A machine launches a softball from ground level. The softball reaches a maximum height of 51.84 meters above the ground

Question

A machine launches a softball from ground level. The softball reaches a maximum height of $51.84$ meters above the ground at $1.8$ seconds and hits the ground at $3.6$ seconds. Which equation represents the height above ground $h$ , in meters, of the softball $t$ seconds after it is launched?

A.

$h = - t^{2} + 3.6$

B.

$h = - t^{2} + 51.84$

C.

$h = - 16 {(t - 1.8)}^{2} - 3.6$

D.

$h = - 16 {(t - 1.8)}^{2} + 51.84$

Accepted Answer

Choice D is correct. An equation representing the height above ground h , in meters, of a softball t seconds after it is launched by a machine from ground level can be written in the form h = - a t - b 2 + c , where a , b , and c are positive constants. In this equation, b represents the time, in seconds, at which the softball reaches its maximum height of c meters above the ground. It's given that this softball reaches a maximum height of 51.84 meters above the ground at 1.8 seconds; therefore, b = 1.8 and c = 51.84 . Substituting 1.8 for b and 51.84 for c in the equation h = - a t - b 2 + c yields h = - a t - 1.8 2 + 51.84 . It's also given that this softball hits the ground at 3.6 seconds; therefore, h = 0 when t = 3.6 . Substituting 0 for h and 3.6 for t in the equation h = - a t - 1.8 2 + 51.84 yields 0 = - a 3.6 - 1.8 2 + 51.84 , which is equivalent to 0 = - a 1.8 2 + 51.84 , or 0 = - 3.24 a + 51.84 . Adding 3.24 a to both sides of this equation yields 3.24 a = 51.84 . Dividing both sides of this equation by 3.24 yields a = 16 . Substituting 16 for a in the equation h = - a t - 1.8 2 + 51.84 yields h = - 16 t - 1.8 2 + 51.84 . Therefore, h = - 16 t - 1.8 2 + 51.84 represents the height above ground h , in meters, of this softball t seconds after it is launched. Choice A is incorrect. This equation represents a situation where the maximum height is 3.6 meters above the ground at 0 seconds, not 51.84 meters above the ground at 1.8 seconds. Choice B is incorrect. This equation represents a situation where the maximum height is 51.84 meters above the ground at 0 seconds, not 1.8 seconds. Choice C is incorrect and may result from conceptual or calculation errors.

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