Help using Pythagorean theorem

Help Using Pythagorean Theorem

Answers

Answer 1

Answer:

x = 1/2

Step-by-step explanation:

Pythagorean Theorem: a^2 + b^2 = c^2, where a and b are the side lengths and c is the hypotenuse.

Substituting in the values:

x^2 + (√2)^2 = (x + 1)^2

Then, we isolate x:

x^2 + 2 = (x + 1)(x + 1) = x^2 + 2x + 1

(Subtract x^2 from both sides)

2 = 2x + 1

(Subtract 1 from both sides, I also flipped the equation)

2x = 1

(Divide both sides by 2)

x = 1/2

To double-check, substitute x with 1/2:

(1/2)^2 + (√2)^2 = (1/2 + 1)^2

Simplify:

1/4 + 2 = 9/4

=> 1/4 + 8/4 = 9/4 (true)


Related Questions

Find the y-intercept of the line y=
5/6 x +5

Answers

Answer: ( 0,-5)

Step-by-step explanation:

y-intercept The value of y at the point where a curve crosses the y-axis.

find the eqautions of the line that passes through points A and B

Answers

What points are you describing?

Find a formula for Sn, n>=1 if Sn is given by: 2/5, 3/9, 4/13, 5/17, 6/21....
Is this supposed to be some kind of geometric series? Not really sure what to do here...

Answers

The given series is not a geometric series as the ratio between consecutive terms is not constant. However, it is an arithmetic series with a common difference of 4 in the denominator and 1 in the numerator.

To find a formula for Sn, we need to first find a general term for the series. We can see that the numerator of each term is increasing by 1, starting from 2. Therefore, the nth term of the numerator is n + 1.

For the denominator, we can see that it is increasing by 4, starting from 5. Therefore, the nth term of the denominator is 4n + 1.

Hence, the general term of the series can be written as (n + 1)/(4n + 1).

To find the formula for Sn, we can use the formula for the sum of an arithmetic series:

Sn = n/2[2a + (n-1)d]

where a is the first term, d is a common difference, and n is the number of terms.

In our case, a = 2/5, d = 4/9, and n is not given. However, we can use the formula for the nth term of an arithmetic series to find n:

(n + 1)/(4n + 1) = 6/21
Solving for n, we get n = 5.

Plugging in the values, we get:

S5 = 5/2[2(2/5) + 4/9(5-1)] = 1.23

Therefore, the formula for Sn is Sn = (n + 1)/(4n + 1) and the sum of the first 5 terms is 1.23.

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(CO 4) In a situation where the sample size was decreased from 39 to 29, what would be the impact on the confidence interval? a. It would become narrower with fewer values b. It would become wider with fewer values c. It would become narrower due to using the z distribution d. It would remain the same as sample size does not impact confidence intervals

Answers

The correct answer is b. It would become wider with fewer values. This is because as the sample size decreases, the variability of the sample mean increases, leading to a wider confidence interval.

The distribution used for the confidence interval calculation (whether z or t) is not impacted by the sample size, only the size of the sample itself affects the confidence interval.

In a situation where the sample size was decreased from 39 to 29, the impact on the confidence interval would be (b) It would become wider with fewer values.

A smaller sample size generally leads to a wider confidence interval, as the decreased sample size provides less information about the overall distribution.

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Mrs. Brown owns a cake shop where she bakes 30 cupcakes per day. In Christmas, as the demand for the cup cakes increases, she increased the number of cupcakes by 5 over the previous day.

Which equation can be used to find the recursive process that describes the number of cupcakes baked by Mrs. Brown after the mth day after 20th of December?
A.
To find the number of cupcakes baked by Mrs. Brown on the mth day, add 30 to the number of cupcakes baked on the (m-1)th day 20th of December. Am = A(m-1) + 30, where Ao = 5
B.
To find the number of cupcakes baked by Mrs. Brown on the mth day, subtract 2 from the number of cupcakes baked on the (m-2)th day 20th of December. Am = A(m-2) - 2, where Ao = 5
C.
To find the number of cupcakes baked by Mrs. Brown on the mth day, subtract 5 from the number of cupcakes baked on the (m-1)th day 20th of December. Am = A(m-1) - 5, where Ao = 30
D.
To find the number of cupcakes baked by Mrs. Brown on the mth day, add 5 to the number of cupcakes baked on the (m-1)th day 20th of December. Am = A(m-1) + 5, where Ao = 30

Answers

The correct equation is D. To find the number of cupcakes baked by Mrs. Brown on the mth day, add 5 to the number of cupcakes baked on the (m-1)th day 20th of December. Am = A(m-1) + 5, where Ao = 30.

