Answer to Question #331050 in Geometry for summer

Question

(08.02 MC)

An hourglass consists of two sets of congruent composite figures on either end. Each composite figure is made up of a cone and a cylinder, as shown below:

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Each cone of the hourglass has a height of 18 millimeters. The total height of the sand within the top portion of the hourglass is 54 millimeters. The radius of both cylinder and cone is 8 millimeters. Sand drips from the top of the hourglass to the bottom at a rate of 10π cubic millimeters per second. How many seconds will it take until all of the sand has dripped to the bottom of the hourglass? (4 points)

Group of answer choices

68.3

38.4

268.8

230.4

Answer 1

For this kind of proble it is very important that you attach the figure because it contains important information to understand the question.

I have attached the figure for better understanding.

1) The top portion (and the bottom is congruent but rotated 180°) of the hourglass is a figure equivalent to a cylinder on top and a cone on bottom.

So the total volume contained in the top portion is the volume of a cylinder + the volume of a cone.

This is how you calculate the volume of the top portion:

1) The height of the cylinder is 54 mm - 18 mm = 36 mm

2) The formula for the volume of a cylinder is $V = \pi r^2 * h$

radius = 8 mm

height = 36 mm

=> $V= \pi * 8^2 * 36 = 2304 \pi (mm)^3$

3) The formula for the volume of a cone is $V ={1 \over 3}\pi * r^2 * h$

radius = 8 mm

height = 18 mm

$V = {1 \over 3}\pi*8^2*18 = 384\pi(mm)^3$

4) The total volume of the top portion is volume of the cylindrical part + volume of the cone:

Total volume = $2304\pi(mm)^3+384\pi(mm)^3=2688\pi(mm)^3$

5) To find the number of seconds it take until all of the sand has dripped to the bottom of the hourglass you have to divide the total volumen of sand by the rate:

time in seconds = total volume of sand / rate of dripping

time in seconds = ${2688π(mm)^3 \over 10π(mm)^3} = 268.8 s$

That is the answer: 268.8 s