1. The flare returns to the water when its height, h(t), is 0.
Set the equation equal to zero: -15t² + 150t = 0.
Factor out the common term: -15t(t - 10) = 0.
Solve for t: t = 0 or t = 10.
The flare is at the water at launch (t=0) and returns at t=10 seconds.
Answer: 10 seconds.
2. The osprey reaches a height of 20 m when h(t) = 20.
Set the equation equal to 20: 5t² - 30t + 45 = 20.
Rearrange into standard form: 5t² - 30t + 25 = 0.
Divide the entire equation by 5: t² - 6t + 5 = 0.
Factor the quadratic: (t - 1)(t - 5) = 0.
Solve for t: t = 1 or t = 5.
Since the osprey is diving towards the water, it will reach 20 m on its way down, which corresponds to the later time.
Answer: 5 seconds.
3. The ball is caught when its height is 3 feet, so set h(t) = 3.
The equation becomes: -16t² + 48t + 3 = 3.
Subtract 3 from both sides: -16t² + 48t = 0.
Factor out the common term: -16t(t - 3) = 0.
Solve for t: t = 0 or t = 3.
The ball is at 3 feet at launch (t=0) and again when caught (t=3).
The ball was in the air for 3 seconds.
This answer is reasonable because the initial velocity is 48 ft/s, and under gravity, a ball thrown straight up with this speed would be airborne for approximately 3 seconds before returning to the same height.
4. The shop starts making a profit when P(t) > 0.
Set the profit function greater than zero: 1125(t - 1)² - 10,125 > 0.
Add 10,125 to both sides: 1125(t - 1)² > 10,125.
Divide both sides by 1125: (t - 1)² > 9.
Take the square root of both sides: |t - 1| > 3.
This gives two inequalities: t - 1 > 3 or t - 1 < -3.
Solve for t: t > 4 or t < -2.
Since t represents years of operation, it must be non-negative. Therefore, t > 4.
The shop starts making a profit after 4 years.
Answer: 4 years.
Parent Tip: Review the logic above to help your child master the concept of quadratic formula word problems worksheets.
