Question

By what factor would you increase the cost of a small pizza if its radius was increased by a factor of 4 times or 5 or 10? Try identify a pattern and why the pattern exists.

4 squared = 16. That means 16 x the cost of the small pizza
5 squared = 25. That means 25 x the cost of the small pizza
10 squared= 100 That means 100 x the cost of the small pizza

We see the pattern of multiplying the cost of the small pizza by the result of the number squared.

Answer 1

The cost of the pizza is proportional to the square of the radius. If we increase the radius by the factor n, the cost will be proportional to the n squared.

$$S_{new} = \pi(nr)^2 = \pi n^2 r^2 = n^2 \pi r^2 = n^2 S$$

Thus, if we multiply the radius by n, we will increase the cost by $n^2$.