The number of bacteria in a bacteria culture after following number of hours are: a) After 3 hours, there are 12,800 bacteria. b) After t hours, there are 200 * 2^(2t) bacteria. c) After 40 minutes, there are 400 bacteria.
Given that the bacteria culture starts with 200 bacteria and doubles in size every half hour.
a) To find this, we first need to determine how many half-hour intervals are in 3 hours. Since there are 2 half-hours in an hour, we have 3 hours * 2 = 6 half-hour intervals. The bacteria doubles in size every half hour, so we have:
200 bacteria * 2^6 = 200 * 64 = 12,800 bacteria
b) To generalize this for any number of hours (t), we need to find how many half-hour intervals are in t hours. That's 2t half-hour intervals. Then we have:
200 bacteria * 2^(2t)
c) First, we need to convert 40 minutes to hours. Since there are 60 minutes in an hour, we have 40/60 = 2/3 hours. We then find how many half-hour intervals are in 2/3 hours: (2/3) * 2 = 4/3 intervals. Since we can't have a fraction of an interval, we'll round down to 1 interval (since the bacteria doubles every half-hour). Then we have:
200 bacteria * 2^1 = 200 * 2 = 400 bacteria
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a rock is dropped from a height of 25 ft, the function h= -16x² + 25 gives the height h of the rock after x seconds. wheb does it hit the ground?
the rock hits the ground after 25/8 seconds or approximately 3.125 seconds.
How to solve the question?
To find when the rock hits the ground, we need to determine the value of x when h equals zero, since at that time the height of the rock will be at ground level.
Setting h=0, we get:
0 = -16x² + 25
Solving for x, we can use the quadratic formula:
x = (-b ± √(b²-4ac))/2a
where a = -16, b = 0, and c = 25.
Plugging in these values, we get:
x = (-0 ± √(0²-4(-16)(25)))/2(-16)
Simplifying:
x = ±√(625)/8
x = ±25/8
Since time cannot be negative, we take the positive value:
x = 25/8
Therefore, the rock hits the ground after 25/8 seconds or approximately 3.125 seconds.
We can also verify our result by graphing the function h= -16x² + 25 and observing where the graph crosses the x-axis, which represents the ground level.
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If a 3×4 matrix has rank 3 , what are the dimensions of its columnspace (e.g., which of R1,R2,…Rn represents the column space) and left nullspace (i.e., for a matrix Am×n, the left null space is the set of all vectors x such that A^T x=0) ?
The left null space is the null space of the transpose of the matrix, A^T. Since the original matrix is 3x4 and has rank 3, its nullity can be calculated as n - rank = 4 - 3 = 1. Thus, the dimension of the left null space is 1, and it is represented by R^1.
If a 3×4 matrix has rank 3, this means that there are 3 linearly independent columns. Therefore, the column space of the matrix is spanned by these 3 columns. In terms of the matrix itself, we can say that the column space is spanned by the columns corresponding to the pivot positions in the matrix after it has been reduced to row echelon form. So, in this case, we would be looking at the columns corresponding to the 3 pivot positions.
To find the left null space of the matrix, we need to find all vectors x such that A^T x = 0. Since A is a 3×4 matrix, its transpose is a 4×3 matrix. So we are looking for vector x that is in R^4 and satisfies the equation A^T x = 0. The left null space is the set of all such vectors.
To find the left null space, we can use the fact that the left null space is orthogonal to the row space of A. The row space of A is spanned by the rows corresponding to the pivot positions in the matrix after it has been reduced to row echelon form. Since the matrix has rank 3, there are only 3 pivot positions, so the row space has dimension 3.
Therefore, we can find a basis for the left null space by finding a basis for the orthogonal complement of the row space. We can use the Gram-Schmidt process to do this. Start with a basis for the row space, and then orthogonalize it by subtracting the projection onto each previous vector in the basis.
Once we have a basis for the left null space, we can determine its dimension. Since the matrix has 4 columns, the left null space has dimension 4 - rank(A) = 4 - 3 = 1. So the left null space is a one-dimensional subspace of R^4.
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A new car is purchased for $29,000 and over time its value depreciates by one half every 3.5 years. What is the value of the car 20 years after it was purchased, to the nearest hundred dollars?
