Question

question 7 of 21 tiana would like to buy cone - shaped hats for her birthday party. she wants hats with the largest volume, since they look bigger. which of the following brands of hat should she buy? brand a: height of cone = 15 in. radius of cone base = 6 in. brand b: height of cone = 18 in. radius of cone base = 5 in.
a. brand a
b. brand b
c. either — the hats have equal volumes

Explanation:

Step1: Recall cone volume formula

The volume \( V \) of a cone is given by \( V = \frac{1}{3}\pi r^2 h \), where \( r \) is the radius of the base and \( h \) is the height.

Step2: Calculate volume of Brand A

For Brand A: \( r = 6 \) in, \( h = 15 \) in.
Substitute into the formula:
\( V_A = \frac{1}{3}\pi (6)^2 (15) \)
\( V_A = \frac{1}{3}\pi (36)(15) \)
\( V_A = \pi (12)(15) \)
\( V_A = 180\pi \) cubic inches.

Step3: Calculate volume of Brand B

For Brand B: \( r = 5 \) in, \( h = 18 \) in.
Substitute into the formula:
\( V_B = \frac{1}{3}\pi (5)^2 (18) \)
\( V_B = \frac{1}{3}\pi (25)(18) \)
\( V_B = \pi (25)(6) \)
\( V_B = 150\pi \) cubic inches.

Step4: Compare volumes

Since \( 180\pi > 150\pi \), \( V_A > V_B \).

Answer:

A. Brand A