A ball is thrown up from the ground. Its height after t seconds is h… Step by Step

Step by step

The question

Question: A ball is thrown up from the ground.

Its height after t seconds is h = -16t^2 + 80t feet.

When is the ball 64 feet high?

1 Find what we know

What do we know?

The height formula: h = −16t² + 80t feet, after t seconds.

The question gives a height: 64 feet.

What do we want?

The times when the height is 64 feet.

2 Name the unknown

The formula already has a letter for the time: t, in seconds.

t = time in seconds

3 Write the equation

In words: the height is the number in the question.

−16t² + 80t = 64

So the equation is −16t² + 80t = 64.

There is t times t in it: a quadratic equation.

Now we solve it.

−16t² + 80t = 64

4 Get 0 on one side

To solve it, we want 0 on the right side.

So we take away 64 on both sides.

We do it on both sides, so they stay equal.

−16t² + 80t − 64 = 64 − 64

Now we have −16t² + 80t − 64 = 0.

−16t² + 80t − 64 = 0

The t² part is negative.

We change every sign on both sides.

0 stays 0.

16t² − 80t + 64 = 0

5 Divide by 16

Easy way first: every number can be divided by 16.

So we divide both sides by 16.

0 divided by 16 is still 0.

16, -80, 64: all divide by 16

Now we have t² − 5t + 4 = 0.

t² − 5t + 4 = 0

6 Factor

We need two numbers.

They multiply to 4, and they add to −5.

? × ? = 4

? + ? = −5

They multiply to a plus number, so the two numbers have the same sign.

They add to a minus number, so both are minus.

× is + → same signs: both −

We list the pairs that make 4, smallest first.

We add each pair.

We need 5.

1 and 4 give 5.

Yes!

With the signs: −1 and −4.

1 × 4: 1 + 4 = 5 ✓

So it is (t − 1)(t − 4) = 0.

(t − 1)(t − 4) = 0

7 One bracket is 0

Two numbers multiply to 0.

That only happens if one of them is 0.

So one bracket must be 0.

A × B = 0 → A = 0 or B = 0

We make each bracket 0.

Two small equations.

t − 1 = 0 or t − 4 = 0

In t − 1 = 0, we add 1 on both sides.

In t − 4 = 0, we add 4 on both sides.

t − 1 + 1 = 0 + 1 or t − 4 + 4 = 0 + 4

So t = 1 or t = 4.

t = 1 or t = 4

8 Answer the question

t = 1 works: going up.

t = 1 ✓

going up

t = 4 works: coming down.

t = 4 ✓

coming down

Careful!

Both times are answers.

It passes 64 feet going up, and again coming down.

Careful: not only t = 1 ✗

It is 64 feet high after 1 and after 4 seconds.

9 Check

Let's check in the story.

Put t = 1 into the height.

We get 64 feet.

t = 1: h = −16(1)² + 80(1) = −16 + 80 = 64

Put t = 4 into the height.

We get 64 feet.

So the answer is right!

t = 4: h = −16(4)² + 80(4) = −256 + 320 = 64

Let's look back.

First: find what we know.

Then: name the unknown.

Then: write the equation.

Then: solve it.

Then: answer the question.

Last: check.

Answer