The value of the car 20 years after it was purchased is approximately $4,100.
What is the meaning of depreciates?Depreciation refers to the decrease in the value of an asset over time due to wear and tear, obsolescence, or other factors. In the context of a car, depreciation means that its value decreases as it is used and ages.
To calculate the value of the car 20 years after it was purchased, we need to find out how many times the value is halved in 20 years. Since 3.5 years is the time it takes for the value to be halved, we can divide 20 by 3.5 to get the number of times the value is halved.
20 / 3.5 = 5.71 (rounded to two decimal places)
So, the value of the car after 20 years would be:
$29,000 / (2^5.71) = $4,090 (rounded to the nearest hundred dollars)
Therefore, the value of the car 20 years after it was purchased is approximately $4,100.
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What's the correct t statistic for the difference between means of independent samples (without pooling)?
t = xbar1 - bar2 / √s₁² / n₁² + s₂² / n₂²
This formula will give you the correct t statistic for comparing the means of two independent samples without assuming equal variances
To calculate the t statistic for the difference between means of independent samples without pooling, you can use the following formula:
t = (xbar1 - xbar2) / √[(s₁² / n₁) + (s₂² / n₂)]
Here,
- xbar1 and xbar2 are the sample means of the two groups,
- s₁² and s₂² are the sample variances for the two groups,
- n₁ and n₂ are the sample sizes for the two groups.
This formula will give you the correct t statistic for comparing the means of two independent samples without assuming equal variances (i.e., without pooling).
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let the discrete random variable x be the number of odd numbers that appear in 16 tosses of a fair die. find the exact value of p(
The exact values of P(X = k) for k = 0, 1, 2, ..., 16 are:
P(X = 0) = 1/65536
P(X = 1) = 1/4096
P(X = 2) = 15/8192
P(X = 3) = 455/65536
P(X = 4) = 3003/262144
P(X = 5) = 1001/65536
P(X = 6) = 2002/65536
P(X = 7) = 1716/65536
P(X = 8) = 6435/262144
P(X = 9) = 5005/262144
P(X = 10) = 3003/262144
P(X = 11) = 455/65536
P(X = 12) = 1001/65536
P(X = 13) = 15/8192
P(X = 14) = 1/4096
P(X = 15) = 1/65536
P(X = 16) = 1/65536
Briefly describe how do you find these answers?The number of possible outcomes when rolling a fair die once is 6, with 3 odd numbers (1, 3, and 5) and 3 even numbers (2, 4, and 6). Therefore, the probability of rolling an odd number is 3/6 = 1/2 and the probability of rolling an even number is also 1/2.
The number of odd numbers that appear in 16 tosses of a fair die is a binomial random variable with parameters n = 16 and p = 1/2. The probability mass function of X, the number of odd numbers, is given by:
P(X = k) = (16 choose k) [tex]*[/tex] (1/2)¹⁶, for k = 0, 1, 2, ..., 16.
To find the exact value of P(X = k), we need to substitute k into this formula and evaluate it. For example:
P(X = 0) = (16 choose 0) [tex]*[/tex] (1/2)¹⁶ = 1/65536
P(X = 1) = (16 choose 1) [tex]*[/tex] (1/2)¹⁶ = 16/65536 = 1/4096
P(X = 2) = (16 choose 2) [tex]*[/tex] (1/2)¹⁶ = 120/65536 = 15/8192
and so on, until
P(X = 16) = (16 choose 16) [tex]*[/tex] (1/2)¹⁶ = 1/65536
Therefore, the exact values of P(X = k) for k = 0, 1, 2, ..., 16 are:
P(X = 0) = 1/65536
P(X = 1) = 1/4096
P(X = 2) = 15/8192
P(X = 3) = 455/65536
P(X = 4) = 3003/262144
P(X = 5) = 1001/65536
P(X = 6) = 2002/65536
P(X = 7) = 1716/65536
P(X = 8) = 6435/262144
P(X = 9) = 5005/262144
P(X = 10) = 3003/262144
P(X = 11) = 455/65536
P(X = 12) = 1001/65536
P(X = 13) = 15/8192
P(X = 14) = 1/4096
P(X = 15) = 1/65536
P(X = 16) = 1/65536
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evaluate the integral using a linear change of variables. z z r (x y)e x 2−y 2 da where r is the polygon with vertices (2, 0), (0, 2), (−2, 0), and (0, −2).2Make sure to include: (A) A transformation or an inverse transformation, where the region transforms to a rectangular region. (B) A transformed rectangular region. (C) The Jacobian of the transformation. (D) An iterated double integral where the bounds and the integrand have been converted. (E) A final answer.