This is because Mrs. Brown increases the number of cupcakes by 5 over the previous day, so each day the number of cupcakes baked increases by 5. The initial value is 30, which is Ao. Therefore, to find the number of cupcakes baked on any given day, we add 5 to the number baked on the previous day.

Therefore, the correct answer is D.

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38. what conditions must be satisfied by b1, b2, b3, b4, and b5 for the overdetermined linear systemx1-x2 =b1x1-3x2 =b2x1+ x2 = b3x1 - 5x2 = b4x1 + 6x2 = b5to be consistent?a) b1 = 10/11r + 1/11s, b2 = 9/11r + 2/11s, b3 = 5/11r + 6/11s, b4=r, b5 = sb) b1 = 10/11r + 1/11s, b2 = 9/11r + 2/11s, b3 = 5/11r + 6/11s, b4=s, b5 = rc) b1 = 9/11r + 2/11s, b2 = 10/11r + 1/11s, b3 = 5/11r + 6/11s, b4=r, b5 = sd) b1 = 5/11r + 6/11s, b2 = 10/10r + 1/11s, b3 = 9/11r + 2/11s, b4=r, b5 = se) b1 = 10/11r + 1/11s, b2 = 2/10r + 9/11s, b3 = 5/11r + 6/11s, b4=r, b5 = s

Answers

The conditions that must be satisfied by b1, b2, b3, b4, and b5 for the overdetermined linear system to be consistent are b1 = 10/11r + 1/11s, b2 = 9/11r + 2/11s, b3 = 5/11r + 6/11s, b4 = r, and b5 = s.

For the system to be consistent, there must be a solution that satisfies all the equations in the system. In an overdetermined system, there are more equations than variables, so not all solutions will satisfy all the equations. Therefore, the system will only be consistent if the equations are not contradictory, meaning there is a common solution to all of them.

In this system, there are two variables, x1 and x2, and five equations. We can write the system in matrix form as Ax = b, where A is the coefficient matrix, x is the variable vector, and b is the constant vector.

⎡1 -1⎤ ⎡x1⎤ ⎡b1⎤

⎢-3 1⎥ x ⎢x2⎥ = ⎢b2⎥

⎢1 -5⎥ ⎣ ⎦ ⎢b3⎥

⎣1 6 ⎦ ⎣b4⎦

⎣b5⎦

To check the consistency of the system, we can use row reduction to determine the echelon form of the augmented matrix [A|b]. If the echelon form has a row of zeros with a non-zero constant on the right-hand side, then the system is inconsistent. Otherwise, the system is consistent.

Performing row reduction on [A|b], we get:

⎡1 0 0 0 10/11r+1/11s⎤

⎢0 1 0 0 9/11r+2/11s ⎥

⎢0 0 1 0 5/11r+6/11s ⎥

⎣0 0 0 1 r ⎦

Since the echelon form does not have a row of zeros with a non-zero constant on the right-hand side, the system is consistent. Therefore, the conditions that must be satisfied by b1, b2, b3, b4, and b5 for the system to be consistent are b1 = 10/11r + 1/11s, b2 = 9/11r + 2/11s, b3 = 5/11r + 6/11s, b4 = r, and b5 = s.

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Find the common ratio of the geometric sequence 16 , − 32 , 64

Answers

Answer:

common ratio r = - 2

Step-by-step explanation:

the common ratio r is calculated as

r = [tex]\frac{a_{2} }{a_{1} }[/tex] = [tex]\frac{-32}{16}[/tex] = - 2

Answer:

-2

Check:

16*-2 is -32

-32 * -2 is 64

I NEED HELP ON THIS ASAP!!!!

