Question

A cannonball is shot upward out of a cannon. The cannonball's height, h (in meters), after t seconds is given by the equation below. What is the maximum height the cannonball will reach?h = 160t - 16t^2Answer choices include: A. 160 metersB. 16 metersC. 400 metersD. 200 meters

Answer 1

A cannonball's height, h, is given by the equation:

`\text{ h = 160t - 16t}^2`

Let's determine the maximum height.

Let's get the first derivative of the equation.

`\text{ h = 160t - 16t}^2`
`\frac{\text{ dh}}{\text{ dt}}\text{ = 160}(1)\text{ - 16}(\text{2})\text{t = 160 - 32t}`

dh/dt represents the speed of the cannonball.

At maximum height, the ball no longer goes up, thus, the speed (dh/dt) is 0.

Let's now determine the time (t) when the cannonball reaches the maximum height.

`\frac{\text{ dh}}{\text{ dt}}\text{ = 160 - 32t}\frac{}{}`
`\text{ 0 = 160 - 32t}`
`\text{ 32t = 160}`
`\frac{\text{ 32t = 160}}{\text{ 32}}`
`\text{ t = 5 seconds}`

Therefore, in 5 seconds, the cannonball reaches its maximum height after being shot.

Let's now determine the maximum height, at t = 5 seconds.

`\text{ h = 160t - 16t}^2`
`\text{ h = 160}(\text{5})\text{ - 16}(\text{5})^2\text{ = 160}(5)\text{ - 16}(\text{25})`
`\text{ h = 800 - 400 = 400 meters}`

Therefore, the maximum height of the cannonball after being shot is 400 meters.

The answer is Choice C: 400 meters.