To evaluate the integral using a linear change of variables, we need to:
(A) Find a transformation: Let u = x + y and v = x - y. The inverse transformation is x = (u + v)/2 and y = (u - v)/2.
(B) Transform the polygon region: The vertices (2, 0), (0, 2), (-2, 0), and (0, -2) transform to (2, 2), (2, -2), (-2, -2), and (-2, 2), forming a rectangular region with u = [-2, 2] and v = [-2, 2].
(C) Compute the Jacobian: J(u, v) = |∂(x, y)/∂(u, v)| = |(1/2, 1/2; 1/2, -1/2)| = 1/2.
(D) Convert the iterated double integral: ∬R e^(x² - y²) dA = ∬_{-2}^2 ∬_{-2}^2 e^((u+v)²/4 - (u-v)²/4) * (1/2) du dv.
(E) The final answer: The integral evaluates to 1/2 ∬_{-2}² ∬_{-2}² e^(uv) du dv = 2π(e⁴ - 1).
The integral evaluates to 2π(e⁴ - 1) using a linear change of variables with the given transformation, transformed region, Jacobian, and converted integral.
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1) If the demand equation for a certain commodity is given by the equation: 550p + q = 86,000 where p is the price per unit; at what price is there unitary elasticity? Round your answer off to two decimal places. p =_____________? (1 point)
The price at which unitary elasticity occurs is $157.14 per unit.
To find the price at which unitary elasticity occurs, we need to first determine the elasticity of demand with respect to price. The elasticity of demand is defined as the percentage change in quantity demanded divided by the percentage change in price.
Let's rearrange the demand equation to solve for q
q = 86,000 - 550p
We can then take the derivative of q with respect to p:
dq/dp = -550
This tells us that the rate of change of quantity demanded with respect to price is a constant -550. To find the price at which unitary elasticity occurs, we need to find the price where the absolute value of the elasticity is equal to 1.
Using the formula for elasticity of demand
e = (dq/dp) × (p/q)
At unitary elasticity, e = -1, so:
-1 = (dq/dp) × (p/q)
Substituting in the expression for dq/dp and q, we get
-1 = (-550) (p / (86,000 - 550p))
Simplifying this equation gives
p = $157.14
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The price at which unitary elasticity occurs is $157.14 per unit.
To find the price at which unitary elasticity occurs, we need to first determine the elasticity of demand with respect to price. The elasticity of demand is defined as the percentage change in quantity demanded divided by the percentage change in price.
Let's rearrange the demand equation to solve for q
q = 86,000 - 550p
We can then take the derivative of q with respect to p:
dq/dp = -550
This tells us that the rate of change of quantity demanded with respect to price is a constant -550. To find the price at which unitary elasticity occurs, we need to find the price where the absolute value of the elasticity is equal to 1.
Using the formula for elasticity of demand
e = (dq/dp) × (p/q)
At unitary elasticity, e = -1, so:
-1 = (dq/dp) × (p/q)
Substituting in the expression for dq/dp and q, we get
-1 = (-550) (p / (86,000 - 550p))
Simplifying this equation gives
p = $157.14
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What is the probability that Niamh chooses B after she had the hint
The probability that Niamh chooses B after she had the hint is given as follows:
0.30 = 30%.
How to calculate a probability?A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.
The sum of all the probabilities is given as follows:
1 = 100%.
Hence, initially, considering that there are four choices, the probability of each is given as follows:
1/4 = 0.25.
A decays by 0.15, while the increase in each of the other probabilities is given as follows:
0.15/3 = 0.05.
Hence the probability of B is given as follows:
p = 0.25 + 0.05
p = 0.3.
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Find x to the nearest degree 
Answer:
X° = 72.6459
Step-by-step explanation:
To solve x we must use tan b/c it contain both side,
which is opposite and adjecent
tan ( x°) =16/5
tan ( x°) =16/5tan ( x°) = 3.2
tan ( x°) =16/5tan ( x°) = 3.2X °= tan^-1(3.2)
tan ( x°) =16/5tan ( x°) = 3.2X °= tan^-1(3.2)X° = 72.6459 round to 72.65°
find the sum of the complex numbers.
