0)^8 (2j+1) = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 = 81
∑_(j=0)^8 (2^j) = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 = 511
Now subtract the two sums: 81 - 511 = -430
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solve each system of inequalities and indicate all the integers that are in the solution set. 2-6y<14 and 1<21-5y
Answers
Answer:
{-1, 0, 1, 2, 3}
Step-by-step explanation:
. 2-6y<14
-6y < 12
y > 12/-6
y > -2.
1<21-5y
-5y > -20
y < 4.
So -2 < y < 4
and the solution set of integers is
{-1, 0, 1, 2, 3}
A sample of size n=150 showed a skewness coefficient of −0.45 and a kurtosis coefficient of +0.85. What is the distribution's shape? Multiple Choice
A. The distribution is normal.
B. The distribution is skewed left and leptokurtic.
C. The distribution is skewed right.
Answers
The distribution is skewed left and leptokurtic is the correct shape of distribution. The correct answer is option B.
Based on the given information, the sample of size n=150 showed a skewness coefficient of -0.45 and a kurtosis coefficient of +0.85. Skewness refers to the degree of asymmetry in a distribution, while kurtosis measures the degree of peakedness or flatness in a distribution. A skewness coefficient of -0.45 indicates that the distribution is skewed to the left. This means that the tail of the distribution is longer on the left side, and the peak of the distribution is shifted to the right. On the other hand, a kurtosis coefficient of +0.85 indicates that the distribution is leptokurtic. This means that the distribution has a sharper peak and heavier tails than a normal distribution.Combining these two pieces of information, we can conclude that the distribution is skewed left and leptokurtic. Therefore, the correct answer is option B: The distribution is skewed left and leptokurtic. It is important to note that the skewness and kurtosis coefficients alone do not provide a complete picture of the distribution's shape. Other factors such as the range, outliers, and the overall pattern of the data should also be taken into consideration when interpreting the shape of a distribution.For more such question on distribution
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if x(t) = 2·tri(t/4)*δ(t – 2), find the values of a. x(1) b. x(–1)
Answers
The values for x(1) and x(-1) are both 0.
To find the values of x(1) and x(-1) given that x(t) = 2·tri(t/4)*δ(t – 2), we will evaluate the function at these points.
a. x(1):
To find the value of x(1), we need to substitute t = 1 into the function:
x(1) = 2·tri(1/4)*δ(1 - 2)
Since δ(1 - 2) is the Dirac delta function at a point different from zero (specifically, -1), its value is 0.
Therefore,
x(1) = 2·tri(1/4) * 0 = 0
b. x(-1):
To find the value of x(-1), we need to substitute t = -1 into the function:
x(-1) = 2·tri(-1/4)*δ(-1 - 2)
Again, since δ(-1 - 2) is the Dirac delta function at a point different from zero (specifically, -3), its value is 0.
Therefore,
x(-1) = 2·tri(-1/4) * 0 = 0
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schema combines all the entities, attributes, and relationships defined in all the external schemas developed for the business.
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A schema is essentially a blueprint or a framework that describes how all the data within a database is organized and structured. Within this schema, entities, attributes, and relationships are defined. Entities refer to objects or concepts within the database, such as customers, orders, or products.
Attributes are the characteristics or properties of these entities, such as a customer's name or an order's date. Relationships describe how these entities are related to or connected to each other, such as a customer placing an order. When external schemas are developed for a business, they define entities, attributes, and relationships specific to a particular aspect of the business. These external schemas are then combined into the overall schema to create a comprehensive view of all the data within the database.
An entity schema is a set of entities and the relationships among them. In an Extreme Scale application with multiple partitions, the following restrictions and options apply to entity schemas: Each entity schema must have a single root defined. This is known as the schema root. an ER model deals with entities and their relationship, whereas a relational schema talks about tuples and attributes. Moreover, an ER model may be easier to understand than a relational schema because we map the cardinalities explicitly (one-to-one, many-to-one, etc.).
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Express the general solution in terms of Bessel functions:
x^2y''+4xy'+(x^2+2)y=0
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The general solution of the given differential equation is expressed in terms of Bessel functions as y(x) = c1 J₀(x) + c2 Y₀(x) - c3 J₁(x) + c4 Y₁(x), where J and Y are Bessel functions of the first and second kind, respectively, and c1, c2, c3, and c4 are constants.
