Question

The relationship between the time since a ball was thrown and its height can be modeled by the equation =−+h equals 32 t minus 16 , t squared , plus 4, where h is the height of the ball after t seconds. Complete the square to find how long it will take the ball to reach a height of 20 ft.

Answer 1

It will take the ball 1 second to reach a height of 20 ft.

Explanation:

From the question,

The relationship between the time since a ball was thrown and its height can be modeled by the equation h equals 32 t minus 16 , t squared , plus 4,

That is,

h = 32t - 16t² + 4

This is the equation

To find how long it will take the ball to reach a height of 20 ft,

Since h is the height, put h to be 20 in the equation

∴20 = 32t - 16t² + 4

Now, to complete the square,

First subtract 4 from both sides

20-4 = 32t - 16t² +4-4

16 = 32t - 16t²

Factorizing the above, we get

16 = 16t(2-t)

Now, divide both sides by 16

16/16 = 16t(2-t)/16

1 = t(2-t)

1= 2t -t²

∴ t²-2t = -1

Then,

Multiply the coefficient of t by half, square and add to both sides,

That is, -2 × 1/2 = -1; (-1)² = 1;

Add 1 to both sides,

t²-2t +1 = -1+1

t²-2t +1 = 0

Then,

t²-2t +1 can be expressed as a perfect square; t²-2t +1 = (t-1)(t-1) = (t-1)²

Then,

(t-1)²=0

∴ t - 1 = 0

t = 1 sec

Hence, it will take the ball 1 second to reach a height of 20 ft.