# Review Problems for the Final

Math 101

These problems are provided to help you study. The presence of a problem on this handout does not imply that there will be a similar problem on the test. And the absence of a topic does not imply that it won't appear on the test.

1. Simplify the following expressions. Express all powers in terms of positive exponents.

(a) .

(b) .

(c) .

(d) .

(e) .

2. y varies inversely with x. Given that when , find x when .

3. P varies directly with and inversely with . Given that when and , find P when and .

4. Calvin Butterball has 29 coins, all of which are dimes or quarters. The value of the coins is $5.45. How many dimes does he have? 5. How many pounds of chocolate truffles worth$2.50 per pound must be mixed with 5 pounds of spackling compound worth $1.70 per pound to yield a mixture worth$2.00 per pound?

6. Calvin leaves the city and travels north at 23 miles per hour. Phoebe starts 2 hours later, travelling south at 17 miles per hour. Some time after Phoebe starts travelling, the distance between them is 166 miles. How many hours has Calvin been travelling at this instant?

7. Calvin can eat 160 double cheeseburgers in 4 hours, while Bonzo can eat 300 double cheeseburgers in 5 hours. If Calvin, Bonzo, and Phoebe work together, they can eat 510 double cheeseburgers in 3 hours. How many hours would Phoebe take to eat 280 double cheeseburgers if she eats alone?

8. The product of two numbers is 156. The second number is 11 more than twice the first. Both numbers are positive. What are the two numbers?

9. Calvin and Bonzo can eat 1440 hamburgers in 6 hours. Eating by himself, it would take Calvin 9 hours longer to eat 1440 hamburgers than it would take Bonzo to eat 1440 hamburgers. How long would it take Bonzo to eat 1440 hamburgers by himself?

10. The hypotenuse of a right triangle is 1 more than 3 times the smallest side. The third side is 1 less than 3 times the smallest side. Find the lengths of the sides.

11. Graph the parabola . Find the roots, and the x and y-coordinates of the vertex.

12. Solve the inequality:

(a) .

(b) .

(c) .

(d) .

13. (a) Write the expression as a single logarithm: .

(b) Write the expression as a single logarithm: .

(c) Write the expression as a single logarithm: .

(d) Write the expression as a single logarithm: .

(e) Write the expression as a single logarithm: .

(f) Write the expression as a single logarithm: .

14. (a) Write the expression as a sum or difference of logarithms, with all variables raised to the first power: .

(b) Write the expression as a sum or difference of logarithms, with all variables raised to the first power: .

(c) Write the expression as a sum or difference of logarithms, with all variables raised to the first power: .

(d) Write the expression as a sum or difference of logarithms, with all variables raised to the first power: .

15. You are given that

(a) Compute .

(b) Compute .

(c) Compute .

16. Suppose and . Find:

(a) .

(b) .

(c) .

17. Solve for x, writing your answer in decimal form correct to 3 places: .

18. Solve for x:

(a) .

(b) .

(c) .

(d) .

(e) .

19. Solve the following equations for x. (Complex number solutions are allowed.)

(a) .

(b) .

(c) .

(d) .

20. Simplify the expression. Complex numbers are NOT allowed.

(a) .

(b) .

(c) .

(d) , assuming that the variables represent nonnegative quantities.

21. Find specific values for a and b which prove that the following statement is not an algebraic identity:

22. (a) Solve: .

(b) Solve: .

23. (a) Combine the fractions over a common denominator: .

(b) Combine the fractions over a common denominator: .

(c) Combine the fractions over a common denominator: .

24. (a) Simplify: .

(b) Simplify: .

(c) Simplify: .

25. Compute (without using a calculator):

(a) .

(b) .

(c) .

(d) .

(e) .

(f) .

26. Suppose that and .

(a) Find .

(b) Find .

(c) Find .

(d) Find .

(e) Find .

(f) Find .

27. Find the inverse function of .

28. Find the inverse function of .

29. Find the inverse function of .

30. Find the domain of the function .

31. Find the domain of the function . (Complex numbers aren't allowed.)

32. Find the domain of the function . (Complex numbers aren't allowed.)

33. Solve for x:

(a) .

(b) .

(c) .

(d) .

(e) .

(f) .

34. Simplify the expressions, writing each result in the form :

(a) .

(b) .

(c) .

(d) .

(e) .

(f) .