To express the general solution in terms of Bessel functions, we first need to determine the characteristic equation of the given differential equation. We assume the solution has the form y(x) = x^r, then differentiate twice to get
y'(x) = rx^(r-1)
y''(x) = r(r-1)x^(r-2)
Substituting these expressions into the given differential equation, we get
x^2y''+4xy'+(x^2+2)y = x^2[r(r-1)x^(r-2)] + 4x[rx^(r-1)] + (x^2+2)x^r = 0
Dividing through by x^2, we get
r(r-1) + 4r + (1+2/x^2) = 0
Simplifying and multiplying by x^2, we get the Bessel equation
x^2y'' + xy' + (x^2 - 1)y = 0
The general solution to this differential equation can be expressed in terms of Bessel functions of the first kind, Jv(x), and second kind, Yv(x), as follows
y(x) = c1J0(x) + c2Y0(x)
where c1 and c2 are constants of integration. Therefore, the general solution to the original differential equation can be expressed as
y(x) = c1J0(x) + c2Y0(x) + c3J1(x) + c4Y1(x)
where c3 and c4 are constants of integration determined by the initial conditions.
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Find the area of this sector.
Give your answer in terms of
π
.
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Answer:245/36 π
Step-by-step explanation: you do 50/360 times π(7)^2
Use the definition of compactness (i.e. the open cover definition) to show that the following sets are not compact, by exhibiting an open cover with no finite sub-cover: (1) The open ball B(x, 1) centered at a given x element R^n with the radius 1 in the Euclidean space R^n; (2) The set A = {(x_1, x_2) element R^2: 0 lessthanorequalto 1, x_2 greaterthanorequalto 0} x_2 greaterthanorequalto 0} in R^2; (3) An infinite set in the metric space (M, d) with the discrete metric d.
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Using the open cover definition of compactness, we can show that (1) open ball B(x, 1), (2) set A in R², and (3) an infinite set in a discrete metric space are not compact by exhibiting open covers with no finite sub-covers.
(1) For the open ball B(x, 1) in Rⁿ, consider the open cover consisting of balls B(x, 1-1/n) for n = 2, 3, 4, ... Since each ball excludes a point on the boundary of B(x, 1), no finite sub-collection can cover B(x, 1).
(2) For the set A in R², consider the open cover consisting of rectangles {(-1/n, 1/n) x (0, 1)} for n = 2, 3, 4, ... No finite sub-collection of these rectangles can cover A, as there will always be a gap along the x₁-axis.
(3) In the metric space (M, d) with a discrete metric d, let S be an infinite subset. The open cover consists of balls B(x, 1/2) centered at each point x in S. Since each ball contains only one point, there cannot be a finite sub-cover for the infinite set S.
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exercise 1.1.10. solve ,dxdt=sin(t2) t, .x(0)=20. it is ok to leave your answer as a definite integral.
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The solution of the differential equation dx/dt = sin(t²)×t with the initial condition x(0) = 20 is [tex]x(t) = 20 + \int_{0}^{t}tsin(t^2) dt[/tex].
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two.
To solve the given differential equation dx/dt = sin(t²)×t with the initial condition x(0) = 20 and leaving the answer as a definite integral, follow these steps:
1. Identify the given differential equation:
dx/dt = sin(t²)×t.
2. Recognize the initial condition:
x(0) = 20.
3. Integrate both sides of the equation with respect to t:
∫dx = ∫sin(t²)×t dt.
4. Apply the initial condition to determine the constant of integration:
x(0) = 20.
5. Write the final solution:
[tex]x(t) = 20 + \int_{0}^{t}tsin(t^2) dt[/tex].
So, the solution is [tex]x(t) = 20 + \int_{0}^{t}tsin(t^2) dt[/tex].
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Find the GLB of the set: {x:|x - 4|<1}. a. -1 b. 1 c. -3 d. 3 e. 0
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The GLB of the set {x : |x - 4| < 1} is option (d) 3
The set {x : |x - 4| < 1} can be written as the open interval (3, 5), which contains all values of x that satisfy the inequality |x - 4| < 1.
The greatest lower bound (GLB), also known as the infimum, is a concept in mathematics that applies to sets of numbers or other mathematical objects that are partially ordered.
To find the GLB (greatest lower bound) of this interval, we need to look for the greatest value that is less than or equal to every element in the interval.
Since the interval contains all real numbers greater than 3 and less than 5, its GLB is 3. Therefore, the answer is option (d) 3.
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find the x-coordinates of the inflection points for the polynomial p(x)= x^5/20
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The inflection point of the polynomial p(x) = [tex]x^5/20[/tex] is at x = 0. This is the only one inflection point.