Answers

Each point (x, y) on the graph of h(x) becomes the point (x - 3, y - 3) on v(x).

Each point (x, y) on the graph of h(x) becomes the point (x + 3, y + 3) on w(x).

What is a translation?

In Mathematics and Geometry, the translation a geometric figure or graph to the left simply means subtracting a digit from the value on the x-coordinate of the pre-image;

g(x) = f(x + N)

On the other hand, the translation a geometric figure to the right simply means adding a digit to the value on the x-coordinate (x-axis) of the pre-image;

g(x) = f(x - N)

Since the parent function is v(x) = h(x + 3), it ultimately implies that the coordinates of the image would created by translating the parent function to the left by 3 units.

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Help please!!

Anything would be much appreciated

Answers

Answer:

a) kinda but not really b) no c) yes

Step-by-step explanation:

a) It's somewhat possible. The mean is the numbers added together divided but the amount so it would be (3(purple)+2(blue)+2(red)+green)/8. It doesn't completely work because they are not numbers.

b)Their median is not possible. It needs to be in order from largest to greatest and that's not possible with words

c) The mode is the most common thing in a set of data. Since this can be applied to words, purple would be the mode.

the sum of two consecutive odd numbers is 56. find the numbers

Answers

Answer: 27, 29

Step-by-step explanation:

Let's say that the 2 numbers are x and x+2

That means that: x+x+2=56

Simplify: 2x+2=56

Solve: 2x=54

x=27

27,29 are the 2 numbers

3 simple math questions for 50 points Please help i have no time for trolls
Thank you!

Answers

The surface area of the sphere, is approximately 172 square inches.

How to calculate the value

It should be noted that the Volume of a sphere = (4/3)πr^3

where r is the radius of the sphere.

Setting Volume of sphere equal to Volume of prism, we get:

(4/3)πr^3 = lwh

Plugging in the given value of r = 3.7 in, we can solve for lwh:

(4/3)π(3.7)^3 = lwh

lwh ≈ 209.7 cubic inches

A = 4πr^2

A = 4π(3.7)^2

A ≈ 171.9 square inches

Rounding this to the nearest square inch, we get:

A ≈ 172 square inches

Therefore, the surface area of the sphere, is approximately 172 square inches.

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(b) group the following numbers according to congruence mod 13. that is, put two numbers in the same group if they are equivalent mod 13. {−63, -54, -41, 11, 13, 76, 80, 130, 132, 137}

Answers

When grouping the given numbers according to congruence mod 13, we find the following groups:

Group 1: {-63}(equivalent to -11 mod 13)

Group 2: {-54, -41}(equivalent to -2 mod 13)

Group 3: {11, 76}(equivalent to 11 mod 13)

Group 4: {13,130}(equivalent to 0 mod 13

Group 5: {80,132}(equivalent to 2 mod 13)

Group 6: {137}(equivalent to 7 mod 13)

Here, we have,

To group the given numbers according to congruence mod 13, we need to find the remainders of each number when divided by 13.

We can find the remainder of a number when divided by 13 by using the modulo operator (%). For example, the remainder of 17 when divided by 13 is 4 (17 % 13 = 4).

Using this method, we can find the remainders of all the given numbers as follows:

=> (-63) % 13= -11

=> -54 % 13 = -2

=> -41 % 13 = -2

=> 11 % 13 = 11

=> 13 %13 = 0

=> (76) % 13 = 11

=> (80) % 13 = 2

=>130 % 13 = 0

=>132 %13 = 2

=>137 % 13 = 7

Now, we can group the numbers according to their remainders as follows:

Group 1: {-63}(equivalent to -11 mod 13)

Group 2: {-54, -41}(equivalent to -2 mod 13)

Group 3: {11, 76}(equivalent to 11 mod 13)

Group 4: {13,130}(equivalent to 0 mod 13

Group 5: {80,132}(equivalent to 2 mod 13)

Group 6: {137}(equivalent to 7 mod 13)

The given numbers have been grouped according to congruence mod 13. Numbers in the same group are equivalent mod 13, i.e., they have the same remainder when divided by 13.