(3+5i)+(10+7i)
Answer:
13 + 12i
Step-by-step explanation:
3 + 5i + 10 + 7i
First, we need to group it and add it.
3 + 10 because they are numbers.
7i + 5i because they have the same variable.
Next,
= 3 + 10 + 5i + 7i
= 13 + 12i
☆Hope this helps!☆
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The sum of complex numbers (3+5i) + (10+7i) is 13+12i.
To find the sum of the two complex numbers (3+5i) and (10+7i), we add their real parts and imaginary parts separately.
Real part of the sum = Real part of (3+5i) + Real part of (10+7i) = 3 + 10 = 13
Imaginary part of the sum = Imaginary part of (3+5i) + Imaginary part of (10+7i) = 5i + 7i = 12i
Therefore, the sum of the two complex numbers is:
(3+5i) + (10+7i) = 13 + 12i
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What is the probability that the spinner will
land on a 5 and then a 1? Write your answer as a
percent
The probability of spinning a 5 first and then a 1 is:
(1/6) * (1/6) = 1/36
Expressed as a percent, this is:
(1/36) * 100% = 2.78%
So the probability of landing on a 5 and then a 1 is 2.78%
Writing Rational Numbers as Repeating Decimals
highlight the number that repeats
When writing a rational number as a decimal, the decimal may either terminate or repeat indefinitely.
If the decimal repeats, there is a pattern of digits that repeat after a certain point. To indicate the repeating pattern, a bar is placed over the digits that repeat. This bar is typically placed over the smallest repeating pattern, which may be one or more digits.
For example, in the decimal representation of 1/3, the digit 3 repeats indefinitely, so the number is written as 0.333... with a bar over the 3. In the decimal representation of 2/7, the pattern 142857 repeats indefinitely, so the number is written as 0.285714285714... with a bar over the repeating pattern.
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Complete Question:
Writing Rational Numbers as Repeating Decimals. Highlight the number that repeats.
Can you help me with this? I don’t understand it..
Note that the histogram showing the age of campers and the frequency of attendance is attached accordingly.
What is histogram?A histogram is a graph that uses rectangles to represent the frequency of numerical data. The vertical axis of a rectangle reflects the distribution frequency of a variable (the quantity or frequency with which that variable appears).
It is used to summarise discrete or continuous data on an interval scale. It is frequently used to depict the key aspects of data distribution in a handy format.
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A company produces two products. Each product can be produced on either of two machines. The time (in hours) required to produce each product on each machine is shown below:Machine 1 Machine 2Product 1 5 4Product 2 8 5Each month, 600 hours of time are available on each machine, and that customers are willing to buy up to the quantities of each product at the prices that are shown below:Demands Prices Month 1 Month2 Month1 Month2Product1 120 200 $60 $15Product2 150 130 $70 $35The company's goal is to maximize the revenue obtained from selling units during the next two months.How many constraints does this problem have?How many decision variables does this problem have?
The decision variables for this problem are:
x1,1 (the number of units of product 1 produced on machine 1)x1,2 (the number of units of product 1 produced on machine 2)x2,1 (the number of units of product 2 produced on machine 1)x2,2 (the number of units of product 2 produced on machine 2)Evaluate decision variables for this problem?This problem has the following constraints:
Production time cannot exceed the available time on each machine:
5x1,1 + 8x2,1 ≤ 600
4x1,2 + 5x2,2 ≤ 600
Production cannot be negative:
x1,1 ≥ 0
x1,2 ≥ 0
x2,1 ≥ 0
x2,2 ≥ 0
Demand must be met for each product:
x1,1 + x1,2 ≥ 120
x2,1 + x2,2 ≥ 150
Demand cannot exceed the maximum demand for each product:
x1,1 + x1,2 ≤ 200
x2,1 + x2,2 ≤ 130
Therefore, this problem has 4 constraints.
The decision variables for this problem are x1,1 (the number of units of product 1 produced on machine 1), x1,2 (the number of units of product 1 produced on machine 2), x2,1 (the number of units of product 2 produced on machine 1), and x2,2 (the number of units of product 2 produced on machine 2).
