Step by step
The question
Question: d varies directly with the square of t.
d = 20 when t = 2.
Write the equation.
1 Write the rule
Varies directly means d is always the same number times t².
When t gets bigger, d gets bigger too.
We call the number k: d = kt².
k stays the same for every pair of numbers.
Books call it the constant of variation.
2 Find k
We put in d = 20 and t = 2.
2 squared is 4, and k times 4 is written 4k.
4k means 4 times k.
We want just k.
To remove times 4, we divide by 4.
We do it on both sides, so they stay equal.
Careful!
k is d divided by t²: 20/4.
Not 4/20.
The d number goes on top.
20/4 on the left.
4k/4 on the right.
4 divided by 4 is 1.
So just k is left.
20 divided by 4 is 5.
So k = 5.
3 Write the equation
We put 5 in place of k.
So the equation is d = 5t².
4 Check
Let's check with the pair we know: put in t = 2.
We get 20, like in the question.
So the equation is right!
Let's look back.
First: write the rule.
Then: find k.
Then: write the equation.
Last: check.