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Find the sum of the series.
[infinity] (−1)^n π^2n
n =0 6^2n(2n)!

Answers

The sum of the given series is 72 / (72 + π^2).

We can use the formula for the sum of an infinite geometric series:

S = a / (1 - r)

where S is the sum, a is the first term, and r is the common ratio. In this case, the first term is (-1)^0 π^0 / (6^0 (2*0)!), which simplifies to 1, and the common ratio is (-1) π^2 / (6^2 (2*1)!), which simplifies to -π^2 / 72. Thus, we have:

S = 1 / (1 + π^2 / 72)

Now, we can simplify the denominator by multiplying the numerator and denominator by 72:

S = 72 / (72 + π^2)

Therefore, the sum of the given series is 72 / (72 + π^2).

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Determine the Longest Common Subsequence and the Longest Common Substring for the following strings: A=(a, c, t, g, a, t, t) and B= (c, g, a, t, g, a). (15+15=30)

Answers

The Longest Common Subsequence (LCS) for strings A=(a, c, t, g, a, t, t) and B=(c, g, a, t, g, a) is (c, t, g, a, t) and the Longest Common Substring (LCSb) is (t, g, a).


1. Create a matrix of size (m+1)x(n+1) where m and n are the lengths of A and B respectively.


2. Initialize the first row and column of the matrix with 0.


3. Iterate through the matrix, comparing characters from A and B.


4. If characters match, update the matrix value as matrix[i-1][j-1] + 1.


5. If characters don't match, update the matrix value as the max(matrix[i-1][j], matrix[i][j-1]).


6. The LCS can be reconstructed by backtracking from the bottom-right corner of the matrix.


7. For LCSb, find the maximum value in the matrix and its position, then backtrack to construct the substring.

This provides the LCS and LCSb as defined above.

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(1 point) find the area lying outside =6sin and inside =3 3sin. area =

Answers

The area lying outside the circle r=6sin and inside the circle r=3+3sin is approximately 21.205 square units.

To solve this problem, we need to first understand what the equations =6sin and =3 3sin represent. These are actually equations of circles in polar coordinates, where r=6sin represents a circle with radius 6 units and centered at the origin, and r=3+3sin represents a circle with radius 3 units and centered at (-3,0) in Cartesian coordinates.

The area lying outside the circle r=6sin and inside the circle r=3+3sin can be found by integrating the equation for the area of a polar region, which is:

A = 1/2 ∫ [f(θ)]^2 - [g(θ)]^2 dθ

where f(θ) and g(θ) are the equations for the outer and inner boundaries of the region, respectively.

In this case, we have:

A = 1/2 ∫ (6sin)^2 - (3+3sin)^2 dθ

A = 1/2 ∫ 36sin^2 - (9+18sin+9sin^2) dθ

A = 1/2 ∫ 27sin^2 - 18sin - 9 dθ

To solve this integral, we can use the half-angle identity for sine, which is:

sin^2 (θ/2) = (1-cos θ)/2

Substituting this identity into our integral, we get:

A = 1/2 ∫ [27(1-cos θ)/2] - 18sin - 9 dθ

A = 1/2 ∫ (13.5-13.5cos θ) - 18sin - 9 dθ

A = 1/2 ∫ -18sin - 22.5cos θ - 9 dθ

Integrating each term separately, we get:

A = -9sin θ - 22.5sin θ - 9θ + C

where C is the constant of integration. To find the bounds of integration, we need to find the values of θ where the two circles intersect. Setting the equations equal to each other, we get:

6sin = 3+3sin

3sin = 3

sin θ = 1

θ = π/2

So the bounds of integration are 0 and π/2. Substituting these values into the equation for the area, we get:

A = -9sin(π/2) - 22.5sin(π/2) - 9(π/2) + C - (-9sin 0 - 22.5sin 0 - 9(0) + C)

A = -13.5π/2

Therefore, the area lying outside the circle r=6sin and inside the circle r=3+3sin is approximately 21.205 square units.

