2. The cost of tuition at a college is \( \$ 12,000 \) and is increasing at a rate of \( 6 \% \) each year. Use an exponential function to find the tuition cost after 4 years.

To find the tuition cost after 4 years, we can use an exponential function. The general form of an exponential function is: \[ y = a \cdot b^x \] where: - \( a \) is the initial value (in this case, the initial tuition cost of $12,000), - \( b \) is the growth factor (in this case, the rate of increase of 6%), - \( x \) is the number of years. Given that the tuition cost is increasing at a rate of 6% each year, we can calculate the growth factor \( b \) as follows: \[ b = 1 + \text{rate of increase} = 1 + 0.06 = 1.06 \] Now, we can substitute the values into the exponential function to find the tuition cost after 4 years: \[ y = 12000 \cdot 1.06^4 \] Let's calculate this using the exponential function. Calculate the value by following steps: - step0: Calculate: \(12000\times 1.06^{4}\) - step1: Convert the expressions: \(12000\left(\frac{53}{50}\right)^{4}\) - step2: Simplify: \(12000\times \frac{53^{4}}{50^{4}}\) - step3: Rewrite the expression: \(50\times 240\times \frac{53^{4}}{50^{4}}\) - step4: Reduce the numbers: \(240\times \frac{53^{4}}{50^{3}}\) - step5: Rewrite the expression: \(10\times 24\times \frac{53^{4}}{50^{3}}\) - step6: Rewrite the expression: \(10\times 24\times \frac{53^{4}}{10^{3}\times 5^{3}}\) - step7: Reduce the numbers: \(24\times \frac{53^{4}}{10^{2}\times 5^{3}}\) - step8: Rewrite the expression: \(8\times 3\times \frac{53^{4}}{10^{2}\times 5^{3}}\) - step9: Rewrite the expression: \(8\times 3\times \frac{53^{4}}{2^{2}\times 5^{2}\times 5^{3}}\) - step10: Rewrite the expression: \(2^{3}\times 3\times \frac{53^{4}}{2^{2}\times 5^{2}\times 5^{3}}\) - step11: Reduce the numbers: \(2\times 3\times \frac{53^{4}}{5^{2}\times 5^{3}}\) - step12: Multiply: \(\frac{6\times 53^{4}}{5^{5}}\) - step13: Simplify: \(\frac{6\times 53^{4}}{3125}\) The tuition cost after 4 years is approximately $15,149.72.