Simplify. The answer should contain only positive exponents.

Y = e^2 -> ln y = z [e^2 > 0 -> y > 0]

Fy (y) = P (Y ≤ y)

= P (e^2 £ y)

= P (Z ≤ ln y)

= Fx (ln y)

Differentiating, fy (y) = 1/y fz (ln y)

= 1/y * [(1 / σ sqrt of 2π) e^(1/2 ((ln y - µ)/a)^2)] , y > 0

Which is the required density function of y = e^2

Final answer:

The student needs to find the density function for y = e^z where z is normally distributed. The density function of a normally distributed variable y is determined by its mean (µ) and variance (σ²). Probabilities can be found using a standard normal distribution table, the z-score transformation, and the PDF equation for a normally distributed variable.

Explanation:

The student is asking to find the density function of y = ez, where z follows a normal distribution with mean μ and variance σ2, denoted as z ~ N(μ, σ2). This transformation results in a lognormally distributed variable since the logarithm of y is normally distributed.

The Probability Density Function (PDF) of a normally distributed random variable y is given by the equation:

f(x) = (1/(σ√(2π))) * e-(x-μ)²/2σ²

If z has a standard normal distribution z ~ N(0, 1), you can find the cumulative probability using the cumulative probability distribution function (CDF), also denoted as Φ(z). To find probabilities for z, such as in the case of finding z0.01 = 2.326, you can use a z-table, a calculator, or computer software.

Conversion of x to a z-score is performed using the formula z = (x - μ) / σ. This transformation allows us to use the table for the standard normal distribution to find associated probabilities.

the answer would be:

g(X)

and

m(X)

it's correct i just did the quiz

Answer:

g(X)

and

m(X)

Step-by-step explanation:

There are 3 movies, and each movie is 2 hours long. Assuming that Brittany watches all three without breaks so the total time consumed watching all 3 movies is:

total hours = 3 movies * (2 hours / movie)

total hours = 6 hours

Add 3 to both sides, so you will have

x^2 - 8x + 13 + 3 = 3

x^2 -8x + 16 = 3

And now the left side is a perfect square trinolmial: (x-4)^2.

Final answer:

The area of the parallelogram with the given vertices is calculated using the base and height found from the coordinates, which yields an area of 90 square units.

Explanation:

The student wants to know the area of a parallelogram with given vertices A(-12, 2), B(6, 2), C(-2, -3), and D(-20, -3). To find this area, we can use the distance formula to calculate the lengths of adjacent sides and then find the product of one side's length and the absolute value of the other side's perpendicular height to it. First, we determine the lengths of sides AB and AD which are parallel to the x and y axes, respectively, thereby simplifying our calculations.

Side AB is horizontal, so we can find its length by the difference in x-coordinates: Length of AB = |xB - xA| = |6 - (-12)| = 18 units. Side AD is vertical, so its length is determined by the difference in y-coordinates: Length of AD = |yD - yA| = |(-3) - 2| = 5 units.

The area of the parallelogram is then the product of the base, AB, and the height, AD, which is equal to 18 units * 5 units = 90 square units.

The area of the parallelogram with given vertices is 90 square units, calculated by multiplying the base length AB (18 units) by the height AD (5 units).

To find the area of a parallelogram with vertices A(−12,  2), B(6,  2), C(−2,  −3), and D(−20,  −3), we can calculate the base and the height of the parallelogram. Since points A and B lie on the same horizontal line (they have the same y-coordinate), AB can be considered as the base. To calculate the length of AB, we simply subtract the x-coordinate of A from that of B:

Base AB = xB - xA = 6 - (−12) = 18 units

Points B and C form one of the sides BC of the parallelogram, but because they do not form a vertical line, we cannot directly take BC as the height. Instead, point D has the same x-coordinate as point A which means line AD is vertical, and thus the height can be found by subtracting the y-coordinate of D from that of A:

Height AD = yA - yD = 2 - (−3) = 5 units

The area of a parallelogram is calculated as the product of its base and height:

Area = Base × Height = 18 units × 5 units = 90 square units

Therefore, the area of the parallelogram is 90 square units.

The coordinates of the other point of intersection with the y-axis will be (0, 9). And the x-intercept is at (3, 0).

Let r be the radius of the circle and the location of the center of the circle be (h, k).

Then the equation of the circle is given as,

(x - h)² + (y - k)² = r²

A circle with center (3,5).

Then the equation of the circle will be

(x - 3)² + (y - 5)² = r²

The circle intersects the y-axis at (0,1).

Then the radius of the circle will be

(0- 3)² + (1 - 5)² = r²

             9 + 16 = r²

                    r² = 25

                      r = 5

Then the equation of the circle will be

(x - 3)² + (y - 5)² = 5²

Then the other y-intercept of the circle (0, a) will be

(0 - 3)² + (a - 5)² = 5²

        9 + (a - 5)² = 25

              (a - 5)² = 16

                 a - 5 = 4

                       a = 9

The intersection of the circle with x-axis at (b, 0) will be

(b - 3)² + (0 - 5)² = 5²

     25 + (b - 3)² = 25

              (b - 3)² = 0

                 b - 3 = 0

                       b = 3

Thus, the coordinates of the other point of intersection with the y-axis will be (0, 9). And the x-intercept is at (3, 0).

The graph is given below.

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Answer: 14 inches

Step-by-step explanation:

We know that the circumference of a circle is given by :_

[tex]\text{Circumference}=2\pi r[/tex], where r is the radius of the circle.

Given : The circumference of a circle [tex]=28\pi\text{ inches}[/tex]

Then , the circumference of the circle is given by :-

[tex]28\pi=2\pi r\\\\\Rightarrow\ r=\dfrac{28}{2}=14[/tex]

Hence, the radius of circle = 14 inches.

To maximize income for Water World, the admission cost should be increased by $0.75, calculated by finding the vertex of the quadratic equation representing the income as a function of price increase and visitor numbers.

To find the amount by which the admission cost to Water World should be increased to obtain maximum income, we can create a quadratic equation to represent the income based on the number of visitors and admission cost.

Step 1: Define variables and equation

Let x represent the increase in dollars to the current admission cost of $7.50. The new admission cost would be (7.50 + x) dollars.

With each $1 increase, the number of visitors decreases by 25. The new average number of visitors per day would be (225 - 25x) visitors.

Step 2: Formulate the income

Income, I, is equal to the admission cost multiplied by the number of visitors, which is I = (7.50 + x)(225 - 25x).

Step 3: Expand and maximize the quadratic equation

I = 1687.5 - 187.5x + 225x - 25x² = -25x² + 37.5x + 1687.5. This is a quadratic equation that opens downward, meaning it has a maximum point at its vertex.

The formula for the x-coordinate of the vertex of a parabola y = ax² + bx + c is -b/(2a). In this scenario, a = -25 and b = 37.5.

Step 4: Find the vertex

The x-coordinate of the vertex (maximum income) is -37.5 / (2 × -25) = 0.75. Therefore, the admission cost should be increased by $0.75 to maximize income.

The number of people who voted in the election is 420 people

Based on the information given,

Winning candidate = 210 votes

Second candidate = 2 × 210/3 = 140 votes

Last candidate = 1 × 210/3 = 70 votes.

The number of people who voted in the election will be the addition of the total votes and this will be:

= 210 + 140 + 70

= 420 votes

Therefore, from the information given, the number of people who voted in the election is 420 people.

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