Question

the following cylinder has a volume of 55π cubic units and a height of 5 units. the cone has the same base area and height, and it has a slant height of 6 units. what is the volume of the cone? choose 1 answer: a) (55/3)π cubic units b) (55/2)π cubic units c) 22π cubic units d) 55π cubic units

Explanation:

Step1: Recall Volume Formulas

The volume of a cylinder is \( V_{\text{cylinder}} = B \times h \), where \( B \) is the base area and \( h \) is the height. The volume of a cone is \( V_{\text{cone}} = \frac{1}{3} \times B \times h \), where \( B \) is the base area and \( h \) is the height.

Step2: Identify Given Information

The cylinder has a volume of \( 55\pi \) cubic units and a height of 5 units. The cone has the same base area (\( B \)) and height (\( h = 5 \)) as the cylinder.

Step3: Relate Cylinder and Cone Volumes

Since the cone has the same base area and height as the cylinder, we can use the relationship between the volume of a cylinder and a cone with the same base and height. The volume of the cone is \( \frac{1}{3} \) of the volume of the cylinder (because \( V_{\text{cone}}=\frac{1}{3}Bh \) and \( V_{\text{cylinder}} = Bh \)).

Step4: Calculate Cone Volume

Given \( V_{\text{cylinder}} = 55\pi \), then \( V_{\text{cone}}=\frac{1}{3}\times V_{\text{cylinder}} \) (since \( Bh = 55\pi \) from the cylinder's volume formula). So \( V_{\text{cone}}=\frac{1}{3}\times55\pi=\frac{55}{3}\pi \) cubic units.

Answer:

A. \(\boldsymbol{\frac{55}{3}\pi}\) cubic units