10. The annual sales for a restaurant are \( \$ 579,000 \) and are increasing at a rate of \( 3 \% \) per year. a) Write the exponential function b) What will be the approximate sales in 9 years? Round your answer to the nearest dollar.

Sure, let's tackle each part of the problem step by step. ### **a) Writing the Exponential Function** When dealing with exponential growth, the general form of the function is: \[ P(t) = P_0 \times (1 + r)^t \] where: - \( P(t) \) is the amount after \( t \) years, - \( P_0 \) is the initial amount, - \( r \) is the growth rate (expressed as a decimal), - \( t \) is the time in years. Given: - **Initial annual sales (\( P_0 \))**: \$579,000 - **Growth rate (\( r \))**: 3% or 0.03 Plugging these values into the formula: \[ S(t) = 579{,}000 \times (1 + 0.03)^t \] \[ S(t) = 579{,}000 \times (1.03)^t \] **So, the exponential function is:** \[ \boxed{S(t) = 579{,}000 \times 1.03^t} \] ### **b) Calculating the Sales in 9 Years** We’ll use the exponential function from part (a) to find the sales after 9 years. \[ S(9) = 579{,}000 \times (1.03)^9 \] First, calculate \( (1.03)^9 \): \[ 1.03^9 \approx 1.304773 \] Now, multiply this factor by the initial sales: \[ S(9) \approx 579{,}000 \times 1.304773 \approx 755{,}182.687 \] Rounding to the nearest dollar: \[ S(9) \approx \$755{,}183 \] **So, the approximate sales in 9 years will be:** \[ \boxed{\$755{,}183} \]