Question

The height in feet of a diver above the surface of a pool is given by the equation h= -16t^2 +20t +12, where t is the time in seconds after the diver jumps.
1. How many seconds after the jumping is the diver’s height 8 meters?
2. How long does it take the diver to reach the water?

Answer 1

#1) 1.4 seconds
#2) 1.7 seconds

For #1, we set the function equal to 8:
8 = -16t² + 20t + 12

To solve a quadratic equation we want it equal to 0, so subtract 8 from both sides:
8 - 8 = -16t² + 20t + 12 - 8
0 = -16t² + 20t + 4

Use the quadratic formula to solve this:
$t=\frac{-b\pm \sqrt{b^2-4ac}}{2a} \\ \\=\frac{-20\pm \sqrt{20^2-4(-16)(4)}}{2(-16)}=\frac{-20\pm \sqrt{400--256}}{-32} \\ \\=\frac{-20\pm \sqrt{400+256}}{-32}=\frac{-20\pm \sqrt{656}}{-32} \\ \\=\frac{-20\pm 25.61}{-32}=\frac{-20+25.61}{-32}\text{ or }\frac{-20-25.61}{-32} \\ \\=\frac{5.61}{-32}\text{ or }\frac{-45.61}{-32}=-0.175\text{ or }1.4$

Since a negative time makes no sense, the answer is 1.4

For #2,
$t=\frac{-20\pm \sqrt{20^2-4(-16)(12)}}{2(-16)}=\frac{-20\pm \sqrt{400--768}}{-32} \\ \\=\frac{-20\pm \sqrt{1168}}{-32}=\frac{-20\pm 34.18}{-32}=\frac{-20-34.18}{-32}\text{ or }\frac{-20+34.18}{-32}=1.7\text{ or }-0.44$

Since negative time makes no sense, the answer is 1.7.

Answer 2

Using the height equation given, we will find that:

a) The diver height is never equal to 8m

b) After 1.69 seconds the diver reaches the water.

So we know that the height, in ft, of a diver above the surface of a pool is given by:

h(t) = -16*t^2 + 20*t + 12

a) First we want to find how many seconds after jumping the height of the diver is equal to 8m

But the equation is in ft, so we need to transform this into ft.

We know that:

1 m = 3.28 ft

Then:

8m = 8*(3.28 ft) = 26.24 ft

Then we just need to solve:

h(t) = 26.24 =  -16t^2 +20t + 12

       0 = -16*t^2 + 20*t + 12 - 26.24

       0 = -16*t^2 + 20*t - 14.4

This is a quadratic equation, the solutions are given by:

$t = \frac{-20 \pm \sqrt{20^2 - 4*(-16)*(-14.4)} }{2*-16} \\\\t = \frac{-20 \pm \sqrt{-521.6} }{-32}$

Notice that we have a negative number inside the square root, this means that we do not have real solutions, thus, the height of the diver is never equal to 8m.

b) Here we must solve:

h(t) = 0 =  -16t^2 +20t + 12

We will get:

$t = \frac{-20 \pm \sqrt{20^2 - 4*(-16)*(12)} }{2*-16} \\\\t = \frac{-20 \pm 34.18 }{-32}$

We only care for the positive time solution, so we will get:

t = (-20 - 34.18)/-32 = 1.69s

So, after 1.69 seconds, the diver will reach the water.

If you want to learn more, you can read:

https://brainly.com/question/16844376