Waiting period. Upstate University currently has a 6,300-car parking capacity for faculty, staff, and students. This year, the university issued 4,272 parking passes. The number of parking passes issued has been growing at a rate of \( 7 \% \) per year. How long will it be before the university will need to add parking spaces? How long will it be before the university will need to add parking spaces? \( \square \) years (Round to twhtdecimal places.)
Real Tutor Solution
Answer
Solution
Reviewed and approved by the UpStudy tutoring team
Beyond the Answer
To determine how long it will take before Upstate University needs to add parking spaces, we can use the formula for exponential growth. The number of parking passes issued is growing at a rate of \( 7\% \) per year. The formula we can use is: \[ P(t) = P_0 \times (1 + r)^t \] Where: - \( P(t) \) is the future number of parking passes, - \( P_0 \) is the current number of parking passes (4,272), - \( r \) is the growth rate (0.07), - \( t \) is the number of years. We want to find \( t \) when \( P(t) = 6,300 \): \[ 6,300 = 4,272 \times (1 + 0.07)^t \] To solve for \( t \), we can rearrange the equation: \[ (1.07)^t = \frac{6,300}{4,272} \] Calculating the right side: \[ \frac{6,300}{4,272} \approx 1.477 \] Now we take the logarithm of both sides: \[ \log((1.07)^t) = \log(1.477) \] Using the power rule of logarithms: \[ t \cdot \log(1.07) = \log(1.477) \] Now we divide both sides by \( \log(1.07) \): \[ t = \frac{\log(1.477)}{\log(1.07)} \] Calculating this gives: \[ t \approx \frac{0.1692}{0.0291} \approx 5.8 \text{ years} \] So, to add parking spaces, it would take approximately \( 5.80 \) years. **Answer:** \( 5.80 \) years