Step by step
The question
Question: h = −16t² + 64t + 80 is the height, in feet, of a ball t seconds after it is thrown from a cliff.
When does the ball hit the ground?
We write the equation big, so we can label it.
1 What the letters mean
h is the height of the ball, in feet.
t is the time, in seconds, after it is thrown.
2 Make a small table
To see the flight, we try t = 0, 1, 2 and so on, until the ball is back on the ground, at height 0.
That is 5 seconds.
t = 0: every part with t is 0.
Only 80 is left: 80.
t = 1: power first.
1 squared is 1.
−16 times 1 is −16.
64 times 1 is 64.
−16 + 64 + 80 = 128.
t = 2: power first.
2 squared is 4.
−16 times 4 is −64.
64 times 2 is 128.
−64 + 128 + 80 = 144.
t = 3: 3 squared is 9.
−16 times 9 is −144.
64 times 3 is 192.
−144 + 192 + 80 = 128.
t = 4: 4 squared is 16.
−16 times 16 is −256.
64 times 4 is 256.
−256 + 256 + 80 = 80.
t = 5: 5 squared is 25.
−16 times 25 is −400.
64 times 5 is 320.
−400 + 320 + 80 = 0.
Look at the heights: 80, 128, 144, 128, 80, 0.
They go up, then come down, like the ball.
3 Find when it lands
On the ground, the height is 0.
We look for h = 0 in the table.
The table has h = 0 at t = 5.
The ball is back on the ground.
Careful!
It starts 80 feet up, not on the ground.
So the way down is longer than the way up: it lands after 5 seconds.
4 Draw the flight
We draw the flight: time t across, height h up.
The h numbers go up in 20s.
Put the points from the table on the grid.
Join them with a smooth curve: the path of the height over time.
It meets the ground, height 0, at t = 5.
5 Say it in words
So the ball hits the ground after 5 seconds.
6 Check
Let's check: put t = 5 in.
5 squared is 25.
We get 0: on the ground.
It works!
Let's look back.
First: what the letters mean.
Then: make a small table.
Then: find when it lands.
Then: draw the flight.
Then: say it in words.
Last: check.