# Exercises in Statistics: Finding Equilibrium Quantity and Equilibrium Price

1. The difference between two numbers is twelve. Three times the smaller number is four less than twice the larger number. Find the smaller number.

2. John has $20,000 to invest. As his financial consultant, you recommend that he invest in Treasury Bills that yield 5%, Treasury bonds that yield 7%, and corporate bonds that yield 9%. John wants to have an annual income of $1,280, and the amount invested in Treasure Bills must be two times the amount invested in corporate bonds. Find the amount of each investment. (Hint: This is the system of three equations and three variables)

3. A tree removal service assesses a $400 fee and then charges $80 per hour for the time on an owner's property. Hint: This is a linear function.

a) Is $750 enough for 5 hours work?

b) Give the domain and range

If:

f (x) = { (x+1@-x^2@√3x)┤

if x < -2

if -2 ≤ x ≤ 1

if x > 1

Find

f(-5)

f (1/2)

f (2.784)

5. Florida Powers & Light Company supplies electricity to residential customers for a fixed monthly customer charge of $5.25 plus 6.787 cents per kilowatt-hour.

a) Write an equation that relates the monthly charge C, in dollars, to the number x of kilowatt-hours used in a month.

b) What is the monthly charge for using 200 kilowatt-hours?

1. A newly issued affirmative action released by industrial organizations has led to an increase in the number of minors in managerial positions. Baan Pu has vacancies for three managers that it will fill from among five minors and eight majors.

What is the probability that no major is selected?

What is the probability that at least one minor is selected?

What is the probability that exactly one minor is selected?

2. Find the derivatives (dy/dx) of the following functions. Please show the steps you take to solve each one.

f (x)= (2x+5)/5x

f (x)= (〖5x〗^2- 3x )2

y3 = x + 1

3. This question requires the creation of a graph in a word document and submitted via the link below.

Let the supply and demand functions for Toyota Altis be given by: S(q) = (3/2)q+1000 and D(q) = 25000 - (3/2)q , where q is the number of cars.

Graph these functions on the same axes. Also, find the equilibrium quantity and the equilibrium price.

4. Find the slope of the tangent line using the definition of the derivative to the function given below. Then find the points on the function where the tangent line is horizontal.

f (X)= (4/(3)) x^3 - 9x

5. Determine the equation of the line through (2, 5) and perpendicular to y = -2.

#### Solution Preview

1. Let x = the smaller number and let y = the larger number. Then from the information provided,

(1) y - x =12, and,

(2) 3x = 2y - 4 .

By adding x to both sides of the first equation we find that y = 12 + x. With this information we may rewrite the second equation (replacing the "y" with "12 + x") to get: 3x = 2(12 + x) - 4. After multiplying the 2 through the parentheses on the right we get that 3x = 24 + 2x - 4, which leads us to 3x = 20 + 2x. Subtracting "2x" from both sides allows us to solve for x: x = 20. Since we found that y = 12 + x, we may plug x = 20 into this equation to get that y = 12 + 20 = 32. Now we have our answers: the smaller number ("x") is 20 and the larger number ("y") is 32.

2. Let x = amount invested in Treasury Bills

Let y = amount invested in Treasury Bonds

Let z = amount invested in Corporate Bonds

Since John has $20,000 to invest it is clear that (1) x+y+z = $20000.

Using what we know about investments and the interest that they generate we may also write that: (2) 0.05x + 0.07y + 0.09z = $1280.

Also x must be twice z; that is, (3) x = 2z.

Since equation (3) ...

#### Solution Summary

The expert finds the equilibrium quantity and equilibrium price.