35. Find the quotient and the remainder when is divided by .

36. Find the quotient and the remainder when is divided by .

37. Solve the following equations, giving exact answers:

(a) .

(b) .

(c) .

(d) .

(e) .

(f) .

(g) .

(h) .

38. In the following problems, complex numbers are allowed.

(a) Simplify .

(b) Simplify .

(c) Simplify .

(d) Rationalize .

(e) Rationalize .

39. Find the equation of the line:

(a) Which passes through the points and .

(b) Which passes through the point and is perpendicular to the line .

(c) Which is parallel to the line and has y-intercept -17.

40. Solve the system of equations for x and y:

41. (a) Simplify and write the result using positive exponents: .

(b) Assuming that all the variables represent positive quantities, simplify and write the result using positive exponents: .

42. (a) Simplify, cancelling any common factors: .

(b) Simplify, cancelling any common factors: .

(c) Simplify, cancelling any common factors: .

43. Find the center,the radius, and the standard equation of the circle whose equation is

# Solutions to the Review Problems for the Final

1. Simplify the following expressions. Express all powers in terms of positive exponents.

(a) .

(b) .

(c) .

(d) .

(e) .

(a)

(b)

(c)

(d)

(e)

2. y varies inversely with x. Given that when , find x when .

y varies inversely with x, so .

when , so

Thus, . Plug in and solve for x:

3. P varies directly with and inversely with . Given that when and , find P when and .

P varies directly with and inversely with , so

when and , so

Hence,

When and ,

4. Calvin Butterball has 29 coins, all of which are dimes or quarters. The value of the coins is $5.45. How many dimes does he have? Let d be the number of dimes and let q be the number of quarters. The first column says . The last column says . The first of these equations gives . Substitute this into to get . Now solve for q: There are 17 quarters and dimes. 5. How many pounds of chocolate truffles worth$2.50 per pound must be mixed with 5 pounds of spackling compound worth $1.70 per pound to yield a mixture worth$2.00 per pound?

Let x be the number of pounds of truffles.

The last line of the table gives

Solve for x:

The mixture needs 3 pounds of truffles.

6. Calvin leaves the city and travels north at 23 miles per hour. Phoebe starts 2 hours later, travelling south at 17 miles per hour. Some time after Phoebe starts travelling, the distance between them is 166 miles. How many hours has Calvin been travelling at this instant?

Let t be the time Calvin has travelled. Then Phoebe has travelled hours. Let x be the distance Calvin has travelled in this time. Then Phoebe has travelled miles.

The first line says . The second line says . Plug into and solve for t:

Calvin has been travelling for 5 hours.

7. Calvin can eat 160 double cheeseburgers in 4 hours, while Bonzo can eat 300 double cheeseburgers in 5 hours. If Calvin, Bonzo, and Phoebe work together, they can eat 510 double cheeseburgers in 3 hours. How many hours would Phoebe take to eat 280 double cheeseburgers if she eats alone?

Let x be the number of burgers Calvin can eat per hour. Let y be the number of burgers Bonzo can eat per hour. Let z be the number of burgers Phoebe can eat per hour.

The first equation says , so .

The second equation says , so .

The last equation says . Divide by 3: . Substitute and :

The third equation says . Substitute : , so hours.

8. The product of two numbers is 156. The second number is 11 more than twice the first. Both numbers are positive. What are the two numbers?

Let x and y be the two numbers.

The product of two numbers is 156, so .

The second number is 11 more than twice the first, so .

Substitute into :

Then

Factor and solve:

is ruled out, because x is supposed to be positive. Therefore, , and .

9. Calvin and Bonzo can eat 1440 hamburgers in 6 hours. Eating by himself, it would take Calvin 9 hours longer to eat 1440 hamburgers than it would take Bonzo to eat 1440 hamburgers. How long would it take Bonzo to eat 1440 hamburgers by himself?

Let c be Calvin's rate, in hamburgers per hour. Let b be Bonzo's rate, in hamburgers per hour. Let t be the amount of time it takes Bonzo to eat 144 hamburgers.

The last equation gives

The first equation gives

The second equation gives

Plug and into and solve for t:

The solution doesn't make sense, since time can't be negative. The solution is 9. Bonzo takes 9 hours.

10. The hypotenuse of a right triangle is 1 more than 3 times the smallest side. The third side is 1 less than 3 times the smallest side. Find the lengths of the sides.