Therefore, this problem has 4 decision variables.
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ASAP!!!!!!! I NEED THIS ANSWERED!!!
Answer:
Total Surface Area is 20
Step-by-step explanation:
The formula for surface are with slant heigh is
SA = a^2 + 2×a×l
a = Base Edge (this case 2)
I = Slant Height (this case 4
2^2 + 2(2)(4) = 4+16=20
21 34 let x be a random variable with pdf f(x)=1/13,21 find p(x>30) (round off to second decimal place).
Let x be a random variable with pdf f(x) = 1/13, 21 P(X > 30) = 0.31.
We are given that X is a random variable with a probability density function (pdf) of f(x) = 1/13 for the interval 21 x 34.
We are asked to find P(X > 30), which means we need to find the probability of the random variable X being greater than 30. To do this, we will calculate the area under the PDF in the interval [30, 34].
Step 1: Determine the width of the interval [30, 34].
Width = 34 - 30 = 4
Step 2: Calculate the area under the PDF in the interval [30, 34].
Since the pdf is a constant value (1/13) within the given interval, we can calculate the area as follows:
Area = f(x) * width
Area = (1/13) * 4
Step 3: Round off the result to the second decimal place.
Area ≈ 0.31 (rounded to two decimal places)
So, P(X > 30) ≈ 0.31.
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Determine whether the sequence, an= cos(n*pi/n+1) converges or diverges. If it converges find the limit.
The limit of the sequence as n approaches infinity is -1. Since the limit exists and is finite, the sequence a_n = cos(n*pi/(n+1)) converges. The limit is -1.
To determine whether the sequence, [tex]an= cos(n*pi/n+1)[/tex] converges or diverges, we first note that[tex]n*pi/n+1 = n*[/tex](pi/(n+1)). As n approaches infinity, pi/(n+1) approaches zero. Thus, we can rewrite the sequence as[tex]an = cos(n*(pi/(n+1)))[/tex]
We know that the cosine function oscillates between -1 and 1, and as n gets larger, the argument [tex]n*(pi/(n+1))[/tex] becomes more and more dense in the interval[tex][0, pi][/tex]. Thus, we can say that the sequence oscillates between -1 and 1 infinitely many times as n approaches infinity.
Therefore, the sequence diverges as it does not approach a single limit value.
To determine if the sequence [tex]a_n = cos(n*pi/(n+1))[/tex]converges or diverges, we need to find the limit as n approaches infinity.
Step 1: Write down the sequence formula:
[tex]a_n = cos(n*pi/(n+1))[/tex]
Step 2: Calculate the limit of the sequence as n approaches infinity:
[tex]lim (n → ∞) cos(n*pi/(n+1))[/tex]
Step 3: Analyze the argument inside the cosine function:
As n approaches infinity, the fraction n/(n+1) approaches 1. Therefore, the argument inside the cosine function [tex](n*pi/(n+1))[/tex] approaches pi.
Step 4: Calculate the limit of the cosine function:
cos(pi) = -1
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A cylinder has a height of 10 centimeters and a radius of 19 centimeters. What is its volume? Use ≈ 3.14 and round your answer to the nearest hundredth.
Answer:
The formula for the volume of a cylinder is given by:
Volume = π * radius^2 * height
Given that the height of the cylinder is 10 centimeters and the radius is 19 centimeters, we can substitute these values into the formula and use the approximation of π as 3.14:
Volume = 3.14 * (19^2) * 10
Calculating the square of the radius:
Volume = 3.14 * 361 * 10
Multiplying the values:
Volume = 11354 * 10
Volume = 113540 cubic centimeters (rounded to the nearest hundredth)
So, the volume of the cylinder is approximately 113540 cubic centimeters.
The volume of a cylinder is calculated using the formula:
Volume = πr²h
where, π = 3.14
radius = 19 cm
Height = 10 cm
Volume = πr²h
= 3.14 × 19² × 10
= 3.14 × 361 × 10
= 11335.40 cm³
if every column of an augmented matrix contains a pivot then the corresponding system is consistent,true or false?