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evaluate the integral. (use c for the constant of integration.) 7x 1 − x4 dx

Answers

To evaluate the integral ∫7x/(1 − x^4) dx, we first need to perform partial fraction decomposition to separate it into simpler fractions. Using algebraic manipulation.

we can rewrite the integrand as: 7x/(1 − x^4) = A/(1 + x) + B/(1 − x) + C/(1 + x^2) + D/(1 − x^2), where A, B, C, and D are constants to be determined. Then, we can multiply both sides by the common denominator (1 − x^4) and solve for the constants by equating coefficients of like terms.



After performing partial fraction decomposition, we get: ∫7x/(1 − x^4) dx = ∫A/(1 + x) dx + ∫B/(1 − x) dx + ∫C/(1 + x^2) dx + ∫D/(1 − x^2) dx, Integrating each of these simpler fractions individually, we get: ∫A/(1 + x) dx = A ln|1 + x| + c1
∫B/(1 − x) dx = −B ln|1 − x| + c2
∫C/(1 + x^2) dx = C arctan(x) + c3
∫D/(1 − x^2) dx = D ln|(1 + x)/(1 − x)| + c4.



where c1, c2, c3, and c4 are constants of integration, Therefore, the final answer to the given integral is: ∫7x/(1 − x^4) dx = A ln|1 + x| − B ln|1 − x| + C arctan(x) + D ln|(1 + x)/(1 − x)| + C, where A, B, C, and D are the constants obtained from partial fraction decomposition, and C is the constant of integration.

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use the laplace transform to solve the given initial-value problem. y' − y = 2 cos(6t), y(0) = 0

Answers

To solve the initial-value problem y' - y = 2 cos(6t), y(0) = 0 using the Laplace transform, we first take the Laplace transform of both sides of the equation.

L[y'] - L[y] = L[2 cos(6t)]

Using the property of the Laplace transform that L[ y' ] = sY(s) - y(0) and L[ cos(6t) ] = s/( s^2 + 36 ), this becomes:

sY(s) - y(0) - Y(s) = 2 * s / ( s^2 + 36 )

Substituting y(0) = 0, we get:

sY(s) - Y(s) = 2 * s / ( s^2 + 36 )

Factoring out Y(s), we get:

( s - 1 ) * Y(s) = 2 * s / ( s^2 + 36 )

Solving for Y(s), we get:

Y(s) = 2 * s / ( ( s - 1 ) * ( s^2 + 36 ) )

Using partial fractions, we can write Y(s) as:

Y(s) = A / ( s - 1 ) + B * s / ( s^2 + 36 )

Multiplying both sides by the denominator on the right-hand side and substituting s = 1, we get:

2 = A / ( 1 - 1 ) + B * 1 / ( 1^2 + 36 )
2 = B / 37

Thus, B = 74.

Substituting B in the previous equation and simplifying, we get:

Y(s) = 2 / ( s - 1 ) + 2s / ( s^2 + 36 )

Taking the inverse Laplace transform of Y(s) using a table or a software, we get:

y(t) = 2 * e^t + sin(6t) / 3

Therefore, the solution to the initial-value problem y' - y = 2 cos(6t), y(0) = 0 using the Laplace transform is y(t) = 2 * e^t + sin(6t) / 3.

There are seven people fishing at Lake Connor three have fishing license and four do not an inspector chooses to do two of the people are random what is the probability that the first person chosen does not have a license and the second one does

Answers

In a case whereby There are seven people fishing at Lake Connor three have fishing license and four do not an inspector chooses to do two of the people are random probability that the first person chosen does not have a license and the second one does is 2/7

How can the probability be determined?

Based on the given information, total number of the people = 7

those with fishing license =3

those without fishing license =4

chance of choosing someone without a license=4/7

chance of choosing someone with a license=3/6

Theerefore probability that the first person chosen does not have a license and the second one does= 4/7 * 3/6 =2/7

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solve the equation. give your answer correct to 3 decimal places. 25,000 = 10,000(1.05)5x

Answers

The solution to the equation 25,000 = 10,000(1.05)5x correct to 3 decimal places is x = 4.017.

