Tony is putting 42 oranges into 4 boxes. Each box holds 12 oranges.

How many more oranges are needed to fill all the boxes?

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.

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

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.

Answer:

D.Non negtaive rational numbers.

Step-by-step explanation:

Given

A store sells self serve frozen yogurt and weight of yogurt taken in ounces.

The function C(w) represents the cost , in dollars .

The given yogurt can be  measured  in gram or kilogram or ounces etc.

The quantity of yogurt can not be negative  because  quantity  is always a positive number . So it can not be a negative number

The quantity is defined as the  amount of  any material or substance .

The quantity is always a positive number . It can not be negative value of a number.

Here, the quantity is taken in ounces  so it is a positive number .

The quantity of any material can be natural number or  positive rational  number .

Hence,the domain for the function C(w)  are non negative rational numbers.

An appropriate domain for the function would be non negtaive rational numbers .

Therefore, the correct answer is option D.

The simplest form of 34 / 710 is 17 / 355.

The division in mathematics is one kind of operation. In this process, we split the expressions or numbers into same number of parts.

Given that the expression 34 / 710.

Now to write the above expression in simple form, we divide the numerator and denominator by 2

34 / 710 = 34/ 2 ÷ 710 / 2

34 / 710 = 17 / 355

Now the above expression is not easily divisible by any number.

Therefore, The simplest form of 34 / 710 is 17 / 355.

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Answer:

Step-by-step explanation:

Given system of linear equations:

y=-1/2x + 4 and

x + 2y=-8.

Let us convert second equation in slope intercept form too.

x + 2y=-8

Subtracting x from both sides, we get

x -x+ 2y=-8-x

2y = -x-8

Dividing both sides by 2, we get

2y/2 = -x/2-8/2

y = -1/2 x - 4.

Let us find slope and y-intercept of each of the equation.

Slope-intercept form of a linear equation is y = mx+b, where m is the slope and b is y-intercept.

On comparing first equation y=-1/2x + 4 by slope-intercept form y = mx+b slope is -1/2 and y-intercept is 4.

For the second equation y = -1/2 x - 4 slope is -1/2 and y-intercept is -4.

Therefore, correct option is 3rd option:

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 number 372 rounded to the nearest 100 would be 400.

When you round a number to a certain digit, you have to check the value of the digit before the place you want to round off to.

1. If the number you are rounding is followed by 5, 6, 7, 8, or 9, round the number up.

2. If the number you are rounding is followed by 0, 1, 2, 3, or 4, round the number down.

Here, the question said nearest hundred, so I had to look at the value in the tens place which is 7.

Hence, we round up to the next hundred.

So, we get;

372 rounded to the nearest 100 is 400.

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Answer:

93 and 110 upper fourth

Step-by-step explanation:

Q3 is at 93 and the max number is at 110 so the upper fourth should fall between 93 and 110, the middle half of the responses  will fall between 70 and 93 and the lower fourth of the responses will fall between 40 and 70

To find the dimensions of the page that will use the least amount of paper, we need to consider the total area of the page and subtract the area of the margins. Set up an equation to represent the given information and solve for the dimensions of the page.

To find the dimensions of the page that will use the least amount of paper, we need to consider the total area of the page and subtract the area of the margins. Let's assume the width of the page is x inches. The total area of the page is then x(x+3) square inches (length times width). The area of the margins at the top and bottom is 2(x+3) square inches (width times depth), and the area of the margins on each side is 3(x) square inches (length times width). We can set up an equation to represent the given information: x(x+3) - 2(x+3) - 3(x) = 64. Simplifying this equation will give us the dimensions of the page that will use the least amount of paper.

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The Answer to the about question rational exponents and radicals is given below.

To solve the first part Let us assume a expression (E) that contains rational exponents

E =  6² / ( 9[tex]\frac{2}{3}[/tex] × 3[tex]\frac{5}{3}[/tex] )

E =  6² / ( (3²)[tex]\frac{2}{3}[/tex] × 3[tex]\frac{5}{3}[/tex] )

E =  6²  / ( 3[tex]\frac{4}{3}[/tex] × 3[tex]\frac{5}{3}[/tex] )

E =   6² / 3[tex]\frac{9}{3}[/tex]

E =   6² / 3³

E = 36/27 = 4/3

E = 4/3

To solve the second part Let us assume a expression (F) that contains radicals.

F = 10 × √(∛(59 +5))

F = 10 × √(∛64)

F = 10 × √(∛(4×4×4))

F = 10 × √4

F = 10 × √(2×2)

F = 10 × 2

F = 20

Thus we have simplified the expression that contains rational exponents and also the expression that contains radicals also.

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To simplify an expression with rational exponents, we multiply the exponents together. To simplify an expression with radicals, we can find the square root of the number.

An expression with rational exponents that can be simplified further is:

To simplify this expression, we can use the rule that states when we raise an exponent to another exponent, we multiply the exponents together. Therefore, (161/4)3/2 = 161/4 * 3/2 = 163/8.

An expression with radicals that can be simplified further is:

To simplify this expression, we can find the square root of 32. The square root of 32 can be simplified as follows: √(32) = √(16 * 2) = √(16) * √(2) = 4√(2).