Question

identify the initial amount ( a ) and the rate of growth ( r ) (as a percent) of the exponential function ( f(t) = 1500(1.074)^t ). evaluate the function when ( t = 5 ). round your answer to the nearest tenth.

( a = square )

( r = square% )

when ( t = 5 ), ( f(5) approx square )

Explanation:

Step1: Identify initial amount \( a \)

The general form of an exponential growth function is \( f(t) = a(1 + r)^t \), where \( a \) is the initial amount. Comparing with \( f(t) = 1500(1.074)^t \), we see \( a = 1500 \).

Step2: Identify growth rate \( r \)

From \( f(t) = a(1 + r)^t = 1500(1.074)^t \), we have \( 1 + r = 1.074 \). Solving for \( r \), we get \( r = 1.074 - 1 = 0.074 \), which is \( 7.4\% \) when converted to a percentage.

Step3: Evaluate \( f(5) \)

Substitute \( t = 5 \) into the function: \( f(5) = 1500(1.074)^5 \). First, calculate \( (1.074)^5 \approx 1.074 \times 1.074 \times 1.074 \times 1.074 \times 1.074 \approx 1.4287 \). Then, multiply by 1500: \( 1500 \times 1.4287 \approx 2143.1 \) (rounded to the nearest tenth).

Answer:

\( a = 1500 \)
\( r = 7.4\% \)
When \( t = 5 \), \( f(5) \approx 2143.1 \)