Which Function Represents the Value of a Computer After t years

**ANSWER KEY** 1. Determine whether the function represents exponential growth or exponential decay. Give the percent rate of change. a) y = 50(1.16)^t 16% growth b) y = 10(0.052)^t 94.8% decay c) y = 2(0.85)^t 15% decay 2. Solve: a) 6^x = 1/36 x = -2 b) 4^(2x) = 8^(x-2) x = 2 c) 9^x = 3^(x+2) x = 2 d) (z^2)^(2x) = (t^5)^(x-2) x = -4 e) 5^(2-3x) = 5^(-4) x = 2 3. A population of 500 turtles is growing each month at a rate of 7.5% a) Write an equation that expresses the number of turtles at time x.   y = 500(1.075)^x b) About how many turtles will there be in 6 months? y(6) ≈ 772 turtles 4. Your new car cost $16000 but it depreciates in value by about 2% each month a) Write an equation that would indicate the value of the car at x months.   y = 16000(0.98)^x b) How much will your car be worth in 2 years? y(24) ≈ $9852.48 5. A town currently has a population of 15,000 but decreases in size by 2% each year a) Write an equation that expresses the number of people living in the town at time x. y = 15000(0.98)^x b) About how many people will be living there in 20 years? y(20) ≈ 6359 people 6. Your new computer cost $1500 but after each year it is only worth 2/3 of the value from the previous year. a) Write an equation that would indicate the value of the computer at x years.   y = 1500(2/3)^x b) How much will your computer be worth in 4 years? y(4) ≈ $296.30 7. You invest $12,000 in an account with 3% annual interest, compounded semi-annually. Assume you don’t touch the money or add money other than the earned interest.   a) Write an equation that models the amount of money,   y, in the account after t years. A = P(1 + r/n)^(nt) A = 12000(1 + 0.03/2)^(2t) b) How much money will you have in the account after 10 years? A(10) ≈ $16,162.26 8. You invest $10,000 in an account with 5.2% interest, compounded annually (n=1). Assume you don’t touch the money or add money other than the earned interest.   a) Write an equation that gives the amount of money, y, in the account after   t years. A = 10000(1.052)^t b) How much money will you have in the account after 5 years? A(5) ≈ $12,884.83 If you would have invested your $10,000 in a different account instead that offered 6% annual interest, compounded monthly, how much money would you have in the account after 5 years?   A = 10000(1 + 0.06/12)^(12*5) A ≈ $13,488.52 9. Simplify the expression. Write your answer using only positive exponents. a. (8x¹⁵y⁴)(16x²y⁸) / 8x⁷y¹² = 16x¹⁷y¹² / 8x⁷y¹² = 2x¹⁰ b. (2p⁴q⁻⁵w)⁻² = 1 / (2p⁴q⁻⁵w)² = 1 / (4p⁸q⁻¹⁰w²) = q¹⁰ / 4p⁸w² c. 2m(mn)⁻³ = 2m / m³n³ = 2 / m²n³ d. (2x³y⁻²)⁻² = 1 / (4x⁶y⁻⁴) = y⁴ / 4x⁶ e. -3x²y²⁴(x⁻⁸y⁸)³ / 3x⁻⁴y⁸ = -3x²y²⁴x⁻²⁴y²⁴ / 3x⁻⁴y⁸ = -x⁻²⁰y⁴⁰ / x⁻⁴y⁸ = -x⁻¹⁶y³² = y³² / x¹⁶ f. (3a²b⁻²)⁻² / (3a⁻³b⁵c⁻²) = 1 / (9a⁴b⁻⁴) * (3a⁻³b⁵c⁻²) = 1 / (3a⁻¹b) = a / 3b 10. Simplify. Leave in simplest radical form. a. √72 = √(36 * 2) = 6√2 b. √540 = √(36 * 15) = 6√15 c. √(3√8)² = 3√8 = 3 * 2√2 = 6√2 d. √(75/15) = √5 11. Evaluate the expression. a. ∛(1/27) = 1/3 b. 16^(3/4) = (√⁴16)³ = 2³ = 8 12. Write an equation in the form y=ab^x that models the given table. a. | x | y | |---|---| | 0 | 2 | | 1 | 10 | | 2 | 50 | | 3 | 250 | | 4 | 1250 | y = 2(5)^x b. | x | y | |---|---| | 0 | 80 | | 1 | 20 | | 2 | 5 | | 3 | 1.25 | | 4 | 0.3125 | y = 80(0.25)^x