MATH 148 FINAL EXAM SAMPLE PROBLEMS

MATH 148
FINAL EXAM SAMPLE PROBLEMS
1. a) Find the midpoint of the line segment connecting the two points, (–52, 17) and (–10, 10).
b) Find the distance from the midpoint found in part a) to the point (–52, 17). (Round to 4
significant figures.)
2. Solve the following systems of equations:
 x 2  y  8 x  41
a) 
 y  x 2  13 x  24
 y  2x 2  9

c) 
1 2
 y   x  3 x  15
4

x 2  y  9
b) 
 y  2x  1
3. Gary leaves New York at 3 p.m. in his old car and averages 45 mph. Jeff Gordon, the famous
NASCAR driver, leaves New York at 10 p.m. in his race car along the same route and averages
180 mph. Let t represent the number of hours after 3 p.m. Write the distances traveled by Gary
and Jeff. How many hours will Gary travel before he is passed by Jeff? Round your answer to 2
decimal places.
4. An alchemist is trying to turn lead into gold. In order to do this, he must combine lead and
copper in such a way that the resulting mixture is
mixture which is
1
7
lead and
6
copper.
7
2
9
lead. He currently has 30 kilograms of a
How many kilograms of lead must he add to the mixture
to get the required “magical” mixture? (Round to the nearest hundredth.) Did the mixture really
turn to gold?
5. A decorator has 92 one-foot-square tiles that will be laid around the edges of a 12-by-15-foot
room. A rectangular rug that is 3 feet longer than it is wide is to be placed in the center area
where there are no tiles. To the nearest quarter foot, find the dimensions of the smallest rug that
will cover the untiled part of the floor. [Assume that all the tiles are used and that none of them
are split.]
6. Draw a complete graph of the equation y  8 x 3  147 x 2  444 x  895 . Be sure to find and label
the roots. Round your answer to 2 decimal places.
7. Give a window for a complete graph of y  0.05 x 3  15 x  16 . Find the zeros and round to 2
decimal places.
8.
a) If H varies directly as T, then H · T is a constant. True
False
b) If two quantities vary inversely, when one is halved the other
i) doubles
ii) quadruples iii) stays unchanged iv) is halved
c) V varies directly as the cube of r. If V = 33.5 when r = 2, then the constant of
proportionality is
i) 268
ii) 8.375
iii) 4.1875
iv) 16.75
MATH 148
FINAL EXAM SAMPLE PROBLEMS
9. A box with a square base and no top is to be made from a 4-foot by 4-foot sheet of aluminum by
cutting out squares from each corner and folding up the sides. If the box is to have maximal
volume, what size squares need to be removed? (Round your answer to 3 decimal places.)
10. Suppose you want to use 50 sq. feet of material to make an open-top box with a square base.
a) Use the formula for the surface area to express the height h of the box in terms of x.
b) Find the dimensions of the box that will produce the maximal volume. Round your answer to 2
decimal places.
11. For every point (x, y) in the first quadrant and is on the graph y = 16 – x2, consider the rectangle
with corners (0, 0), (x, 0), (0, y), and (x, y). For which value of x does the rectangle have maximal
area? Round to 2 decimal places.
12. Find the minimum distance from the graph of the equation y = x3 + x2 – 6x to the point (–2, 1).
Round to 2 decimal places.
13.
a) Find the equation of a parabola with a zero at 2 and -5 and a y-intercept at (0, -15).
b) Find the equation of a parabola with vertex at (2, 4) and y-intercept at (0, 0).
 3x  2 if x  1
. Find the domain and range of the function and graph the
14. Let f  x   
if x  0
3x  2
function. Label the points for f(–1), f(0), and f(1).
15. Find the largest possible domain and range for each of the functions:
a) f ( x )  log(3  x )
b) g ( x)  e3 x  2
c) h( x )  3  x
16. Use the properties of logarithms to rewrite each expression. Assume x > 0.
a) 2 log( x  2)
b) ln(4ex )
c) log(2 x  2)  log( x  1)
