softball 9 25

Question

Susan throws a softball upward into the air at a speed of 32 feet per second from a
152-foot
platform. The distance upward that the ball travels is given by the function
d(t)=−16t2+32t+152.
What is the maximum height of the? softball? How many seconds does it take to reach the ground after first being thrown? upward? ? (Round your answer to the nearest? tenth.)


The maximum height of the softball is
_____ feet.

Answer 1

Shawn,

 

The function d(t) is a quadratic function. It's graph is a parabola which opens down and the vertex is the highest point on the graph. The vertex will have coordinates (t,d), where d is the maximum height and t is the time it takes the softball to reach that height.

 

To find the vertex, you have options. You could complete the square or use the vertex formula. The vertex formula works like this: t = −b/(2a) and d = d(−b/(2a)).

 

In this problem a = −16, b = 32, and c = 152. Using these values, t = −32/(2·(-16)) = 1 second, and d = d(1) = −16 + 32 + 152 = 168 ft.

 

The maximum height of the softball is 168 feet.

 

To find how long it takes for the ball to reach the ground, you get d = 0 in the function and solve for t.

 

0 = −16t² + 32t + 152

 

Divide both sides by −8:

 

0 = 2t² − 4t − 19.

 

This cannot be factored. (You can try, but you will not succeed.) You can solve using the quadratic formula.

 

t = (−b ± √(b² − 4ac))/(2a) = (4 ± √((−4)² −4(2)(−19)))/4 = (4 ± √152)/4 = (4 ± 2√38)/4 = (2 ± √38)/2.

 

Notice there are two solutions. If you work them out with a calculator, you will see that the solution with the minus sign is negative and therefore useless. The solution with the positive sign is valid, and is

 

t = (2 ± √38)/2 ≈ 4.1 sec. (This was rounded to one decimal place as requested.)

 

The ball will take approximately 4.1 seconds to reach the ground after being thrown upward.