Answer: The given statement "if every column of an augmented matrix contains a pivot then the corresponding system is consistent" is true. This is because when every column of an augmented matrix contains a pivot, it implies that there are no free variables in the system of equations represented by the matrix.
Step-by-step explanation: Since every variable has a pivot in the augmented matrix, there is a unique solution to the system of equations. This is the definition of a consistent system - one that has at least one solution. In summary, the statement is true because the presence of a pivot in every column of an augmented matrix guarantees a unique solution to the system of equations, which is the definition of a consistent system.
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in a poisson distribution, the: a. median equals the standard deviation. b. mean equals the variance. c. mean equals the standard deviation. d. none of these choices.
The correct answer is d. none of these choices.
In a Poisson distribution, the mean is equal to the variance. The median may or may not be equal to the standard deviation, as it depends on the specific values and shape of the distribution.
The Poisson distribution is a random distribution. It gives the probability of an event occurring at any time (k) at a given time or place. The Poisson distribution has only one parameter, the number of events, λ (lambda).
For example, a call center receives an average of 180 calls per hour, 24 hours a day. The call is free; accepting one does not change the outcome of the next coming. The number of calls received per minute follows a Poisson probability distribution with an average of 3: the most common numbers are 2 and 3, but 1 and 4 are also possible with a probability of as little as zero, and the result is very small maybe 10. Another example is the number of radio disturbance events during the observation period.
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Which statement correctly compares functions f and g? function f function g An exponential function passes through (minus 1, 5), and (2, minus 1.5) intercepts axis at (1, 0), and (0, 2) Function g is a decreasing exponential function with a y-intercept of 5 and no x-intercept. A. They have different end behavior as x approaches -∞ and different end behavior as x approaches ∞. B. They have the same end behavior as x approaches -∞ but different end behavior as x approaches ∞. C. They have different end behavior as x approaches -∞ but the same end behavior as x approaches ∞. D. They have the same end behavior as x approaches -∞ and the same end behavior as x approaches ∞.
This text presents information about two exponential functions f and g. Function f passes through the points (-1, 5) and (2, -1.5), and intercepts the x-axis at (1, 0) and the y-axis at (0, 2). Function g is a decreasing exponential function with a y-intercept of 5 and no x-intercept. The text asks to compare the end behavior of these two functions as x approaches negative and positive infinity. End behavior refers to the behavior of the function as x approaches either positive or negative infinity.
suppose that x ⇠ unif(10) and y ⇠ unif(10) are independent discrete rvs. find p (xy = 36)
The probability that xy = 36 is 3/100.
Since x and y are discrete uniform random variables over {1,2,3,4,5,6,7,8,9,10}, we have:
P(x = i) = 1/10 for i = 1,2,...,10
P(y = j) = 1/10 for j = 1,2,...,10
We need to find P(xy = 36), which means that xy = 36. Since x and y can only take on integer values between 1 and 10, the only possible pairs of (x,y) that satisfy xy = 36 are (6,6) and (9,4) (or (4,9)).
Therefore:
P(xy = 36) = P((x=6) and (y=6)) + P((x=9) and (y=4)) + P((x=4) and (y=9))
= P(x=6) * P(y=6) + P(x=9) * P(y=4) + P(x=4) * P(y=9)
= (1/10)(1/10) + (1/10)(1/10) + (1/10)*(1/10)
= 3/100
So the probability that xy = 36 is 3/100.
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(1 point) consider the linear system y⃗ ′=[−36−24]y⃗ . y→′=[−3−264]y→. find the eigenvalues and eigenvectors for the coefficient matrix.
The eigenvalues are λ1 = -3 and λ2 = -4, and the corresponding eigenvectors are:
| 2 | | 1 |
| -3 | | -2 |
The coefficient matrix for the linear system is A = [−36−24], or
| -3 -2 |
| -6 -4 |
We must resolve the characteristic equation det(A - I) = 0 to get the eigenvalues, where I is the identity matrix of the same size as A:
[tex]| -3 -2 | | λ 0 | | -3-λ -2 |[/tex]
| -6 -4 | - | 0 λ | = | -6 -4-λ|
Expanding the determinant and setting it to zero, we get:
(-3-λ)(-4-λ) - (-2)(-6) = 0
λ^2 + 7λ + 12 = 0
(λ+3)(λ+4) = 0
Therefore, the eigenvalues are λ1 = -3 and λ2 = -4.
