Question

While standing on a 192-foot cliff, Anna throws a rock into the air and lets it fall to the bottom of the cliff. The height of the rock after x seconds can be modeled by h(x) = -16x^2 + 32x + 192. Determine the time interval over which the height of the rock will be at least 192 feet.

a) x < 3 seconds
b) 2 ≤ x ≤ 6 seconds
c) x > 6 seconds
d) 3 ≤ x ≤ 6 seconds

Answer 1

The time interval over which the height of the rock will be at least 192 feet is x ≤ 2 seconds.

Step-by-step explanation:

To determine the time interval over which the height of the rock will be at least 192 feet, we need to find the values of x that satisfy the inequality h(x) ≥ 192. Substituting the equation h(x) = -16x^2 + 32x + 192, we get:

-16x^2 + 32x + 192 ≥ 192

Simplifying the inequality:

-16x^2 + 32x ≥ 0

Factoring out -16x:

-16x(x - 2) ≥ 0

Setting each factor equal to 0 and solving for x:

x = 0 or x - 2 = 0

x = 0 or x = 2

Therefore, the time interval over which the height of the rock will be at least 192 feet is x ≤ 2 seconds.