Question

When a baseball is hit by a batter, the height of the ball, h(t) , at time t, t=0 , is determined by the equation h(t)=-16t²+64t+4. If t is in seconds, for which interval of time is the height of the ball greater than or equal to 52 feet?

a. 1 < t<3
b. t ≤ 1
c. 0 < t <1
d. 1≤ t≤ 3

Answer 1

Correct answer is option d: 1 ≤ t ≤ 3.

The interval during which the baseball's height is greater than or equal to 52 feet is determined by solving the inequality -16t² + 64t + 4 ≥ 52 and is found to be 1 ≤ t ≤ 3 seconds.

Step-by-step explanation:

To determine the intervals of time during which the baseball's height is greater than or equal to 52 feet, we need to solve the inequality using the given quadratic equation for the height of the baseball, which is h(t) = -16t² + 64t + 4. We set the inequality h(t) ≥ 52:

• -16t² + 64t + 4 ≥ 52
• -16t² + 64t + 4 - 52 ≥ 0
• -16t² + 64t - 48 ≥ 0

Dividing the entire inequality by -16 (and flipping the inequality sign because we are dividing by a negative number) simplifies it to:

• t² - 4t + 3 ≤ 0

Factoring the quadratic we get:

• (t - 1)(t - 3) ≤ 0

This gives us the roots t = 1 and t = 3.

The sign of the expression changes at these roots, so we test values between the roots to determine the interval where the inequality holds true.

By testing points in the intervals (0, 1), (1, 3), and (3, ∞), we find that the inequality holds true in the interval 1 ≤ t ≤ 3 seconds.

Therefore, the correct answer is option d: 1 ≤ t ≤ 3.