To solve this equation, we can first divide both sides by 10,000 to get:

2.5 = 1.05^(5x)

Next, we can take the natural logarithm of both sides:

ln(2.5) = ln(1.05^(5x))

Using the logarithmic identity ln(a^b) = b*ln(a), we can simplify the right side of the equation:

ln(2.5) = 5x*ln(1.05)

Finally, we can solve for x by dividing both sides by 5ln(1.05) and rounding to 3 decimal places:

x = ln(2.5) / (5*ln(1.05)) = 4.017

Therefore, the solution to the equation is x = 4.017, correct to 3 decimal places. This means that after 5 years of an initial investment of $10,000 at an annual interest rate of 5%, the investment will be worth $25,000.

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Rectangle TUVW is on a coordinate plane at T (a, b), U (a + 2, b + 2), V (a + 5, b − 1), and W (a + 3, b − 3). What is the slope of the line that is parallel to the line that contains side UV?

a. −2
b. 2
c. −1
d. 1

Answers

Answer:

  c.  -1

Step-by-step explanation:

You want the slope of the line parallel to UV, where U=(a +2, b +2) and V = (a +5, b -1).

Slope

The slope of UV is given by ...

  m = (y2 -y1)/(x2 -x1)

  m = ((b -1) -(b +2))/((a +5) -(a +2)) = -3/3 = -1

The parallel line will have the same slope.

The slope of the line parallel to UV is -1, choice C.

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The amount of snowfall in feet in a remote region of Alaska in the month of January is a continuous random variable with probability density function
f(x)= 6/125 (5x−x^2); (0≤ x ≤ 5)
Find the amount of snowfall one can expect in any given month of January in Alaska.

Answers

one can expect about 16.67 feet of snowfall in any given month of January in this remote region of Alaska.

To find the expected amount of snowfall in any given month of January in Alaska, you need to calculate the expected value (E) of the continuous random variable with the given probability density function f(x) = 6/125(5x - x^2), where 0 ≤ x ≤ 5.

The expected value (E) is found using the following formula:

E(X) = ∫[x * f(x)]dx, with integration limits from 0 to 5.

For this problem, we need to evaluate:

E(X) = ∫[x * (6/125)(5x - x^2)]dx from 0 to 5.

Upon integrating, you get:

E(X) = (6/125) * [5/3 * x^3 - x^4/4] evaluated from 0 to 5.

Now, substitute the limits:

E(X) = (6/125) * [5/3 * (5^3) - (5^4)/4 - (0)]

E(X) = (6/125) * [5/3 * 125 - 625/4]

E(X) = (6/125) * [625/3 - 625/4]

E(X) = (6/125) * (625/12)

E(X) = 50/3 ≈ 16.67 feet

So, one can expect about 16.67 feet of snowfall in any given month of January in this remote region of Alaska.

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In the diagram below of right triangle ABC, CD is
the altitude to hypotenuse AB, CB = 6, and AD = 5.
C
A
5
What is the length of BD?
1) 5
2) 9
3) 3
4) 4

Answers

The volume of the prism is determined as 120 in³.

What is the volume of the triangular prism?

The volume of the triangular prism is calculated by applying the following formula as shown below;

V = ¹/₂bhl

where;

b is the base of the prismh is the height of the priml is the length of the prism

The volume of the prism is calculated as follows;

V = ¹/₂ x 6 in x 4 in x 10 in

V = 120 in³

,

Thus, the volume of the prism is a function of its base, height and length.

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let bold r left parenthesis t right parenthesis equals t bold i plus t cubed bold j plus t bold k the tangential component of acceleration is

Answers

The tangential component of acceleration is:
Bold a subscript T left parenthesis t right parenthesis equals 18 t cubed divided by square root of 1 plus 9 t to the power of 4 plus 1

To find the tangential component of acceleration:

We first need to find the velocity vector.

