Pre calculus word problem

Question

A ball is thrown directly upward from a height of 7ft with an initial velocity of 24ft/sec the function s(t)= -16t^2 +24t+7 gives the height of the ball in feet t seconds after it has been thrown. Determine the time at which the ball reaches its maximum height and find the maximum height. The ball reaches its maximum height of _ ft _ sec(s) after the ball is thrown? Type it in integer or decimal form).

Answer 1

Hi Aireeyon!

The maximum height will occur in the middle of the graph. Notice that the graph's equation is:

s(t) = -16t² + 24t + 7

This is a parabola, and the middle will be located at the parabola's vertex (the tip of the parabola).

To find the vertex we use the following formulas:

The x-value (the time) of the vertex is x = -b/2a

The y-value (the height) of the vertex can be found by plugging the x-value into the given equation.

Let's find the x-value! We were given -16t² + 24t + 7. The coefficients (the numbers) are a = -16, b = 24, and c = 7. So,

x = -(24)/2(-16) = -24/-32 = 3/4.

The x-value is 3/4 seconds.

To find the y-value, we plug in the x-value we just found:

y = -16(3/4)² + 24(3/4) + 7 = -9 + 18 + 7 = 16.

The y-value is 16 ft.

So, the vertex is located at (3/4, 16). This means that the ball reaches a maximum height of 16 ft, 3/4 (or 0.75) seconds after it is thrown.

Hope that helps!