To find the eigenvectors corresponding to each eigenvalue, we solve the system (A - λI)v = 0, where v is a non-zero vector. For λ1 = -3, we have:
[tex]| -3 -2 | | v1 | | 0 |[/tex]
[tex]| -6 -4 | - | v2 | = | 0 |[/tex]
which simplifies to the equation -3v1 - 2v2 = 0, or v2 = -3/2 v1. Choosing v1 = 2, we get v2 = -3, so the eigenvector corresponding to λ1 is:
| 2 |
| -3 |
For λ2 = -4, we have:
[tex]| -3 -2 | | v1 | | 0 |[/tex]
[tex]| -6 -4 | - | v2 | = | 0 |[/tex]
which simplifies to the equation -4v1 - 2v2 = 0, or v2 = -2v1. Choosing v1 = 1, we get v2 = -2, so the eigenvector corresponding to λ2 is:
| 1 |
| -2 |
Therefore, the eigenvalues are λ1 = -3 and λ2 = -4, and the corresponding eigenvectors are:
| 2 | | 1 |
| -3 | | -2 |
respectively.
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7.Consider the graph of figure ABCD.
Imagine that figure ABCD
is rotated 90∘
clockwise about the origin, to create figure A′B′C′D′.
Match each point of the image to its coordinate.
The new position of the rectangle after the 90-degree clockwise rotation will be A'B'C'D' shown in figure.
Define the term translation?A translation is a geometric transformation in which each point in a figure or space moves in the same direction.
If a point A (h, k) is rotated about the origin through 90° in clockwise direction. So, the new position will become A' (k, -h).
After rotating the rectangle ABCD 90° clockwise about the origin, the new position of the rectangle will be A'B'C'D'. The coordinates of the new vertices can be found by applying a 90° clockwise rotation transformation to each of the original coordinates of the vertices.
Coordinates of original vertices are A (1, 5), B (4, 5), C (1, 2), and D (4, 2) the coordinates of the new vertices A'B'C'D' can be calculated as follows:
A (1, 5) ⇒ A' (5, -1)
B (4, 5) ⇒ B' (5, -4)
C (1, 2) ⇒ C' (2, -1)
D (4, 2) ⇒ D' (2, -4)
So, the new position of the rectangle after the 90-degree clockwise rotation will be A'B'C'D'.
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true or false if wealth has increasing marginal utility for an individual, that individual is said to be risk-averse.
False. If wealth has increasing marginal utility for an individual, it implies that the person derives greater satisfaction from each additional unit of wealth.
False. If wealth has increasing marginal utility for an individual, it implies that the person derives greater satisfaction from each additional unit of wealth. However, risk-averse individual typically experiences diminishing marginal utility of wealth, which means they derive less satisfaction from each additional unit of wealth. Risk-averse individuals are more cautious with their decisions, preferring lower-risk options to avoid potential losses .unwilling to take risks or wanting to avoid risks as much as possible
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Switch to the Cost Estimates worksheet. In cell A9, create a formula using the AVERAGE function that calculates the average of the values in the range A5:A7, then copy your formula to cell 09. In cell A10, create a formula using the MAX function that identifies the maximum value in the range A5:A7 and then copy your formula to cell D10. In cell A11, create a formula using the MIN function that identifies the minimum value in the range A5:A7 and then copy your formula to cell 011.In cell B13, create a formula using the VLOOKUP function that looks up the value from cell A11 in the range A5:B7, returns the value in column 2, and specifies an exact match. Copy the formula to cell E13. Switch to the Profit Projections worksheet. In cell H5, use the TODAY function to insert the current date.
The current date using the TODAY function, you can enter the formula "=TODAY()" in cell H5. This will display the current date in the cell.
Switch to the Cost Estimates worksheet. In cell A9, create a formula using the AVERAGE function that calculates the average of the values in the range A5:A7, then copy your formula to cell 09.
cell A10, create a formula using the MAX function that identifies the maximum value in the range A5:A7 and then copy your formula to cell D10. In cell A11, create a formula using the MIN function that identifies the minimum value in the range A5:A7 and then copy your formula to cell 011.