Let s be the length of the smallest side, let t be the length of the third side, and let h be the length of the hypotenuse. By Pythagoras' theorem,

The hypotenuse of a right triangle is 1 more than 3 times the smallest side, so .

The third side is 1 less than 3 times the smallest side, so .

Plug and into and solve for s:

The possible solutions are and . Now is ruled out, since a triangle can't have a side of length 0. Therefore, is the only solution. The other sides are

11. Graph the parabola . Find the roots, and the x and y-coordinates of the vertex.

The roots are and .

The x-coordinate of the vertex is . The y-coordinate is

12. Solve the inequality:

(a) .

(b) .

(c) .

(d) .

(a) gives , so the roots are and .

The solution is .

(b) equals 0 when and is undefined when . Set up a sign chart with these break points:

The solution is , or in interval notation, .

(c) Solve the corresponding equation:

Draw the picture:

Since the absolute value expression is on the big side of the " ", I want the outside intervals: or , or in interval notation, .

(d) equals 0 when or and is undefined when . Set up a sign chart with these break points:

The solution is or , or in interval notation, .

13. (a) Write the expression as a single logarithm: .

(b) Write the expression as a single logarithm: .

(c) Write the expression as a single logarithm: .

(d) Write the expression as a single logarithm: .

(e) Write the expression as a single logarithm: .

(f) Write the expression as a single logarithm: .

(a)

(b)

(c)

(d)

(e)

(f)

14. (a) Write the expression as a sum or difference of logarithms, with all variables raised to the first power: .

(b) Write the expression as a sum or difference of logarithms, with all variables raised to the first power: .

(c) Write the expression as a sum or difference of logarithms, with all variables raised to the first power: .

(d) Write the expression as a sum or difference of logarithms, with all variables raised to the first power: .

(a)

(b)

I got the last expression using the fact that , because .

(c)

(d)

15. You are given that

(a) Compute .

(b) Compute .

(c) Compute .

(a) Compute .

(b) Compute .

(c) Compute .

16. Suppose and . Find:

(a) .

(b) .

(c) .

(a) .

(b) .

(c) .

17. Solve for x, writing your answer in decimal form correct to 3 places: .

18. Solve for x:

(a) .

(b) .

(c) .

(d) .

(e) .

(a) Square both sides and multiply out:

The possible solutions are and .

Check: For ,

For ,

The only solution is .

(b)

Check: gives

The solution is .

(c)

The possible solutions are and .

Check: If ,

If ,

The solutions are and .

(d)

The possible solutions are and .

Check: If ,

If , both sides of the equation are undefined.

The only solution is .

(e) Square both sides:

Then

Square both sides:

Then

Factor and solve:

Check: gives

gives

The solution is .

19. Solve the following equations for x. (Complex number solutions are allowed.)

(a) .

(b) .

(c) .

(d) .

(a) Write the equation as . Let .

The possible solutions are and .

gives , or . And gives , or .

The solutions are and .

(b) Let .

The possible solutions are and .

gives , or . And gives , or .

The solutions are and .

(c) Write the equation as

Let .

The possible solutions are and .

gives , or . And gives , or .

The solutions are and .

(d) Write the equation as

Let . Then

20. Simplify the expression. Complex numbers are NOT allowed.

(a) .

(b) .

(c) .

(d) , assuming that the variables represent nonnegative quantities.

(a)

(b)

(c)

(d)

21. Find specific values for a and b which prove that the following statement is not an algebraic identity:

If and , then

Since , for and .

22. (a) Solve: .

(b) Solve: .

(a) Factor the denominator on the left:

Multiply both sides by to clear denominators:

Then

Check: gives

The solution is .

(b)

The possible solutions are and , but causes division by 0 in the original equation. Hence, the only solution is .

23. (a) Combine the fractions over a common denominator: .

(b) Combine the fractions over a common denominator: .

(c) Combine the fractions over a common denominator: .

(a)

(b)

(c) First, using factoring by grouping, I have

Then

24. (a) Simplify: .

(b) Simplify: .

(c) Simplify: .

(a)

(b)

(c)

25. Compute (without using a calculator):

(a) .

(b) .

(c) .

(d) .

(e) .

(f) .

(a)

(b)

(c)

(d)

(e)

(f)

26. Suppose that and .

(a) Find .

(b) Find .

(c) Find .

(d) Find .

