Question

A french fry stand at the fair serves their fries in paper cones. The cones have a radius of 222 inches and a height of 666 inches. It is a challenge to fill the narrow cones with their long fries. They want to use new cones that have the same volume as their existing cones but a larger radius of 444 inches. What will the height of the new cones be?

Answer 1

To find the height of the new cone with a larger radius that maintains the same volume, we calculate the volume of the original cone and solve for the new height using the cone volume formula.

Step-by-step explanation:

The student wants to find out the new height of the cone with a larger radius while maintaining the same volume as the original cone. The volume of a cone is given by the formula V = 1/3 πr²h. To find the height of the new cone, we'll use the volume of the original cone as the constant value.

Let's first calculate the volume of the original cone:

1. V_original = 1/3 π(222²)(666)

Now, let's set up an equation with the new radius to solve for the new height (h_new):

1. 1/3 π(444²)(h_new) = V_original
2. h_new = (V_original) / (1/3 π(444²))

By substituting the V_original from step 1 into step 3, we can solve for h_new. The result will be the height of the new cone with a radius of 444 inches that has the same volume as the original cone.

Answer 2

The height of the new cone will be calculated as follows:
volume of cone=1/3πr²h
volume of the old cone is:
1/3×π×2²×6
=12 π in³

given that the new cone and old one have the same volume and the new cone has a radius of 4 inches, the the height will be:
12π=1/3×π×4²×h
solving for h we get:
36=16h
thus
h=2.25 inches
thus the height of the new cone is 2.25 inches