Questions: d the value of r, where r represents and interest rate in decimal form. Give your answer as a percent, rounded to arest hundredth. SHOW ALL WORK! 24) 90,000=600(1+r)^120 r= %

Solution Steps

To solve for \( r \) in the equation \( 90,000 = 600(1 + r)^{120} \), we need to isolate \( r \). This involves the following steps:

1. Divide both sides of the equation by 600.
2. Take the 120th root of both sides to eliminate the exponent.
3. Subtract 1 from both sides to solve for \( r \).
4. Convert \( r \) from decimal form to a percentage and round to the nearest hundredth.

We start with the equation given in the problem: \[ 90,000 = 600(1 + r)^{120} \]

Divide both sides by 600 to isolate the term containing \( r \): \[ \frac{90,000}{600} = (1 + r)^{120} \] This simplifies to: \[ 150 = (1 + r)^{120} \]

To solve for \( 1 + r \), take the 120th root of both sides: \[ 1 + r = 150^{\frac{1}{120}} \]

Subtract 1 from both sides to find \( r \): \[ r = 150^{\frac{1}{120}} - 1 \]

Convert \( r \) from decimal form to a percentage: \[ r \approx 0.04263930755917267 \] Thus, \[ r \times 100 \approx 4.263930755917267 \] Rounding to the nearest hundredth gives: \[ r \approx 4.26\% \]

Final Answer