1. Suppose that a random variable X has a probability density function given by f(x) = {ax³e-x/2, x>0 0, elsewhere. a) Find the value of a that makes f(x) a probability density function (pdf). [3]

Answer:

10.5ft2

Step-by-step explanation:

Given:

The population of the bacteria at the beginning = 80

the population grows according to a continuous exponential growth model.

after 14 days, there are 216 bacteria. ​

y = the number of bacteria after time t

So, the general relation between y and t will be:

We need to find the values of a and b

At t = 0 y = 80

So,

When t = 14 , y = 216

So,

so, the function will be:

Part b: we need to find the number of bacteria after 23 days

So, substitute with t = 23

so,

so, after 23 days the number of bacteria = 409

Answer:

Step-by-step explanation:

Rewrite in slope-intercept form.

The slope-intercept form is  

y = m x + b , where  m  is the slope and  b  is the y-intercept.  y  = m x + b

Reorder  2  and  2 x .  y = 2 x + 2

Use the slope-intercept form to find the slope and y-intercept.

Slope:  

2

y-intercept:  

2

Answer:

280 seats total

Step-by-step explanation:

90/100 = 252/x

To find x, cross multiply.

90x = 25,200

Divide both sides by 90 to get the variable alone.

x = 280 seats

a. The equation representing the cost of buying and operating the Super Cool X1400 is: C = 800 + 60m. b. The equation representing the cost of buying and operating the Efficient Energy X2000 is: C = 1200 + 40m. c. Your family would have to use the Efficient Energy model for 10 months to compensate for the additional cost of the original purchase. d. Figuring your family will only use the air conditioner for four months each year, you would have to wait 5 years to start saving money overall.

a. The cost of buying the Super Cool X1400 is a one-time expense of $800. Additionally, the cost of operating it is $60 per month. Therefore, the equation representing the cost C as a function of months m is C = 800 + 60m.

b. The cost of buying the Efficient Energy X2000 is a one-time expense of $1200. The operating cost is $40 per month. Hence, the equation representing the cost C as a function of months m is C = 1200 + 40m.

c. To compensate for the additional cost of the original purchase (which is $400), we equate the two cost equations and solve for m:

800 + 60m = 1200 + 40m

20m = 400

m = 20

Therefore, your family would have to use the Efficient Energy model for 20 months (or 10 months more than the Super Cool X1400) to compensate for the additional cost.

d. Considering your family only uses the air conditioner for four months each year, we divide the total number of months (20) by four to find the number of years needed to start saving money overall:

20 months / 4 months/year = 5 years

Hence, you would have to wait 5 years to start saving money overall by choosing the Efficient Energy X2000.

a. The cost of buying and operating the Super Cool X1400 is represented by the equation C = 800 + 60m.

b. The cost of buying and operating the Efficient Energy X2000 is represented by the equation C = 1200 + 40m.

c. Your family would have to use the Efficient Energy model for 10 months to compensate for the additional cost of the original purchase.

d. Assuming your family uses the air conditioner for four months each year, you would have to wait 5 years to start saving money overall by choosing the Efficient Energy X2000.

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complete question- Your family plans to buy a new air conditioner. They can buy the Super Cool X1400 for $800, or they can buy the Efficient Energy X2000 for $1200. Both models will cool your home equally well, but the Efficient Energy model is less expensive to operate. The Super Cool X1400 will cost $60 per month to operate, while the Efficient Energy X2000 costs only $40 per month to operate. a. Write an equation to represent the cost of buying and operating the Super Cool X1400 where C= cost and m= months. b. Write an equation to represent the cost of buying and operating the Efficient Energy X2000. c. How many months would your family have to use the Efficient Energy model to compensate for the additional cost of the original purchase? d. Figuring your family will only use the air conditioner for four months each year, how many years ill you have to wait to start saving money overall?

The weight of an object is the amount of gravitational force applied on the object by the earth. The weight of the oranges to be used is 6.25 pounds.

The level of influence of the earth's gravitational pull on an object determines its weight. This depends also on the size of the mass of the object.

Given that Bella bought 15.625 pounds of oranges.

Let the weight of the oranges required to prepare fruit salad be represented by W. Then the weight of the oranges required is 0.4 of the weight of oranges bought, so that;

W = 0.4 * 15.625

   = 6.25

W = 6.25

Thus, the weight of the oranges required to be used for fruit salad is 6.25 pounds.

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the amount of juice left in Luna's glass is 1/2.

We have been given that Luna mixes 3/4 cup of orange juice with 3/8 cup of cranberry juice.

First of all, let us find the total amount of juice Luna had by adding the amount of orange juice to amount of cranberry juice.

Total amount of juice Luna had = 3/4 + 3/8

                                                 = 9/8

Since Luna gave 5/8 cup of juice to Mags, so let us subtract amount of juice given to Mags from the total amount of juice to find the amount of juice left in Luna's glass.

