Problem 1: Investment Inequality
Alice invested $5000 in a fund that offers an annual return rate of at least 4%. She wants to know how many years it will take for her investment to grow to at least $7000. If \( A(t) = 5000(1 + 0.04)^t \), determine \( t \).
Solution: To find the minimum \( t \): \[ 5000(1 + 0.04)^t \geq 7000 \] \[ (1.04)^t \geq \frac{7000}{5000} = 1.4 \] Taking the logarithm: \[ t \log(1.04) \geq \log(1.4) \] \[ t \geq \frac{\log(1.4)}{\log(1.04)} \approx 10.25 \]
Thus, it will take at least 11 years.
Problem 2: Travel Time Inequality
Emma is planning a road trip. She estimates that she needs to average at least 55 miles per hour to reach her destination in under 6 hours. If her trip is 320 miles long, determine whether her speed plan is sufficient.
Solution: To find the maximum time: \[ \frac{320}{\text{speed}} < 6 \quad \Rightarrow \quad \text{speed} > \frac{320}{6} \approx 53.33 \] Since she needs an average of 55 miles per hour, her speed plan is sufficient.
Problem 3: Competition Points Inequality
A basketball player scores points in games and needs at least 120 points to qualify for the next level. If she already has scored 85 points, at least how many points does she need to score in the next games \( x \) to qualify, assuming she will play \( y \) more games?
Solution: To find the required points: \[ 85 + x \geq 120 \quad \Rightarrow \quad x \geq 120 - 85 = 35 \] Thus, she needs to score at least 35 points in the next games combined, splitting it over \( y \) games results in: \[ \frac{35}{y} \text{ points per game} \]
She must average at least \(\frac{35}{y}\) points in her remaining games.
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