Problem Possible Score Number Points Total 73

Transcription

1 Mth 138 Exam III Fall 017 Name: Instructor: Section: Problem Possible Score Number Points Total 73 Directions Please Read Carefully! You have 50 minutes to take this exam. Make sure to use correct mathematical notation. To receive full credit on a problem, you will need to justify your answers carefully unless indicated otherwise unsubstantiated answers will receive little or no credit. Please be sure to write neatly illegible answers will receive little or no credit. Round your answer if indicated; otherwise, give an exact, simplified answer. Approved calculators are permitted. Good Luck!!!

2 1. or false (circle if always true, otherwise circle ). No justification needed. No partial credit. (a) ln e x = x (b) log(x) log(y) = log(x + y) (c) Given the quadratic ax + bx + c the x coordinate of the vertex is at b a. (d) The graph of a quadratic function is an example of a one-to-one (invertible) function. (e) The functions f(x) = x + 3 and g(x) = x + 3 are inverse functions. Solution: (a), (b), (c), (d), (e). Page

3 . Consider the quadratic function, f(x), that has a vertex of (3, 4) and passes through the point (5, 0). (a) Determine the equation for f(x). [4] Solution: Since the vertex is at (3, 4) then, f(x) = a(x 3) + 4. To determine a we set x = 5 and y = 0, that is, Therefore a = 1 and the function is 0 = 4a + 4 f(x) = (x 3) + 4. (b) Sketch the graph of f(x) and label any x or y-intercepts. [4] y x Page 3

4 Solution: y (1, 0) (5, 0) x (0, 5) -6 Page 4

5 3. A breeder of horses wants to fence in two rectangular grazing areas along a river with 600 meters of fence as shown in the picture below. (a) Determine an expression for the area A of the grazing land in terms of the width, [4] w, of the rectangle. Solution: The perimeter is P = l + 3w = 600 Hence, L = 600 3w. Therefore the area is A(w) = w(600 3w) = 3w(00 w). (b) Determine the largest area that the breeder can enclose? [4] Solution: Based on the previous part we see that a maximum occurs when w = 100. This is an area of A(100) = = 30000m Page 5

6 4. Solve x 3 = 5x for x. Solution: x = 1/ or Solve x = 5 x 1 for x. Solution: x = 5 ± 17 4 Page 6

7 6. A lawn mower generates a noise intensity of 10 watts per square meter. Find the decibel level of the sound of a lawn mower using the formula ( ) I D = 10 log 10, 10 1 where D is the decibel level and I represents the intensity of sound waves. Solution: 100 decibels 7. In 1963, radioactive strontium-90 was released during atmospheric testing of nuclear weapons and got into the bones of people alive at the time. If someone living in 1963 absorbed 14 micrograms of strontium-90, and the half-life is 31 years, how much will remain in their body in 00? (round to the nearest tenth) Solution: A = 14 ( ) 57/31 1 = 39.7 Page 7

8 8. Find the formula for the inverse function of f(x) = 3(x + 1) 3 4. Solution: f 1 (x) = ( ) 1/3 x Solve for the unknown value in each of the following: Round your answer to four decimal places. (a) e 4x = 8 Solution: x = (b) 5(ln x) + 6 = 14 Solution: x = (c) log 4 (x + 8) log 4 (x + ) = Solution: x = 8/5 or x = 1.6 Page 8

9 10. Zelda invested $1000 in an account that pays 4.5% interest compounded continuously. (a) Write a formula for A(t) that gives the amount of money in Zelda s account after t [4] years. Solution: A(t) = 1000e 0.045t (b) How much will be in Zelda s account in 10 years? [4] Solution: $1, (c) How long, to the nearest tenth of a year, will it take the money in Zelda s account [4] to double? Solution: 15.4 years Page 9