Hence, Amount of juice left in Luma glass is,

= 9/8 - 5/8

= 4/8

= 1/2

Therefore, the amount of juice left in Luna's glass is 1/2.

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I just checked with a calculator, and the mean stays the same.

So the answer is the first option! :D

the volume of the solid is (32/5)π cubic units.

The volume of the solid that lies under the paraboloid z=x²+y², above the xy-plane, and inside the cylinder x²+y²=2x is given by the following formula below: V=∫∫R (x²+y²) dA, where R is the projection of the solid on the xy-plane. In this case, the projection is a circle centered at (1, 0) with radius 1. This can be obtained by completing the square in the equation of the cylinder: x²-2x+y²=0⇔(x-1)²+y²=1. Hence, we can write the integral as V=∫0²π) ∫0¹ r³ dr dθ=π/2. Answer: π/2.
To compute the volume of the solid, we can use polar coordinates. The equation of the cylinder x² + y² = 2x in polar coordinates is r² = 2r×cos(θ), which simplifies to r = 2cos(θ). The volume can be found by integrating the paraboloid function z = r² with respect to r and θ.

The volume V is given by the double integral:

V = ∬(r²) * r dr dθ

The limits of integration for r are from 0 to 2cos(θ) and for θ are from 0 to 2π. Evaluating the double integral, we get:

V = ∫(0 to 2π) ∫(0 to 2cos(θ)) (r³) dr dθ

After evaluating the inner integral, we have:

V = ∫(0 to 2π) [(1/4)r⁴]_(0 to 2cos(θ)) dθ

Which simplifies to:

V = ∫(0 to 2π) (4cos⁴(θ)) dθ

Evaluating the final integral, we get:

V = (32/5)π

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Answer:

3rd option

Step-by-step explanation:

=)

Answer:

Sure.

Step-by-step explanation:

So X is a variable meaning its a letter that represents a number. When there are a number and a variable (or a letter) like this 2x or 9y it means you are going to multiply the number by the variable. In this case it is 7 times x is equal to 21 so all we have to do is find what times 7 is equal to 21 and that number is 3. So if 7*x=21 is 7*3=21 then x=3. If you have any questions ill be more than open to answering them, hopefully this helped. Have a good day/night.

Answer:

5

Step-by-step explanation:

1200cm3 = 12cm * 20cm * Xcm

1200/12 = 100 = 20*X

100/20 = 5 = X

Answer:

1.79% chance or \(\frac{6}{343}\)

Step-by-step explanation:

7+3+4=14

chance of yellow \(\frac{3}{14}\)

chance of orange \(\frac{4}{14}\)

\(\frac{3}{14} * \frac{4}{14} *\frac{4}{14} = \frac{6}{343}\)

Answer:

9.54 - 4.59 =4.95

A short computer code !!

Answer: 8 Oranges

Step-by-step explanation:

Given information

3 Oranges = $1.23

Total cost = $3.28

Determine the unit price of an orange

Unit price = Cost ÷ Number of Oranges

Unit price = 1.23 ÷ 3

Unit price = $0.41 / orange

Determine the number of oranges bought

Number of orange × Unit price = Total cost

N × (0.41) = (3.28)

Divide 0.41 on both sides

N = 3.28 ÷ 0.41

\(\Large\boxed{Number~of~oranges=8}\)

Hope this helps!! :)

Please let me know if you have any questions

The probability θ of teen recidivism based on a study is: the posterior mean is 0.3208. Mode is 0.313 and the standard deviation of the probability estimate is 0.0635.

Probability is the ability to happen. The subject of this area of mathematics is the occurrence of random events. From 0 to 1 is used to express the value. To forecast how probable occurrences are to occur, probability has been introduced in mathematics. The degree to which something is likely to happen is essentially the definition of probability. This is the fundamental theory of probability, which is also applied to the probability distribution, and from which you will discover the likelihood of results for a random experiment.

The posterior mean is given as:

E [Theta| y] = (a + y) / (a + y) + (b + n - y) = 17 /53 = 0.3208.

The mode for the estimate is:

Mode =  (a + y - 1) / (a + y - 1) + (b + n - y -1) = 16/51 = 0.313.

The standard deviation is:

√ (a + y) (b + n - y) / [(a + y) + (b + n - y)]² [(a + y) + (b + n - y) + 1] =  √17(36) / 53(53) (54) = 0.0635.

The plot of the functions are as shown.

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The complete question is:

To determine whether the set S is linearly independent or linearly dependent.The set is linearly independent. This is because the only way to make a linear combination of vectors equal to the zero vector is to have all the coefficients equal to zero.

