Question

show all work for this question.

1. a store sells colored craft sand in the three different containers below.

container 1: a square prism with a base length of 4 inches and a height of 7.5 inches.
container 2: a cylinder with a diameter of 5 inches and a height of 6 inches.
container 3: a cone with a diameter of 7.5 inches and a height of 8.5 inches.
if the containers are filled to the top, which container will hold the most sand? justify your answer.

Explanation:

Step1: Calculate Volume of Container 1

Volume of square prism: $V_1 = l^2h$
$l=4$ in, $h=7.5$ in
$V_1 = 4^2 \times 7.5 = 16 \times 7.5 = 120$ cubic inches

Step2: Calculate Volume of Container 2

First find radius: $r = \frac{d}{2} = \frac{5}{2} = 2.5$ in
Volume of cylinder: $V_2 = \pi r^2 h$
$h=6$ in
$V_2 = \pi \times (2.5)^2 \times 6 = \pi \times 6.25 \times 6 = 37.5\pi \approx 117.81$ cubic inches

Step3: Calculate Volume of Container 3

First find radius: $r = \frac{d}{2} = \frac{7.5}{2} = 3.75$ in
Volume of cone: $V_3 = \frac{1}{3}\pi r^2 h$
$h=8.5$ in
$V_3 = \frac{1}{3} \times \pi \times (3.75)^2 \times 8.5 = \frac{1}{3} \times \pi \times 14.0625 \times 8.5 = \frac{119.53125\pi}{3} \approx 125.07$ cubic inches

Step4: Compare the three volumes

$V_3 \approx 125.07 > V_1 = 120 > V_2 \approx 117.81$

Answer:

Container 3 (the cone) will hold the most sand, as its volume of approximately 125.07 cubic inches is larger than the square prism's 120 cubic inches and the cylinder's approximately 117.81 cubic inches.