Cylinder A is eight time as tall as cylinder B, but the diameter of cylinder B is four times the diameter of the cylinder A. Which cylinder has larger volume? How many times as large ?
the volumes are
A:πr^2(8h) = 8πr^2h
B:π(4r)^2h = 16πr^2h
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To determine which cylinder has a larger volume and how many times larger it is, we can use the formulas for the volume of a cylinder.
The formula for the volume of a cylinder is:
Volume = π * r^2 * h
Where:
- π (pi) is a mathematical constant approximately equal to 3.14159
- r is the radius of the base of the cylinder
- h is the height of the cylinder
Let's denote the height of Cylinder B as hB, and the height of Cylinder A as hA.
Given that Cylinder A is eight times as tall as Cylinder B, we can say that hA = 8 * hB.
We're also given that the diameter of Cylinder B is four times the diameter of Cylinder A. Since the diameter is twice the radius, we can say that the radius of Cylinder B (rB) is 4 times the radius of Cylinder A (rA). Mathematically, this can be written as rB = 4 * rA.
Now, to find the ratio of the volumes, we need to compare the volumes of Cylinder A and Cylinder B.
Volume of Cylinder A:
VA = π * rA^2 * hA
Volume of Cylinder B:
VB = π * rB^2 * hB
We can substitute the given relationships into these equations to compare the volumes.
Volume of Cylinder A:
VA = π * rA^2 * (8 * hB)
Volume of Cylinder B:
VB = π * (4 * rA)^2 * hB
Simplifying further:
VA = π * 64 * rA^2 * hB
VB = π * 16 * rA^2 * hB
Comparing the ratios:
VA / VB = (π * 64 * rA^2 * hB) / (π * 16 * rA^2 * hB)
= 64 / 16
= 4
Therefore, Cylinder A has a volume that is 4 times larger than Cylinder B.