A linear combination of vectors is the sum of a scalar multiple of each vector in the set. We must check if the equation a(-2,2,4) + b(1,9,-2) + c(2,3,-3) = (0,0,0) has only the trivial solution, i.e., a=b=c=0. This gives us the system of equations,-2a + b + 2c = 01a + 9b + 3c = 02a - 2b - 3c = 0We can solve the system of equations by using Gauss-Jordan elimination. The augmented matrix for the system is:[-2 1 2 0][1 9 3 0][2 -2 -3 0]Let's use elementary row operations to simplify the matrix.

We can swap the first and second rows since the first element in the second row is 1.[1 9 3 0][-2 1 2 0][2 -2 -3 0]We can then add twice the first row to the third row to eliminate the leading coefficient in the third row.[1 9 3 0][-2 1 2 0][0 16 3 0]We can then add nine times the first row to the second row to eliminate the leading coefficient in the second row.[1 9 3 0][0 17 15 0][0 16 3 0]We can then add -16/17 times the second row to the third row to eliminate the leading coefficient in the third row.[1 9 3 0][0 17 15 0][0 0 -117/17 0]We see that the only solution is a=0, b=0, and c=0. Therefore, the set S is linearly independent.

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Answer:

11/12

Step-by-step explanation:

To show that the matrix A = [XX - X'Z(ZZ)^(-1)Z'X] is positive definite, we need to demonstrate two properties: (1) A is symmetric, and (2) all eigenvalues of A are positive.

Symmetry: To show that A is symmetric, we need to prove that A' = A, where A' represents the transpose of A. Taking the transpose of A: A' = [XX - X'Z(ZZ)^(-1)Z'X]'. Using the properties of matrix transpose, we have:

A' = (XX)' - [X'Z(ZZ)^(-1)Z'X]'. The transpose of a sum of matrices is equal to the sum of their transposes, and the transpose of a product of matrices is equal to the product of their transposes in reverse order. Applying these properties, we get: A' = X'X - (X'Z(ZZ)^(-1)Z'X)'.  The transpose of a transpose is equal to the original matrix, so: A' = X'X - X'Z(ZZ)^(-1)Z'X. Comparing this with the original matrix A, we can see that A' = A, which confirms that A is symmetric.  Positive eigenvalues: To show that all eigenvalues of A are positive, we need to demonstrate that for any non-zero vector v, v'Av > 0, where v' represents the transpose of v. Considering the expression v'Av: v'Av = v'[XX - X'Z(ZZ)^(-1)Z'X]v

Expanding the expression using matrix multiplication : v'Av = v'X'Xv - v'X'Z(ZZ)^(-1)Z'Xv. Since X and Z have full column rank, X'X and ZZ' are positive definite matrices. Additionally, (ZZ)^(-1) is also positive definite. Thus, we can conclude that the second term in the expression, v'X'Z(ZZ)^(-1)Z'Xv, is positive definite.Therefore, v'Av = v'X'Xv - v'X'Z(ZZ)^(-1)Z'Xv > 0 for any non-zero vector v. Implications in econometrics: In econometrics, positive definiteness of a matrix has important implications. In particular, the positive definiteness of the matrix [XX - X'Z(ZZ)^(-1)Z'X] guarantees that it is invertible and plays a crucial role in statistical inference.

When conducting econometric analysis, this positive definiteness implies that the estimator associated with X and Z is consistent, efficient, and unbiased. It ensures that the estimated coefficients and their standard errors are well-defined and meaningful in econometric models. Furthermore, positive definiteness of the matrix helps in verifying the assumptions of econometric models, such as the assumption of non-multicollinearity among the regressors. It also ensures that the estimators are stable and robust to perturbations in the data. Overall, the positive definiteness of the matrix [XX - X'Z(ZZ)^(-1)Z'X] provides theoretical and practical foundations for reliable and valid statistical inference in econometrics.

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Answer:

1/4

Step-by-step explanation:

there are 24 boats in total. out of those 24 boats, six are from rhode island. so 6/24 boats are from rhode island. simplified it will be 1/4 :)

162 m^2

you can make a triangle in the corner with legs length s/2 and the angles are 45 degrees. in a 45 45 90 triangle, the legs are h/sqrt(2) where h is the hypotenuse. this means that s/2=9/2 sqrt(2), s=9 sqrt(2)

and s^2=162

Answer:

|2x + 1| = 7

|2x − 1| = 7

2x + 1 = -7 or 2x + 1 = 7

Step-by-step explanation:

or if u have no time

C, E, F.

These might not be in order but hey they are correct amiright :/

(a) The real discrete Fourier transform (DFT) is calculated for the given data set to analyze the helicopter's acoustic signature.

(b) To obtain the ck values and illustrate the relative size of each term in the Fourier series, we calculate the magnitude of each coefficient and plot their amplitudes on a bar graph against the corresponding frequency component, k.