Taking the derivative of the position vector gives us:
bold v left parenthesis t right parenthesis equals bold i plus 3 t squared bold j plus bold k
The tangential component of acceleration is the component of acceleration that is parallel to the velocity vector.

Taking the derivative of the velocity vector gives us:
bold a left parenthesis t right parenthesis equals 0 bold i plus 6 t bold j plus 0 bold k
So the tangential component of acceleration is:
bold a subscript T left parenthesis t right parenthesis equals bold a left parenthesis t right parenthesis dot bold v left parenthesis t right parenthesis divided by the magnitude of bold v left parenthesis t right parenthesis
Since the velocity vector is:
bold v left parenthesis t right parenthesis equals bold i plus 3 t squared bold j plus bold k
The dot product of bold a and bold v is:
bold a left parenthesis t right parenthesis dot bold v left parenthesis t right parenthesis equals 0 times 1 plus 6 t times 3 t squared plus 0 times 1 equals 18 t cubed
The magnitude of the velocity vector is:
magnitude of bold v left parenthesis t right parenthesis equals square root of 1 plus 9 t to the power of 4 plus 1
So the tangential component of acceleration is:
bold a subscript T left parenthesis t right parenthesis equals 18 t cubed divided by square root of 1 plus 9 t to the power of 4 plus 1

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Which one is the correct answer?

Answers

Answer:

its 6/6

Step-by-step explanation:

Answer: C

Step-by-step explanation:

Because all of the numbers are lower than 7 on a 1 to 6 dice.

Are the following statements true or false? 1. For any scalar c, u^T (cv) = c(u^Tv) 2. Let u and be non zero vectors: If the distance from u to is equal to the distance from U to -V, then U and v are orthogonal: 3. For square matrix A_ vectors in R(A) are orthogonal to vectors in N(A): 4. v^Tv = Ilvll^2. 5. If vectors V1,....,vp, Yp span subspace W and If x is orthogonal to each vj for j = 1,.....,P then X is in W^⊥

Answers

Hence, x is orthogonal to any vector in W, and hence x is in W^⊥

For any scalar c, u^T (cv) = c(u^Tv)

True. This follows from the distributive property of matrix multiplication and the fact that scalar multiplication is commutative.

Let u and v be non-zero vectors: If the distance from u to v is equal to the distance from u to -v, then u and v are orthogonal.

True. This statement can be restated as saying that u lies on the perpendicular bisector of the line segment connecting v and -v. Since the perpendicular bisector is a line perpendicular to this line segment, it follows that u is orthogonal to both v and -v, and hence orthogonal to their sum, which is the zero vector.

For square matrix A, vectors in R(A) are orthogonal to vectors in N(A).

True. The range of a matrix A consists of all vectors b that can be expressed as b = Ax for some vector x, whereas the null space of A consists of all vectors x such that Ax = 0. If v is in R(A) and w is in N(A), then v = Ax for some x, and we have w^T v = w^T Ax = (A^T w)^T x = 0, since A^T w is in N(A) by the definition of the null space. Hence, v is orthogonal to w.

v^Tv = Ilvll^2.

True. This follows from the definition of the Euclidean norm, which is given by ||v|| = sqrt(v^T v). Hence, ||v||^2 = v^T v.

If vectors v1,....,vp span subspace W and if x is orthogonal to each vj for j = 1,.....,p, then x is in W^⊥.

True. Let v1,....,vp be a basis for W, and let x be orthogonal to each vj. Then, any vector w in W can be expressed as w = c1v1 + ... + cpvp for some scalars c1,....,cp. Since x is orthogonal to each vj, we have x^T w = c1 x^T v1 + ... + cp x^T vp = 0. Hence, x is orthogonal to any vector in W, and hence x is in W^⊥.

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6. (a) is there a smallest real number a for which x 26 x is big-o of a x ? explain your answer. (b) is there a smallest integer number a for which x 26 x is big-o of a x ? explain your answer.

Answers

(a) Yes, there is a smallest real number a for which x^26 is big-O of ax. To find this value, we can use the limit definition of big-O notation.

