If the length of the radius of a circle is a third of its original size, what will the area of the new circle be?

If the length of the base's radius and height of a cone is doubled, what will the surface area of the new cone be?

We can find the new area by noting that the area will change by a factor of when we change the dimensions of the cone. In this case we are changing two dimensions of the cone and so the new area will be:

The value of comes from the word “doubled” in the question: the value of is 2.

So the new area of the cone will be if we double the height and the base's radius of the cone.

Therefore the surface area of the new cone will be 4 times the original surface area.

If the height of a prism is doubled, how much will its volume increase by?

We do not know if we have a rectangular prism or a triangular prism. However we do know that the volume of a prism is given by:

Now we are changing just one dimension of the prism: the height. Therefore the new volume is given by:

Therefore the volume of the prism doubles if the height is doubled.

If the length of each side of a triangular prism is quadrupled, what will the volume of the new triangular prism be?