To analyze the helicopter's acoustic signature, the real DFT is computed for the provided data set. The DFT transforms the time-domain measurements of acoustic pressure into the frequency domain, revealing the different frequencies present and their corresponding amplitudes. This analysis helps in understanding the spectral characteristics of the helicopter's acoustic signature and identifying prominent frequency components.

Using the Fourier series representation, the amplitudes (ck's) of the different frequency components in the Fourier series are determined. These amplitudes represent the relative sizes of each term in the series, indicating the contribution of each frequency component to the overall acoustic signature. By plotting the amplitudes on a bar graph, the relative strengths of different frequency components become visually apparent, enabling a clear comparison of their importance in characterizing the helicopter's acoustic signature.

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Answer:

Parents usually have a bigger budget bc they can work , teens on the other hand depending on age cant

Step-by-step explanation:

Answer:

The answer to the question is A.

Step-by-step explanation:

Plug in both equations to the graph!

Answer:  A

Step-by-step explanation:

In order to find the intersections you must graph the lines.

slope - intercept form:

y=mx + b

b is where the line hits the y-axis.

y=3x-4

y=-2x+2

The y-intercepts for each line is -4 and +2 respectively

The only graph that has lines going through those points at y-axis is A

An HTML (Hypertext Markup Language) element consists of several parts that work together to define its behavior and appearance on a webpage. The basic structure of an HTML element includes:

Opening tag: The first part of an HTML element is the opening tag, which is enclosed in angle brackets ("<" and ">") and indicates the beginning of the element. It includes the name of the element, such as "p" for a paragraph or "img" for an image.

Attributes: After the name of the element, you can specify one or more attributes that modify the element's behavior or appearance. Attributes consist of a name and a value, separated by an equal sign. For example, the "src" attribute specifies the source URL of an image element.

Content: The content of an HTML element is the information that appears between the opening and closing tags. For example, the content of a paragraph element would be the text that appears within it.

Closing tag: The closing tag is similar to the opening tag but includes a forward slash ("/") before the element name to indicate the end of the element. For example, the closing tag for a paragraph element would be "</p>".

Here's an example of a complete HTML element:

<p class="intro">This is a paragraph element with a class attribute.</p>

In this example, "p" is the name of the element, "class" is an attribute that specifies the CSS class of the element, and "This is a paragraph element with a class attribute." is the content of the element.

a. No, the function is not continuous at x = 1. b. lim(x->1+) f(x) = lim(x->1+) (2 - x) = 2 - 1 = 1 c. the function is not continuous at x = 1.

a. No, the function is not continuous at x = 1

b. To calculate the limit along the left and right path/approach, we need to evaluate the function as x approaches 1 from the left and from the right.

Approaching from the left:

lim(x->1-) f(x) = lim(x->1-) (x - 1) = 1 - 1 = 0

Approaching from the right:

lim(x->1+) f(x) = lim(x->1+) (2 - x) = 2 - 1 = 1

c. To calculate the value of the limit along the top and bottom path/approach, we need to evaluate the function as x approaches 1 from values above and below 1.

Approaching from above:

lim(x->1) f(x) = lim(x->1) (2 - x) = 2 - 1 = 1

Approaching from below:

lim(x->1) f(x) = lim(x->1) (x - 1) = 1 - 1 = 0

At this stage, we can see that the limit of the function as x approaches 1 depends on the direction of approach. The limit from the left and the limit from below are both 0, while the limit from the right and the limit from above are both 1. Since these two sets of limits are different, the limit does not exist at x = 1. Therefore, the function is not continuous at x = 1.

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∠4 and ∠1 are opposite angles, and since opposite angles are equal, ∠4 =  2x - 8

(if u need a specific number)

∠1 is corresponding to ∠5, and since corresponding angles are equal, ∠5 = 2x-8

∠7 is supplementary to ∠5, and since supplementary angles add up to 180, x + 14 + 2x - 8 = 180

-> 3x + 6 = 180
x = (180-6)/3 = 58.

Since ∠4 = 2x - 8 -> ∠4 = 2 x 58 - 8 = 108.

The proof starts by assuming n is a composite integer and proceeds to show that there must exist a prime divisor of n that is less than or equal to √n by contradiction.

To produce a proof that if n is a composite integer, then n has a prime divisor less than or equal to √n, the steps should be arranged in the following order:

Assume n is a composite integer.

Express n as a product of its prime factors.

Suppose all prime factors of n are greater than √n.

Take the product of all prime factors of n.

The product obtained in step 4 is greater than n.

This contradicts the fact that n is a composite integer.

Therefore, the assumption made in step 3 is false.

There must exist at least one prime factor of n that is less than or equal to √n.

Hence, if n is a composite integer, then n has a prime divisor less than or equal to √n.


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