Question

identify the initial amount a and the rate of growth r (as a percent) of the exponential function $y = 12(1.05)^t$. evaluate the function when $t = 5$. round your answer to the nearest tenth.

$a = \square$

$r = \square$%

when $t = 5$, $y \approx \square$

Explanation:

Step1: Identify initial amount \( a \)

The general form of an exponential growth function is \( y = a(1 + r)^t \), where \( a \) is the initial amount. Comparing with \( y = 12(1.05)^t \), we see \( a = 12 \).

Step2: Identify growth rate \( r \)

From \( y = a(1 + r)^t \) and \( y = 12(1.05)^t \), we have \( 1 + r = 1.05 \). Solving for \( r \): \( r = 1.05 - 1 = 0.05 \), which is \( 5\% \) when converted to a percent.

Step3: Evaluate \( y \) when \( t = 5 \)

Substitute \( t = 5 \) into \( y = 12(1.05)^t \). So \( y = 12(1.05)^5 \). Calculate \( (1.05)^5 \approx 1.27628 \). Then \( y \approx 12 \times 1.27628 \approx 15.31536 \), rounded to the nearest tenth is \( 15.3 \).

Answer:

\( a = \boldsymbol{12} \)
\( r = \boldsymbol{5}\% \)
When \( t = 5 \), \( y \approx \boldsymbol{15.3} \)