Question

number six
the function $y = 50(1.3)^x$ was written on the board. henry said the growth rate was 130%. explain the error henry made.

Explanation:

Step1: Recall exponential growth formula

The general form of an exponential growth function is \( y = a(1 + r)^x \), where \( a \) is the initial amount, \( r \) is the growth rate (as a decimal), and \( x \) is the time variable.

Step2: Analyze the given function

The given function is \( y = 50(1.3)^x \). Comparing with the general form \( y = a(1 + r)^x \), we can rewrite \( 1.3 \) as \( 1 + r \). So, \( 1 + r = 1.3 \).

Step3: Solve for the growth rate \( r \)

Subtract 1 from both sides: \( r = 1.3 - 1 = 0.3 \). To convert this to a percentage, we multiply by 100, so \( r = 0.3\times100\% = 30\% \).

Step4: Identify Henry's error

Henry thought the growth rate was 130%, but he confused the growth factor (\( 1.3 \)) with the growth rate. The growth factor is \( 1 + r \), so the growth rate is \( 30\% \), not \( 130\% \).

Answer:

Henry confused the growth factor (\( 1.3 \)) with the growth rate. The correct growth rate is \( 30\% \) (since \( 1.3 = 1 + 0.3 \), and \( 0.3\times100\% = 30\% \)), not \( 130\% \).