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It was just learned that if the length scale factor of two similar figures and one of the areas of the figures are known, then the unknown area can be found. Next, consider if both areas but only the side length of one figure are known. It will be possible to solve for the other similar figure's corresponding side length.

The diagram shows two similar figures. Figure $A$ has an area of $9$ square inches, and Figure $B$ has an area of $25$ square inches.

The scale factor of two similar figures can be used to find the volume of one of the figures when the volume of the other figure is known.

The suitcase company Case-O-La produces snazzy suitcases of various sizes. When a large-sized suitcase is bought, the company offers its cabin-sized version at a discounted price to the same customer.

Solution

The suitcases can be considered as two similar rectangular prisms with heights $27$ and $18$ inches.

Similar solids have the same shape and all of their corresponding sides are proportional. The ratio of the corresponding linear dimensions of the similar solids is the scale factor.

$$\begin{array}{c}\frac{\text{Height of small suitcase}}{\text{Height of big suitcase}}=\frac{18}{27}\end{array}$$

If the scale factor of two similar solids is $a:b,$ then the ratio of their corresponding volumes is $a^3:b^3.$ Now, raise the scale factor to the third power to obtain the ratio of the volumes.

$\frac{a}{b}=\frac{18}{27}$

ReduceFrac

$\frac{a}{b}=\frac{a/9}{b/9}$

$\frac{a}{b}=\frac{2}{3}$

RaiseEqn

${\text{LHS}}^3={\text{RHS}}^3$

${\left(\frac{a}{b}\right)}^3={\left(\frac{2}{3}\right)}^3$

PowQuot

${\left(\frac{a}{b}\right)}^m=\frac{a^m}{b^m}$

$\frac{a^3}{b^3}=\frac{2^3}{3^3}$

CalcPow

Calculate power

$\frac{a^3}{b^3}=\frac{8}{27}$

FracToScale

$\frac{a}{b}=a:b$

$a^3:b^3=8:27$

The ratio of the volumes is $8:27.$ Now, let $V_1$ be the volume of the small suitcase. Since the volume of the big suitcase is $90$ liters, the ratio of $V_1$ to $90$ is $8:27.$

$\frac{V_1}{90}=\frac{8}{27}$

Solve for $V_1$

MultEqn

$\text{LHS}\cdot 90=\text{RHS}\cdot 90$

$V_1=\frac{8}{27}\cdot 90$

MoveRightFacToNum

$\frac{a}{c}\cdot b=\frac{a\cdot b}{c}$

$V_1=\frac{720}{27}$

CalcQuot

Calculate quotient

$V_1=26.6666666\dots$

RoundDec

Round to $1$ decimal place(s)

$V_1\approx 26.7$

The volume of the small suitcase is about $26.7$ liters.

After reading a physics magazine, Mark feels confident in estimating the radius of the Sun. To do so, he will use the volumes of the Sun and Earth, which are $1.41\times 10^{18}$ and $1.08\times 10^{12}$ cubic kilometers, respectively.

If the radius of the Earth is about $6300$ kilometers, help Mark find the radius of the Sun.

A three-dimensional figure is called a composite solid if it is the combination of two or more solids. Like similar solids, if the corresponding linear measures of two composite solids are proportional, the composite solids are said to be similar. Therefore, their length scale factor can be determined and used to find certain characteristics of the shapes.

The amount of material used to construct the larger silo is three times that of the smaller one. That is, the surface area of the larger silo is three times as large as the surface area of the smaller silo. These two silos can be considered to be similar solids. Furthermore, each silo is composed of a cone and a cylinder as shown.

Solution

The given silos are similar composite solids. Since they are similar, their corresponding linear measures are proportional. Therefore, each silo can be considered as a whole. To find the volume of the smaller silo, these steps will be followed.

• The ratio of their surface areas will be used to determine the length scale factor.
• The length scale factor will be used to determine the volume scale factor.
• Finally, the volume scale factor will be used to determine the volume of the smaller silo.

It is given that the surface area of the larger silo is three times as large as the surface area of the smaller silo.

To find the length scale factor, consider its relationship to the surface areas. Recall that for areas of similar figures, the ratio of their areas is equal to the square of the ratio of their corresponding side lengths. Refer to the smaller silo's side length as $a$ and the larger silo's side length as $b.$

$$\begin{array}{ccc}\underline{\text{Length Scale Factor}}& & \underline{\text{Area Scale Factor}}\\ \frac{a}{b}& \Rightarrow & {\left(\frac{a}{b}\right)}^2\end{array}$$

Since the surface area of the larger silo is three times the surface area of the smaller silo, the ratio of the surface area, small to large, is $1:3.$ With that information, the following equation can be expressed.

$$\begin{array}{c}{\left(\frac{a}{b}\right)}^2=\frac{1}{3}\end{array}$$

Next, this equation can be simplified to solve for the length scale factor. Begin by taking the square root of both sides of the equation.

${\left(\frac{a}{b}\right)}^2=\frac{1}{3}$

Solve for $\frac{a}{b}$

SqrtEqn

$\sqrt{\text{LHS}}=\sqrt{\text{RHS}}$

$\sqrt{{\left(\frac{a}{b}\right)}^2}=\sqrt{\frac{1}{3}}$

SqrtPowToNumber

$\sqrt{a^2}=a$

$\frac{a}{b}=\sqrt{\frac{1}{3}}$

SqrtQuot

$\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$

$\frac{a}{b}=\frac{\sqrt{1}}{\sqrt{3}}$

CalcRoot

Calculate root

$\frac{a}{b}=\frac{1}{\sqrt{3}}$

Now, the length scale factor can be used to find the volume scale factor. To do so, the length scale factor needs to be raised to the third power.

$$\begin{array}{ccc}\underline{\text{Length Scale Factor}}& & \underline{\text{Volume Scale Factor}}\\ \frac{a}{b}=\frac{1}{\sqrt{3}}& \Rightarrow & {\left(\frac{a}{b}\right)}^3={\left(\frac{1}{\sqrt{3}}\right)}^3\end{array}$$

Finally, knowing that the volume of the larger silo is about $6750$ cubic meters, the volume of the smaller silo $V_s$ can be calculated. Similar to the areas, the ratio of the volumes should be equal to the volume scale factor.

$\frac{V_s}{6750}={\left(\frac{1}{\sqrt{3}}\right)}^3$

Solve for $V_s$

PowQuot

${\left(\frac{a}{b}\right)}^m=\frac{a^m}{b^m}$

$\frac{V_s}{6750}=\frac{1^3}{(\sqrt{3}{)}^3}$

BaseOne

$1^a=1$

$\frac{V_s}{6750}=\frac{1}{(\sqrt{3}{)}^3}$

MultEqn

$\text{LHS}\cdot 6750=\text{RHS}\cdot 6750$

$V_s=\frac{6750}{(\sqrt{3}{)}^3}$

UseCalc

Use a calculator

$V_s=1299.038105\dots$

RoundInt

Round to nearest integer

$V_s\approx 1299$

The capacity of the smaller silo is about $1299$ cubic meters.

In this course, the relationships between the length scale factor, area scale factor and volume scale factor have been discussed. If the scale factor between two similar figures is $\frac{a}{b},$ then the ratio for their areas and volumes can be expressed as the table shows.

Considering these expressions, the challenge presented at the beginning can be solved with more confidence.

Emily knows that the models in the museum are similar pyramids and the scale factor between the corresponding side lengths is $1:2.$