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 = -16t² + 96t feet.

When does it land?

There are four choices.

First we solve it, the normal way.

Then we pick.

1 Find what we know

What do we know?

The height formula: h = −16t² + 96t feet, after t seconds.

It lands when it is back on the ground: the height is 0.

What do we want?

The time when it lands.

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 0.

−16t² + 96t = 0

So the equation is −16t² + 96t = 0.

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

Now we solve it.

−16t² + 96t = 0

The t² part is negative.

That is harder to work with.

So we multiply both sides by −1.

Every sign changes, and −1 times 0 is still 0.

16t² − 96t = 0

4 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, -96, 0: all divide by 16

Now we have t² − 6t = 0.

t² − 6t = 0

5 Factor

There is no plain number.

Both parts have t in them.

t² is t times t, and 6t is t times −6.

So we take t out: t times (t − 6).

t² = t × t, −6t = t × (−6)

So it is t(t − 6) = 0.

t(t − 6) = 0

6 One bracket is 0

t times the bracket is 0.

That only happens if one of them is 0.

A × B = 0 → A = 0 or B = 0

We make each part 0.

Two small equations.

t = 0 or t − 6 = 0

t = 0 is done already.

In t − 6 = 0, we add 6 on both sides.

t = 0 or t − 6 + 6 = 0 + 6

So t = 0 or t = 6.

t = 0 or t = 6

7 Answer the question

t = 0: that is the moment it is thrown.

So it is not the answer.

t = 0 ✗

that is the moment it is thrown

Careful!

At t = 0 the height is 0 too.

But that is when it is thrown, not when it lands.

Careful: t = 0 is the throw, not the landing

t = 6 works: it lands.

t = 6 ✓

it lands

It lands after 6 seconds.

8 Check

Let's check in the story.

Put t = 6 into the height.

We get 0, on the ground.

So the answer is right!

t = 6: h = −16(6)² + 96(6) = −576 + 576 = 0

9 Pick the choice

Our answer is 6 s.

That is choice B.

B) 6 seconds

10 Why not the others

A is 3 seconds.

Wrong: that is the top: halfway.

A) 3 seconds ✗ that is the top: halfway

C is 16 seconds.

Wrong: 16 is the number in front of t², not a time.

C) 16 seconds ✗ 16 is the number in front of t², not a time

D is 96 seconds.

Wrong: 96 is the number in front of t, not a time.

D) 96 seconds ✗ 96 is the number in front of t, not a time

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