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Difficulty: MediumExponential Functions and Equations

The value of a certain investment is modeled by the function f(t)=500(1.21)t2f(t) = 500(1.21)^{\frac{t}{2}}, where f(t)f(t) represents the value of the investment, in dollars, tt years after the initial investment was made. Which of the following is the annual growth rate, as a percentage, of the investment?

  1. A
    21%21\%
  2. 10%10\%Answer
  3. C
    10.5%10.5\%
  4. D
    46.41%46.41\%

Answer

The annual growth rate of the investment is 10%10\%.
To find the annual growth rate, we need to rewrite the function f(t)=500(1.21)t2f(t) = 500(1.21)^{\frac{t}{2}} in the form f(t)=P(1+r)tf(t) = P(1 + r)^t, where PP is the initial value, 1+r1 + r is the annual growth factor, and rr is the annual growth rate. Applying the power of a power rule for exponents, we can rewrite the expression as f(t)=500(1.2112)tf(t) = 500\left(1.21^{\frac{1}{2}}\right)^t. Since raising a number to the power of 12\frac{1}{2} is equivalent to taking the square root, this simplifies to f(t)=500(1.21)t=500(1.1)tf(t) = 500(\sqrt{1.21})^t = 500(1.1)^t. The annual growth factor is 1.11.1, which can be written as 1+0.11 + 0.1. Therefore, the annual growth rate rr is 0.10.1, which corresponds to 10%10\%.

Step-by-Step Solution

1
Identify the given exponential function and rewrite the expression to isolate the annual growth factor.
The function is f(t)=500(1.21)t2f(t) = 500(1.21)^{\frac{t}{2}}. Using exponent rules, we can rewrite the term (1.21)t2(1.21)^{\frac{t}{2}} as (1.2112)t\left(1.21^{\frac{1}{2}}\right)^t.
Isolating the variable tt in the exponent allows us to identify the annual growth factor.
2
Calculate the base of the exponential term.
1.2112=1.21=1.11.21^{\frac{1}{2}} = \sqrt{1.21} = 1.1. Thus, the rewritten function is f(t)=500(1.1)tf(t) = 500(1.1)^t.
Simplifying the base gives us the actual growth factor per unit of time (per year).
3
Determine the percentage growth rate from the growth factor.
The growth factor is 1.11.1, which is equal to 1+0.11 + 0.1. The growth rate rr is 0.10.1. Converting 0.10.1 to a percentage gives 0.1×100%=10%0.1 \times 100\% = 10\%.
The growth rate is the decimal increase per year, which we convert to a percentage.

Key Concept

Manipulating exponential expressions to identify rates of growth or decay.
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Topics

Exponential Functions and Equations

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