Question

identify the initial amount a and the rate of growth r (as a percent) of the exponential function $y = 10(1 + 0.4)^t$. 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 \( y = 10(1 + 0.4)^t \), we can see that \( a = 10 \).

Step2: Identify growth rate \( r \) (as a percent)

In the function \( y = a(1 + r)^t \), \( r \) is the growth rate in decimal form. Here, \( 1 + r = 1 + 0.4 \), so \( r = 0.4 \). To convert to a percent, multiply by 100: \( r = 0.4 \times 100 = 40\% \).

Step3: Evaluate the function at \( t = 5 \)

Substitute \( t = 5 \) into the function \( y = 10(1 + 0.4)^t \). First, calculate \( (1 + 0.4)^5 \). \( 1.4^5 = 1.4 \times 1.4 \times 1.4 \times 1.4 \times 1.4 \). \( 1.4 \times 1.4 = 1.96 \), \( 1.96 \times 1.4 = 2.744 \), \( 2.744 \times 1.4 = 3.8416 \), \( 3.8416 \times 1.4 = 5.37824 \). Then multiply by 10: \( y = 10 \times 5.37824 = 53.7824 \). Rounding to the nearest tenth gives \( y \approx 53.8 \).

Answer:

\( a = 10 \)
\( r = 40\% \)
When \( t = 5 \), \( y \approx 53.8 \)