(e) Find .

(f) Find .

(a)

(b)

(c)

(d)

(e)

(f)

If you needed to simplify this, here's what you'd do:

27. Find the inverse function of .

Write . Swap x's and y's:

Solve for y:

The inverse function is .

28. Find the inverse function of .

Write .

Swap x's and y's:

Solve for y:

The inverse function is .

29. Find the inverse function of .

Write .

Swap x's and y's:

Solve for y:

The inverse function is .

30. Find the domain of the function .

Factoring the bottom, I get .

and are not in the domain, because those values cause division by 0.

Hence, the domain is .

Note: In inequality notation, this is or or . In interval notation, this is .

31. Find the domain of the function . (Complex numbers aren't allowed.)

I can't take the square root of a negative number, so can't be negative. Thus, is bad, and this is the same as . So numbers less than -3 aren't in the domain.

Hence, the domain is . In interval notation, this is .

32. Find the domain of the function . (Complex numbers aren't allowed.)

Factoring , I get .

First, and are not in the domain, because those values cause division by zero.

I must also throw out values of x for which , since I can't take the square root of a negative number. To find out which values I need to throw out, I solve the inequality using a sign chart.

Thus, for .

I've found that I need to throw out , , and . Thus, the bad points are .

The domain is the good points: or , or in interval notation, .

33. Solve for x:

(a) .

(b) .

(c) .

(d) .

(e) .

(f) .

(a)

The solutions are and .

(b)

Then

(c)

Then

(d)

Factor and solve:

(e)

(f)

34. Simplify the expressions, writing each result in the form :

(a) .

(b) .

(c) .

(d) .

(e) .

(f) .

(a)

(b)

(c)

(d)

(e)

(f)

35. Find the quotient and the remainder when is divided by .

The quotient is and the remainder is 42.

36. Find the quotient and the remainder when is divided by .

Notice that I "padded" with a " " term to help keep the terms beneath it lined up.

The quotient is and the remainder is .

37. Solve the following equations, giving exact answers:

(a) .

(b) .

(c) .

(d) .

(e) .

(f) .

(g) .

(h) .

(a)

Let . The equation becomes . Factor and solve:

gives . Then .

gives . Then .

The solutions are and .

(b)

(c)

The solution is .

(d)

(e)

Factor and solve:

can't be plugged into the original equation, because you can't take the log of a negative number.

gives

The only solution is .

(f) Let . The equation becomes

The solutions are and .

The solutions are and .

(g)

The possible solutions are and . However, if you plug into the original equation, you get the log of a negative number. On the other hand, checks. Hence, the only solution is .

(h)

The possible solutions are and . However, if you plug into the original equation, you get the log of a negative number. On the other hand, checks. Hence, the only solution is .

38. In the following problems, complex numbers are allowed.

(a) Simplify .

(b) Simplify .

(c) Simplify .

(d) Rationalize .

(e) Rationalize .

(a)

(b)

(c)

(d)

(e)

39. Find the equation of the line:

(a) Which passes through the points and .

(b) Which passes through the point and is perpendicular to the line .

(c) Which is parallel to the line and has y-intercept -17.

(a) The slope is . The point-slope form for the equation of the line is

(b)

The slope of the given line is . The line I want is perpendicular to the given line, so the line I want has slope -4 (the negative reciprocal of ). The point-slope form for the equation of the line is

(c)

The given line has slope 2. The line I want is parallel to the given line, so it also has slope 2. Since it has y-intercept -17, the equation is .

40. Solve the system of equations for x and y:

Multiply the second equation by 2, then subtract it from the first:

Plug this into the first equation: . Then

The solution is , .

41. (a) Simplify and write the result using positive exponents: .

(b) Assuming that all the variables represent positive quantities, simplify and write the result using positive exponents: .

(a)

(b)

42. (a) Simplify, cancelling any common factors: .

(b) Simplify, cancelling any common factors: .

(c) Simplify, cancelling any common factors: .

(a)

(b)

(c)

43. Find the center,the radius, and the standard equation of the circle whose equation is

Complete the square in x and in y.

Half of -4 is -2, and . I need to add 4 to the x-stuff to complete the square.

Half of 6 is 3, and . I need to add 9 to the y-stuff to complete the square.

The center is and the radius is 5.

The best thing for being sad is to learn something. - Merlyn, in T. H. White's The Once and Future King

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