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