Remember that the volume of a three dimensional shape is the amount of space that the shape takes up. Using the applet below, explore how changes in dimensions can affect the volume.

Exploration

You are at the local hardware store to buy a can of paint. After settling on one product, the salesman offers to sell you a can of the same paint, but it will have either double the height or double the radius (your choice) for twice the price. Assuming all cans of paint are filled to the brim, is it worth taking him up on his offer?

If so, would you get more paint for each dollar if you chose the can that was double the radius or the can that was double the height?

To see how changes in height and radius affect the volume of a can to different extents, try the following interactive. You can vary the height and radius by moving the sliders around.

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As the height and radius increase, the volume of a can also increases. Conversely, as the height and radius decrease, the volume of a can also decreases.

Whether you want to find the volume of a can of paint so you know if you are getting a good price or find out the available space that can be filled by grain in a silo, the concept of volume is used often in daily life.

With a given volume and some of the dimensions provided, we can also determine the missing dimensions of a solid.

Examples

Example 1

The planet Mars has a radius of 3 400\text{ km.} What is the volume of Mars? Write your answer in scientific notation to three decimal places.

Worked Solution

Create a strategy

Use the formula of the volume of sphere.

Apply the idea

We have been given the radius of planet Mars which is 3400\text{ km.}

\displaystyle V	\displaystyle =	\displaystyle \frac{4}{3}\pi r^3	Use the formula
	\displaystyle =	\displaystyle \frac{4}{3}\pi \times 3400^3	Substitute r=3400
	\displaystyle \approx	\displaystyle 164\,636\,210\,200	Evaluate
	\displaystyle =	\displaystyle 1.646 \times 10^{11} \text{ km}^3	Write in scientific notation

Example 2

A podium is formed by sawing off the top of a cone. Find the volume of the podium. Round your answer to two decimal places.

An 8m-podium formed by sawing off the top of a cone with radius of 8 m and 4 m. Ask you teacher for more information.

Worked Solution

Create a strategy

The volume of podium is the difference of the volume of the large cone and the volume of the small cone at the top. Use the Pythagorean theorem to find the perpendicular heights of the cones.

Apply the idea

Two right triangles formed with a common side length. Ask you teacher for more information.

Based on the diagram, the radii of the small and large cones are r_1=2 m and r_2=4 m, respectively.

The slant height of the large cone is 16\text{ cm} and the slant height of the smaller cone is 8\text{ cm}

To find the perpendicular heights of the cones we can use the Pythagorean theorem.

For the smaller cone, let a be the perpendicular height:

\displaystyle 8^2	\displaystyle =	\displaystyle 2^2+a^2	Use the Pythagorean formula
\displaystyle a^2	\displaystyle =	\displaystyle 8^2-2^2	Isolate a^2 on one side
\displaystyle \sqrt{a}	\displaystyle =	\displaystyle \sqrt{8^2-2^2}	Square root both sides
	\displaystyle =	\displaystyle \sqrt{60}	Evaluate
\displaystyle V_1	\displaystyle =	\displaystyle \dfrac{1}{3}\pi \times 2^2 \times \sqrt{60}	Find the volume of the small cone
	\displaystyle =	\displaystyle \dfrac{4\sqrt{60} \pi}{3}	Simplify

For the large cone, let b be the perpendicular height:

\displaystyle 16^2	\displaystyle =	\displaystyle 4^2+b^2	Use the Pythagorean formula
\displaystyle b^2	\displaystyle =	\displaystyle 16^2-4^2	Isolate b^2 on one side
\displaystyle \sqrt{b}	\displaystyle =	\displaystyle \sqrt{16^2-4^2}	Square root both sides
	\displaystyle =	\displaystyle 2\sqrt{60}	Evaluate
\displaystyle V_2	\displaystyle =	\displaystyle \dfrac{1}{3}\pi \times 4^2 \times 2\sqrt{60}	Find the volume of the large cone
	\displaystyle =	\displaystyle \dfrac{32\sqrt{60} \pi}{3}	Simplify

\displaystyle \text{Volume of podium }	\displaystyle =	\displaystyle \dfrac{32\sqrt{60} \pi}{3}-\dfrac{4\sqrt{60} \pi}{3}	Subtract the smaller volume from the larger volume
	\displaystyle =	\displaystyle 227.12\text{ m}^3	Evaluate

Idea summary

The volume of a three dimensional shape is the amount of space that is taken up by the shape.

We can use the formulas for volume and the dimensions that we are given, to solve for the missing dimensions.

Outcomes

8.G.C.9

Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

10.04 Problem solving with volume

Introduction

Ideas

Problem solving with volume