To find the x-coordinates of the inflection points for the polynomial p(x) = [tex]x^5/20[/tex], we'll need to follow these steps:
1. Find the first derivative, p'(x), to determine the slope of the function.
2. Find the second derivative, p''(x), to determine the concavity of the function.
3. Set p''(x) equal to zero and solve for x to find the inflection points.
Step 1: Find the first derivative, p'(x):
p'(x) = [tex]d(x^5/20)/dx = (5x^4)/20 = x^4/4[/tex]
Step 2: Find the second derivative, p''(x):
p''(x) = [tex]d(x^4/4)/dx = (4x^3)/4 = x^3[/tex]
Step 3: Set p''(x) equal to zero and solve for x:
[tex]x^3[/tex] = 0
x = 0
There is only one inflection point, and its x-coordinate is 0.
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In the equation y = ab(x-h)+ k how does the value of a affect the graph?
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The answer of the given question based on the graph is the value of 'a' affects the graph by determining the steepness of the curve.
What is Slope?Slope is a measure of the steepness of a line or a curve. It is defined as ratio of vertical change (rise) between two points to horizontal change (run) between same two points. The slope of a line is constant, while the slope of a curve may change from point to point.
In the equation y = ab(x-h)+k, the value of 'a' affects the graph by determining the steepness of the curve.
If 'a' is positive, the graph will slope upwards as 'x' increases. The larger the value of 'a', the steeper the slope of the curve will be. On the other hand, if 'a' is negative, the graph will slope downwards as 'x' increases. Again, the larger the absolute value of 'a', the steeper the slope of the curve will be.
In general, the value of 'a' controls the vertical scaling of the curve, while the value of 'b' controls the horizontal scaling, and 'h' and 'k' control the horizontal and vertical translations of the curve, respectively. Changing the value of 'a' will stretch or compress the curve vertically, but will not affect the position of the curve on the x-axis.
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find a particular solution to ″ 6′ 9=^−3/^3
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The particular solution is [tex]3x^(-1) - 1/27 + 3(9)^(-2)[/tex] based on integration.
To find a particular solution to given equatio we need to integrate twice. First, we integrate with respect to x to get [tex]-3x^(-2)[/tex].
Then, we integrate again with respect to x to get 3x^(-1) + C1, where C1 is a constant of integration.
Next, we use the initial condition 6′ 9 to solve for C1. Taking the derivative of [tex]3x^(-1) + C1[/tex], we get [tex]-3x^(-2)[/tex]. Plugging in x = 9, we get [tex]-3(9)^(-2) = -1/27[/tex].
Therefore, [tex]-1/27 = -3(9)^(-2) + C1[/tex], and solving for C1, we get[tex]C1 = -1/27 + 3(9)^(-2)[/tex].
Thus, the particular solution is [tex]3x^(-1) - 1/27 + 3(9)^(-2)[/tex].
Hi! It seems there might be a typo in your question, making it difficult to understand the exact problem you need help with. However, I will try to address the terms "solution" and "particular."
A "solution" refers to the result or answer obtained when solving an equation, problem, or system of equations. It is the value or values that satisfy the given conditions or equations.
A "particular solution" is a specific instance of a solution, usually when there are multiple solutions or when dealing with differential equations. It is a single example of a valid answer that meets the given criteria.
If you can provide more clarification on your question, I would be happy to help you find the particular solution you're looking for!
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Which equations represent circles that have a diameter of 12 units and a center that lies on the y-axis? Select two options. x2 + (y – 3)2 = 36 x2 + (y – 5)2 = 6 (x – 4)² + y² = 36 (x + 6)² + y² = 144 x2 + (y + 8)2 = 36
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The equations that represent the circle with diameter 12 are x² + (y - 6)² = 36 and x² + (y + 6)² = 36.
What is equation of circle?A circle can be represented in polar coordinates by the equation r = a, where an is the circle's radius. In polar coordinates, the circle's centre is found at the origin (0, 0).
We use the links between polar and rectangular coordinates to translate this equation to rectangular coordinates:
X=r cos(theta) and Y=r sin (theta)
When we add r = a to these equations, we obtain:
X = cos(theta) and Y = sin (theta)
Hence, the equation of a circle in rectangular coordinates with radius "a" and origin-based centre.
The standard form of the equation of circle is given as:
(x - h)² + (y - k)² = r²
Here, (h , k) are the center and r is the radius.
For diameter = 12 we have radius = 6. Thus, the square of the radius is 36.
The equations representing this radius are:
x² + (y - 6)² = 36 and x² + (y + 6)² = 36
Hence, the equations that represent the circle with diameter 12 are x² + (y - 6)² = 36 and x² + (y + 6)² = 36.
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