Question

A catapult launches a boulder with an upward velocity of 120 ft/s. The height of the boulder, h,

in feet after t seconds is given by the function h=-16t^2 + 120t + 10. How long does it take to
reach maximum height? What is the boulder's maximum height? Round to the nearest
hundredth, if necessary.
a. Reaches a maximum height of 235.00 feet in 3.75 seconds.
b. Reaches a maximum height of 10.00 feet in 7.50 seconds.
c. Reaches a maximum height of 7.58 feet in 3.75 seconds.
d. Reaches a maximum height of 15.16 feet in 7.5 seconds.

Answer 1

The boulder's maximum height is:

a. Reaches a maximum height of 235.00 feet in 3.75 seconds.

Explanation:

The easiest way to solve the unknown is to replace the time in the given equation, with the values ​​given in the answer, look for the highest value and check if the height value decreases as the time increases or decreases.

First the equation provided is taken:

• h = -16t ^ 2 + 120t + 10

The time values ​​in the options are 3.75 seconds and 7.5 seconds, therefore we proceed to replace this formula with those values:

• h (3.75) = - 16 (3.75) ^ 2 + 120 (3.75) + 10 = 235
• h (7.5) = - 16 (7.5) ^ 2 + 120 (7.5) + 10 = 10

Since a higher height value is obtained with 3.75 seconds (235 feet), this time will be taken as appropriate, however, it is best to check that by using more time and less time, the height decreases from that value, so we will take two very close times: 3.74 seconds and 3.76 seconds:

• h (3.74) = - 16 (3.74) ^ 2 + 120 (3.74) + 10 = 234.9984
• h (3.76) = - 16 (3.76) ^ 2 + 120 (3.76) + 10 = 220.5884

As you can see, if slightly higher or lower time values ​​are taken, the height of the rock still decreases from 235 feet, so we can be sure that the time in which it reaches its maximum height is 3.75 seconds.