1. The flare returns to the water when h(t) = 0.
Set -15t² + 150t = 0.
Factor: -15t(t - 10) = 0.
Solutions: t = 0 (launch) or t = 10.
Answer: 10 seconds.
2. The osprey reaches 20 m when h(t) = 20.
Set 5t² - 30t + 45 = 20.
Rearrange: 5t² - 30t + 25 = 0.
Divide by 5: t² - 6t + 5 = 0.
Factor: (t - 1)(t - 5) = 0.
Solutions: t = 1 or t = 5.
Since the osprey is diving, it reaches 20 m on the way down at t = 5 seconds.
Answer: 5 seconds.
3. The ball is caught at 3 feet, so set h(t) = 3.
Set -16t² + 48t + 3 = 3.
Simplify: -16t² + 48t = 0.
Factor: -16t(t - 3) = 0.
Solutions: t = 0 (launch) or t = 3.
The ball was in the air for 3 seconds.
This is reasonable because a pop-up with an initial velocity of 48 ft/s would take about 3 seconds to return to near its launch height.
4. The shop starts making a profit when P(t) > 0.
Set 1125(t - 1)² - 10,125 = 0.
Add 10,125: 1125(t - 1)² = 10,125.
Divide by 1125: (t - 1)² = 9.
Take square root: t - 1 = ±3.
Solutions: t = 4 or t = -2.
Since t ≥ 0, t = 4.
Answer: 4 years.
Parent Tip: Review the logic above to help your child master the concept of quadratic formula word problems worksheet.
