ace canning company is redesigning its product. The New cans will have twice the radius, but the same height of the original cans. Which statement is true about the volume of the new cans compared to the volume of the original cans? Show your work
A. The volume of the new and original cans will be equal
b. The volume of the new cans will be twice the volume of the original cans
c. The volume of the new cans will be four times the volume of the original cans
the volume is the base area times the height
recall that area grows as the square of the linear scale factor (2)
thus, the base area grows by a factor of 2^2 = 4
since the height remains the same, and the new radius is twice the old radius, the new volume also grows by a factor of 4.
To show this, look at the ratio of volumes (new/old):
(π(2r)^2 h)/(πr^2 h) = (4πr^2h)/(πr^2h) = 4
To determine which statement is true about the volume of the new cans compared to the volume of the original cans, we need to use the formula for the volume of a cylinder, which is V = πr^2h, where V is the volume, r is the radius, and h is the height.
Let's assume the radius of the original cans is represented by r1 and the height is represented by h1. Since the new cans will have twice the radius, the new radius will be 2r1. The height of the new cans remains the same.
Now let's calculate the volumes of both the original and new cans.
The volume of the original cans (V1) is given by V1 = πr1^2h1.
The volume of the new cans (V2) is given by V2 = π(2r1)^2h1.
Expanding the equation for V2, we have V2 = π4r1^2h1.
Now we can compare the volume of the new cans (V2) with the volume of the original cans (V1).
If we divide V2 by V1, we get (π4r1^2h1) / (πr1^2h1).
Simplifying the equation, we find V2/V1 = 4.
Therefore, the volume of the new cans will be four times the volume of the original cans.
So, the correct answer is:
c. The volume of the new cans will be four times the volume of the original cans.