For Math Spolitgenus Only

Math 181 Final Exam - Page 2 of 12 Due: 6/12/2017

Directions:

· You must solve all problems using methods described in this course; guess and check is not an appropriate method for solving equations.

· For all financial math problems, list the formula that you used (from the last page of the exam) and the values for each of the variables. Do not just write your answer!

1. (5 points) Find the value of c for which the lines cx+3y = 1 and 2x−5y = 8 are perpendicular.

2. (5 points) The Hulk is crossing a bridge over the a canyon through which a river is flowing. He flips a coin off the bridge with his big green thumb. The distance h in feet of the coin above the water t seconds after The Hulk has flipped it is given by

h = −16t2 + 96t + 314.

What is the greatest height the coin reaches above the water?

3. (10 points) A rental car charge is an initial fee plus $32 per day. After 4 days, the charge is $204.

(a) Find a linear function C(d) which gives the cost in dollars for the car after d days.

(b) Use your linear model to determine the cost of renting the car for 8 days.

4. (10 points) Consider the box shown below.

The girth of a box is the perimeter of the base. The base of the box below is 3 times its width. The combined height and girth of the box is 80 feet.

(a) Let w be the width of the box. Find a formula, V (w), for the volume as a function of the width.

(b) What is the domain of the volume function with the assumption that all side lengths need to be positive?

5. (7 points) Poiseuille’s Law gives the rate of flow, R, of a gas through a cylindrical pipe in terms of the radius of the pipe, r, for a fixed drop in pressure between the two ends of the pipe. In a given pipe where the flow rate is proportional to the fourth power of the radius, the model would be

R(r) = kr4

where k is the constant of proportionality.

(a) Given that the flow rate is 20 cm3/s in a pipe of radius 3 cm for a certain gas, find the constant of proportionality k.

(b) Using the given model and the result from part (a), determine the radius of the cylindrical pipe iif the flow rate is 836.7 cm3/s.

6. (10 points) How much more money will you earn in an account that compounds interest continuously than in an account that compounds monthly if you invest $1,800 for a period of 33 years and the annual interest rate is 6.2%.

7. (10 points) You won $1,500,000 in the lottery. You choose to receive your total winnings by getting an equal payment at the end of each month for 10 years. If the annual interest rate is 4.2% and interest is compounded monthly, how much should the lottery commission invest today in order to pay you your monthly winnings as described?

8. (10 points) The population of Philadelphia in 1776 was 40,000. By 1865, the population had grown to approximately 600,000.

(a) Assume that the population can be modeled with exponential growth by p(t) = p0ekt, where t is the number of years since 1776. Determine the values of p0 and k.

(b) Using your model from part (a), what is the projected population in the year 2018?

9. (10 points) Consider the following system of equations:

Write the system as an augmented matrix and perform elementary row operations (Gaussian Elimination) to solve the system. For full credit, you must label your row operations in each step.

10. (6 points) A dietician is planning a meal that supplies 18 g of Protein, 334 mg of Calcium, and 85 mg of Vitamin C. Three foods will be used.

· Food 1: provides 17 g of Protein, 180 of Calcium, and 0 mg of Vitamin C.

· Food 2: provides 1 g of Protein, 10 mg of Calcium, and 135 mg of Vitamin C.

· Food 3: provides 8 g of Protein, 300 mg of Calcium, and 0 mg of Vitamin C.

Let x the the amount of Food 1 to be eaten, y the the amount of Food 2 to be eaten, and z the the amount of Food 3 to be eaten. Set up a system of linear equations which can solve this problem. You do not need to solve your system of equations.

11. (7 points) Consider the following system of equations:

 3x2 + 4x3  3x1 − 7x2 + 8x3  3x1 − 9x2 + 6x3 The reduced row echelon form of the system is:	− 6x4 − 8x4 − 12x4	= = =	−5 9 15
1 0 0 4 0 1 0 −2  0 0 1 0

This system of equations has infinitely many solutions. Let x4 = t. What are the solutions to the system?

x1 =

x

2

=

x

3

=

x

4

=

12. (10 points) Consider the following system of equations:

(a) If the system were written as a matrix equation AX = B, what are the following?

A = X = B =

(b) Solve for X by using the inverse of the coefficient matrix A.

Potentially Useful Formulas

I = Prt

A = P + I = P + Prt = P(1 + rt)

A = Pert

re = er − 1

n = mt