In cell B13, create a formula using the VLOOKUP function that looks up the value from cell A11 in the range A5:B7, returns the value in column 2, and specifies an exact match. Copy the formula to cell E13. Switch to the Profit Projections worksheet. In cell H5, use the TODAY function to insert the current date.
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Express the following Cartesian coordinates in polar coordinates in two ways. (-6, 2√3) Select all that apply. A. (4 √3, 3 π/4) B. (3 √3, 3 π/4) C. (-3, √3, 7 π/4) D. (4 √3, 5 π/6) E. (-4 √3, 7 π/4) F. (-4 √3, 11 π/6) G. (3 √3, 5 π/6) H. (-3 √3, 11 π/6)
The polar coordinates are (4√3, 5π/6). The correct answer is D. (4√3, 5π/6). The other given options are incorrect.
To convert Cartesian coordinates (-6, 2√3) to polar coordinates, we use the formulas:
r = √(x^2 + y^2)
θ = tan^-1 (y/x)
Plugging in the values, we get:
r = √((-6)^2 + (2√3)^2) = √(36 + 12) = 2√13
θ = tan^-1 (2√3/-6) = -π/3
However, since the point is in the second quadrant, we need to add π to the angle, giving us:
θ = -π/3 + π = 2π/3
Therefore, the polar coordinates of (-6, 2√3) can be expressed in two ways:
A. (4 √3, 3 π/4)
B. (3 √3, 3 π/4)
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For the sequence (5, 8, 11, 14, 17,...), answer the following question.
1.What is the second term of the sequence?
Answer:8
Step-by-step explanation:5 is 1st term 8 is second term 11 is third term…
Help, please. I'm stuck.
CD is the altitude to side AB of right [tex]\triangle[/tex]ABC, where m[tex]\angle[/tex]ACB = [tex]90^o[/tex] The value of BC is 7.28 units.
What is value?Value in math is a concept that describes the magnitude, or size, of a number. It can refer to absolute value, which is the actual number, or it can refer to relative value, which is the number compared to other numbers. Value is important in math because it is used to compare and measure different quantities. For example, in addition and subtraction, the value of the numbers being added or subtracted determines the answer. In multiplication, the value of the factors determines the product. Value is also important for performing calculations, such as finding averages, which requires knowledge of numbers and their relative values.
The given triangle is a right triangle, with ∠acb as the right angle. Using the Pythagorean Theorem, we can find the length of the side BC. The Pythagorean Theorem states that the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
Therefore, BC² = AC² + BD²
Substituting the given values in the equation,
BC² = 52 + (5 1/3)²
Simplifying the equation,
BC² = 25 + 27.69
Therefore, BC² = 52.69
Taking the square root of both sides,
BC = √52.69
Therefore, BC = 7.28 units.
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free motions of a mass–spring systems are modeled as nonhomogeneous linear odes, true or false?
The required answer is free motions of a mass–spring systems are modeled as nonhomogeneous linear false.
The free motions of a mass-spring system are actually modeled as homogeneous linear ordinary differential equations. Nonhomogeneous linear ordinary differential equations can arise when there are external forces or inputs acting on the system.
A differential equation is homogeneous if it is a homogeneous function of the unknown function and its derivatives. In the case of linear differential equations, this means that there are no constant terms. The solutions of any linear ordinary differential equation of any order may be deduced by integration from the solution of the homogeneous equation obtained by removing the constant term.
Hooke's equation holds (to some extent) in many other situations where an elastic body is deformed, such as wind blowing on a tall building, and a musician plucking a string of a guitar. An elastic body or material for which this equation can be assumed is said to be linear-elastic or Hookean.
Hooke's law is only a first-order linear approximation to the real response of springs and other elastic bodies to applied forces. It must eventually fail once the forces exceed some limit, since no material can be compressed beyond a certain minimum size, or stretched beyond a maximum size, without some permanent deformation or change of state. Many materials will noticeably deviate from Hooke's law well before those elastic limits are reached.
Free motions of a mass-spring systems are modeled as homogeneous linear ordinary differential equations . nonhomogeneous. This is because in free motion, there is no external force acting on the system, making the equation homogeneous.
A linear differential equation can be represented as a linear operator acting on y(x) where x is usually the independent variable and y is the dependent variable. Therefore, the general form of a linear homogeneous differential equation .
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