MAT Practice (solutions) 1. Find an algebraic formula for a linear function that passes through the points ( 3, 7) and (6, 1).

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1 MAT Practice (solutions) 1. Find an algebraic formula for a linear function that passes through the points ( 3, 7) and (6, 1). Answer: y = Let f(x) = 8x 120 (a) What is the y intercept of the graph of f. Answer: (0, 120) (b) Find all x intercepts of the graph of f. Answer: (15, 0) (c) What kind of function is f: (i) linear (ii) quadratic (iii) exponential (iv) logarithmic (v) other Answer: linear (d) What is the slope of the graph of f. Answer: 8 3. Suppose that a valuable gold coin is worth $120 today and appreciates in value by $7.50 each year. Let V be the value of the coin and t be the number of years passed. (a) Find a formula for V as a function of t. Answer: V = t (b) What is the vehicle worth after 5 years? Answer: $157.5 (c) Roughly when will the coin be worth $200? Answer: In 10.7 years (d) What is V over any input interval? What does this number t measure (i.e., what are appropriate units for it)? Answer: V = 7.5. The units for this are dollars per year. t 1

2 4. Solve the system of equations Answer: (6, 2) 5. Let g(x) = 6x x + 80 x + 7y = 20 3x 5y = 8 (a) What is the y intercept of the graph of g. Answer: (0, 80) (b) What is the vertex of the graph of g? ( 25 Answer: 12, 2545 ) 24 (c) Is the graph of g concave up or concave down? Answer: concave down (d) Find all x intercepts of the graph of g. Answer: x = 25 ± , (e) What kind of function is g: (i) linear (ii) quadratic (iii) exponential (iv) logarithmic (v) other Answer: quadratic 6. What are the x intercepts of the function h(x) = 5(x 2)(x 7)? Answer: x = 2, 7 7. Label the curves in the plot below appropriately (A) y = (x + 3) 2 10 (B) y = (x + 5)(x 5), (C) y = (x 1) 2 5, (D) y = 2x Answer: (D) is upper left, (C) is upper right, (B) is lower left, and (A) is lower right. 2

3 The height of an object in feet dropped from an airplane t seconds after the drop is given by the function h(t) = t 2 (a) Compute h(4) and explain what this value means. Answer: h(4) = means that four seconds after the drop, the object is feet up. (b) Sove the equation h(t) = and interpret your solutions. Answer: The solutions are t = ± Only the solution t = has meaning for the model it is the time after the drop when the object will be feet up. (c) Is h concave up or concave down? Answer: concave down (d) When does the object hit the ground? Answer: seconds after the drop. 9. The value of an account in t years is given by f(t) = 1000( t ) dollars. (a) What is the initial value of the account? Answer: $1000 (b) What is the annual percentage growth rate of the account? Answer: 5.95% 3

4 (c) What is the continuous growth rate of the account? (HINT: write the exponential in the form ae kt.) Answer: 5.78 % 10. After a vaccination program is begun, the incidence of a certain viral infection in a city declines at a rate of 18% each year people are infected when the vaccination program begins. (a) Construct a model y = f(t) that gives the number of infected people t years after the vaccination program begins. Answer: f(t) = 10000(.82) t (b) How many people does this model predict will be infected four years after the vaccination program begins? Answer: f(4) = 4521 people. (c) How long will it take for the number of infected people to reach 2000? Answer: 8.11 years 11. An investment is initially valued at $800 and grows at a constant rate of 4% each year. (a) Determine an algebraic expression for f(t) the value of the investment after t years. Answer: f(t) = 800(1.04) t (b) How much is the investment worth in 12 years? Answer: $ Lake Bigfish had no trout in it until it was stocked with 2500 trout. (a) Suppose that each year after this stocking the trout population grew by 110 trout. Determine a function f(t) that gives the number of trout in the lake t years after the initial stocking. What kind of function is f? Answer: f(t) = t (b) Suppose that each year after this stocking the trout population grew by 4%. Determine a function g(t) that gives the number of 4

5 trout in the lake t years after the initial stocking. What kind of function is g? Answer: g(t) = 2500(1.04) t. This is an exponential function. 13. Suppose $2000 is deposited in an account that makes 5% interest compounded quarterly. How much will the account be worth in 8 years? Answer: $ Label the curves in the plot below appropriately (A) y = 50(1.02) t, (B) y = 50(1.04) t, (C) y = 50(.97) t, (D) y = 100(1.02 t ) y t 15. Suppose a radioactive substance decays at a continuous rate of 14% per year and 50 grams of the substance are present in the year (a) Determine a formula for the amount of the substance t years after the year 2000 Answer: 50e.14t (b) How much of the substance will be present in the year 2012? Answer: 9.32 grams 16. The population of a city t years after 2010 is modeled by the function f(t) = 5000e.087t 5

6 (a) Is the city growing or shrinking? Answer: It is shrinking. (b) The graph of f(t) is below. Mark on this graph a point that indicates when the cities population is or indicate that there is no such point If an account earns 12% interest compounded continuously, what is the effective annual rate (i.e., by what percentage does the account grow each year)? Answer: 12.75% ( ) Simplify the expression ln. e 8x ( ) 1 Answer: ln = 8x e 8x 19. Simplify the expression ln ( e 8x). Answer: ln ( e 8x) = 8x 20. Solve the equation ln(4x 1) = 2. Give your answer as a decimal accurate to at least three decimal places. Answer: x = e A certain city has a population of 50,000 in the year 2000 and grows by 3.5% per year. 6

7 (a) Determine a growth model f(t) that gives the population of the city t years after Answer: f(t) = 50000(1.035) t (b) What does the model estimate that the population of the city will be in the year 2020? Answer: (c) When does the model estimate the population of the city will reach 2 million? Answer: In the year In 1991, the body of a man was found in melting snow in the Alps of Northern Italy. An examination of the tissue sample revealed that 46% of the carbon-14 present in his body at the time of his death had decayed. the half-life of carbon-14 is about 5728 years. How long ago did this man die? Answer: 1991 is 6417 years after the man s death. 7