Step by step
The question
Question: h = −5t² + 30t is the height, in meters, of a rocket t seconds after it is launched.
When does the rocket hit the ground?
We write the equation big, so we can label it.
1 What the letters mean
h is the height of the rocket, in meters.
t is the time, in seconds, after it is launched.
2 Make a small table
To see the flight, we try t = 0, 1, 2 and so on, until the rocket is back on the ground, at height 0.
That is 6 seconds.
t = 0: every part with t is 0.
So h is 0.
t = 1: power first.
1 squared is 1.
−5 times 1 is −5.
30 times 1 is 30.
−5 + 30 = 25.
t = 2: power first.
2 squared is 4.
−5 times 4 is −20.
30 times 2 is 60.
−20 + 60 = 40.
t = 3: 3 squared is 9.
−5 times 9 is −45.
30 times 3 is 90.
−45 + 90 = 45.
t = 4: 4 squared is 16.
−5 times 16 is −80.
30 times 4 is 120.
−80 + 120 = 40.
t = 5: 5 squared is 25.
−5 times 25 is −125.
30 times 5 is 150.
−125 + 150 = 25.
t = 6: 6 squared is 36.
−5 times 36 is −180.
30 times 6 is 180.
−180 + 180 = 0.
Look at the heights: 0, 25, 40, 45, 40, 25, 0.
They go up, then come down, like the rocket.
3 Find when it lands
On the ground, the height is 0.
We look for h = 0 in the table.
After the start, the table has h = 0 at t = 6.
The rocket is back on the ground.
Careful!
At t = 0 the height is 0 too: that is the start, on the ground.
It lands at the second 0 in the table: t = 6.
4 Draw the flight
We draw the flight: time t across, height h up.
The h numbers go up in 10s.
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 = 6.
5 Say it in words
So the rocket hits the ground after 6 seconds.
6 Check
Let's check: put t = 6 in.
6 squared is 36.
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.