17. Due to increases in tuition, it is expected that in 18 years, the average cost of attending a state
university for one year will be $24,000. How much should you invest today in an account that
earns 6% compounded quarterly so that you have $24,000 in 18 years? (Round to the nearest
cent.)
18. A certain type of bacteria reproduces exponentially. A population of 40 bacteria grows to 200
bacteria in 4 hours. How many hours will it take for the bacteria population of 40 to reach 400?
(Answer accurate to the nearest hour.)
2000
19. The number of people who have heard a rumor after t hours is given by N t  
. How
1  499e 0.3t
long will it take for 200 people to hear the rumor? Find the answer to 2 decimal place accuracy.
20. An earthquake has a magnitude of 6.7 on the Richter Scale. Find the intensity.
MATH 148
FINAL EXAM SAMPLE PROBLEMS
21. Solve: a) log(x) + log(x – 3) = 1
b) 2 log(x – 2) = 4
c) log(x+2) + log(x+2) = 2
22. I found on my calculator that the solution to the equation 3 · 2x = 17 to be approximately x ≈
2.502500341. Find the exact solution to that equation. (No decimal approximations accepted.)
23.
a) Find the decimal approximation for x: log 3 100  x .
2
b) Solve 4 x 16 x  64 algebraically without using logarithms.
ex
c) Solve
 5 x using logarithms. Check your answer by looking at the graph.
3
24. Triangle ABC has the following measures: A = 70°, B = 80°, and a = 2. Triangle FDE has the
following measures: D = 80°, E = 30°, and d = 2.13. These two triangles are
i) similar by SSS
ii) similar by AA
iii) similar by SAS iv) not similar
25. Solve these using the picture below and the right triangle definition for the trigonometric
functions. (Other methods will receive no credit.)
A
c
b
C
B
a
4
, c  15 . Find the exact value of a.
5
1
b) cos A  , c  6 . Find the exact value of a.
3
c) Find the measure of angle B. Answer accurate to 1 decimal place. (Use the information in part a).)
a) sin A 
26. Given that the two triangles below are similar, find a and α. (Answer accurate to 3 significant
figures.)
2
5.3
α
a
4
7
MATH 148
FINAL EXAM SAMPLE PROBLEMS
27. The following information is about the triangles ABC and A'B'C'.
a = 2, b = 4, C = 55°
a' = 4, b' = 8, C' = 55°
a. Are the two triangles similar? Justify your answer.
b. Find the length of side c, angle A, and angle A'.
28. A 12 foot long ladder leans on a wall of a building. The ladder makes an angle of 19° with the
vertical wall. Draw a picture that models this situation and compute how far the bottom of the
ladder is from the wall, accurate to 3 significant figures.
29. A man 6.2 feet tall stands 12 feet away from a streetlight and casts a 5-foot-long shadow. How
tall is the streetlight?
30. A team of surveyors have been hired to measure the distance across a canyon. Using a tree at
point T on the opposite site of the canyon as a reference point, they established points A, B, and C
and found the following distances:
AB = 12.25 ft BC = 6.5 ft
AC = 15 ft
T
B
C
A
(a) Find the measure of angle ABC. Round your answer to 5 significant figures.
(b) Find the distance TC across the canyon to the nearest foot.
31. Find the value of A in A  2 h(r1  r2 ) when r1  1.03 cm, r2  8.1 cm, and h  13.48 cm. Be
sure to give your answer using significant figures.
32.
a) If the central angle θ = 288°, then the sector formed by θ has two-fifths the area of the
entire circle.....................................................................................................True False
b) In a circle with radius 7 cm an arc of the circle has length 22 cm. Then the central angle θ
is approximately
i) 3.14°
ii) 18.23°
iii) 90°
iv) 180°
c) A wheel with a diameter of 12 inches is spinning at 10 revolutions per minute. In one
minute, a point on the edge of the circle would have traveled a distance of approximately
i) 37.7 inches
ii) 377 inches
iii) 188.5 inches
iv) 754 inches