Question

1. find the volume of each cylinder in terms of π. which cylinder has the greater volume? cylinder a: area of base = 6π ft², height = 10 ft cylinder b: circumference = 6π ft, height = 6 ft

Explanation:

Step1: Recall Volume of Cylinder Formula

The volume \( V \) of a cylinder is given by \( V = \text{Base Area} \times \text{height} \), or \( V=\pi r^{2}h \). For Cylinder A, we know the base area and height. For Cylinder B, we know the circumference and height, so we first find the base area from the circumference.

Step2: Calculate Volume of Cylinder A

Given base area \( A_A = 6\pi \, \text{ft}^2 \) and height \( h_A = 10 \, \text{ft} \). Using \( V = A \times h \), we get:
\( V_A=6\pi\times10 = 60\pi \, \text{ft}^3 \)

Step3: Find Radius of Cylinder B from Circumference

The circumference \( C \) of a circle is \( C = 2\pi r \). Given \( C_B = 6\pi \, \text{ft} \), we solve for \( r \):
\( 6\pi=2\pi r \)
Divide both sides by \( 2\pi \): \( r=\frac{6\pi}{2\pi}=3 \, \text{ft} \)

Step4: Calculate Base Area of Cylinder B

Base area \( A_B=\pi r^{2} \), with \( r = 3 \, \text{ft} \):
\( A_B=\pi(3)^{2}=9\pi \, \text{ft}^2 \)

Step5: Calculate Volume of Cylinder B

Given height \( h_B = 6 \, \text{ft} \), using \( V = A \times h \):
\( V_B=9\pi\times6 = 54\pi \, \text{ft}^3 \)

Step6: Compare Volumes

Compare \( V_A = 60\pi \) and \( V_B = 54\pi \). Since \( 60\pi>54\pi \), Cylinder A has a greater volume.

Answer:

Cylinder A has a greater volume. The volume of Cylinder A is \( 60\pi \, \text{ft}^3 \) and the volume of Cylinder B is \( 54\pi \, \text{ft}^3 \).