Aball is thrown into the air with an upward velocity of 24 ft/s. its height h in feet after t seconds is given by the function h = –16t2 + 24t + 7. a. in how many seconds does the ball reach its maximum height? round to the nearest hundredth if necessary. b. what is the ball’s maximum height?
a. 1.5 s; 7 ft

b. 0.75 s; 34 ft

c. 0.75 s; 13 ft

d. 0.75 s; 16 ft

The answer is D. 0.75 s; 16 ft

Step-by-step explanation:

The function has the following form, which belongs to the quadratic function: and where:

The graph of a quadratic function is a curve called a parabola, which has the shape similar to the letter U.  The value "a" indicates whether the shape of the curve goes up or down. If its value is positive, it takes an upward form, and if its value is negative it goes down. And the value "c" indicates where it intersects the y axis, or h in this particular case.

I add the image of the graph obtained for this case.

As you can see, the value of "a" is negative and the curve goes down. And the curve intersects 7 to the y axis (height). You can also see that there is a maximum height (y axis), which corresponds to a time, and from which the rest of the heights are smaller. This is the maximum height at which the ball arrives, which is what you want to know, along with the corresponding time.

For its calculation, the simplest is first to determine in how many seconds does the ball reach its maximum height. For that the following calculation is applied, which corresponds to the vertex of the curve. Because in summary the maximum point (x, y) (or (t, h) for this case) that you want to know is the vertex of the function, that is, where there is a change in curvature:

So: t=0.24 seconds

Now, to calculate the maximum height simply replace the value of t in the given function:

h= 16 ft

Finally, you can say that in 0.75 seconds the ball reaches a maximum height of 16 ft.