We want to find a value of a such that x^26 is less than or equal to ax multiplied by some constant C, for all x greater than some value N. Mathematically, we can write this as:

x^26 <= Cax, for all x >= N

Dividing both sides by x and taking the limit as x approaches infinity, we get:

lim x->inf (x^25 / a) <= C

This limit exists only if a is greater than zero, so let's assume that. Then we can simplify the left-hand side of the inequality as:

lim x->inf x^25 / a = inf

So for any value of C, we can always find a value of N such that x^26 is less than or equal to ax multiplied by C, for all x greater than or equal to N. Therefore, we can say that x^26 is big-O of ax, for any positive real number a, and there is no smallest such value of a.

(b) No, there is no smallest integer number a for which x^26 is big-O of ax. The proof is similar to part (a), but we need to show that for any positive integer a, there exists a constant C such that x^26 is not less than or equal to ax multiplied by C, for infinitely many values of x.

To do this, we can choose x to be a power of 2, say x = 2^k. Then we have:

x^26 = (2^k)^26 = 2^(26k)

ax = a * 2^k

So we want to find a value of a and a constant C such that:

2^(26k) > Ca * 2^k, for infinitely many values of k

Dividing both sides by 2^k, we get:

2^(25k) > Ca, for infinitely many values of k

But this is true for any value of a greater than 2^(25), since 2^(25k) grows faster than Ca for large enough values of k. Therefore, for any integer value of a greater than 2^(25), there exist infinitely many values of k for which x^26 is not less than or equal to ax multiplied by some constant C. Hence, x^26 is not big-O of ax for any integer value of a less than or equal to 2^(25), and there is no smallest such value of a.

(a) No, there isn't a smallest real number 'a' for which x^26x is big-O of ax. This is because x^26x has a higher growth rate than ax for any real number 'a'. As 'x' becomes larger, the term x^26x will always grow faster than ax, no matter the value of 'a'.

(b) Yes, there is a smallest integer number 'a' for which x^26x is big-O of ax. The smallest integer 'a' would be 1, because if we let 'a' be any integer smaller than 1, ax will have a lower growth rate than x^26x. When 'a' is equal to 1, we have x^26x = O(x), which means x^26x grows at most as fast as x, and there's no smaller integer 'a' for which this is true.

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Which expression is equivalent to x^5 × x^2?​

Answers

Answer:

no choices given but it is x^

Step-by-step explanation:

when the bases are the same and you are multiplying, add the powers.

which is the area of the region in quadrant i bounded by y = 2x2 and y = 2x3?

Answers

The area of the region in the given quadrant i is 1/3 square units.

How to find the area of the region in quadrant?

To find the area of the region in quadrant i bounded by y = 2x2 and y = 2x3, we need to first find the x-coordinates where these two curves intersect.

Setting 2x2 equal to 2x3, we get:
2x2 = 2x3

Dividing both sides by 2x2 (which is non-zero since we are only considering quadrant i), we get:
x3 = x2

So the curves intersect at the point (0,0) and (1,2).

To find the area of the region between these curves in quadrant i, we can integrate the difference between the two curves with respect to x, from x = 0 to x = 1:

∫[0,1] (2x3 - 2x2) dx
= [x4 - 2/3 x3] from 0 to 1
= (1 - 2/3) - (0 - 0)
= 1/3

Therefore, the area of the region in quadrant i bounded by y = 2x2 and y = 2x3 is 1/3 square units.

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iii) Find the values of x
when y = 1
0.5
+

Answers

Please post the full question by replying to my answer

(maybe you can like it so that I'll know you have updated the question)

Which of the following are correct statements? Check all that apply.
A. A segment can be named only one way.
B. A segment can be named in more than one way.
C. A segment has two endpoints.
D. A segment has only one endpoint.
OE. A segment does not continue forever.

Answers

B. A segment can be named in more than one way.
C. A segment has two endpoints.
E. A segment does not continue forever.

A segment can be named in more than one way because it can be named based on either of its endpoints. A segment has two endpoints, and it does not continue forever. However, a segment can only be named one way if the same endpoint is used as the starting point for the naming.
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