Step by step
The question
Question: h = −5t² + 20t + 25 is the height, in meters, of a stone t seconds after it is thrown from a cliff.
When does the stone hit the ground?
We write the equation big, so we can label it.
1 What the letters mean
h is the height of the stone, in meters.
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 stone is back on the ground, at height 0.
That is 5 seconds.
t = 0: every part with t is 0.
Only 25 is left: 25.
t = 1: power first.
1 squared is 1.
−5 times 1 is −5.
20 times 1 is 20.
−5 + 20 + 25 = 40.
t = 2: power first.
2 squared is 4.
−5 times 4 is −20.
20 times 2 is 40.
−20 + 40 + 25 = 45.
t = 3: 3 squared is 9.
−5 times 9 is −45.
20 times 3 is 60.
−45 + 60 + 25 = 40.
t = 4: 4 squared is 16.
−5 times 16 is −80.
20 times 4 is 80.
−80 + 80 + 25 = 25.
t = 5: 5 squared is 25.
−5 times 25 is −125.
20 times 5 is 100.
−125 + 100 + 25 = 0.
Look at the heights: 25, 40, 45, 40, 25, 0.
They go up, then come down, like the stone.
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 stone is back on the ground.
Careful!
It starts 25 meters 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 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 = 5.
5 Say it in words
So the stone 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.