body{margin:0;} noscript{z-index: 10000; position: fixed; display: block; height: 100%; width: 100%; background: #fff;} noscript .fa{font-size: 40px; display:block; color: #daa520;} noscript div{margin-top: 6%; padding-left: 20px; padding-right: 20px; text-align: center;} ! You must have JavaScript enabled to use this site.

Big Ideas Math: Modeling Real Life, Grade 8

BI

Big Ideas Math: Modeling Real Life, Grade 8 View details

Practice Test

Continue to next subchapter

Exercise 7 Page 458

Remember that the ratio between the volumes of two similar solids is equal to the ratio of any linear measures raised to the third power.

About 42.7 in^3

We have two similar waffle cones where the corresponding linear measures have a ratio of 3 to 4. We want to find the volume of the larger cone given that the volume of the smaller cone is 18 cubic inches. Let's begin by writing the ratio as a fraction. 3 to 4 = 3/4The cones are similar so this ratio is true for all of the linear measures of the cones. Let's use the height h as an example. \begin{gathered} \dfrac{{\color{#0000FF}{h_\text{small}}}}{{\color{#009600}{h_\text{large}}}}= \dfrac{{\color{#0000FF}{3}}}{{\color{#009600}{4}}} \end{gathered} Now, we can find the volume of the larger cone by using the fact that the ratio between the volumes of two similar solids is equal to the ratio of any linear measures raised to the third power.

\dfrac{{\color{#FF0000}{V_\text{small}}}}{V_\text{large}}=\bigg(\dfrac{{\color{#0000FF}{h_\text{small}}}}{{\color{#009600}{h_\text{large}}}}\bigg)^3

SubstituteValues

Substitute values

\dfrac{{\color{#FF0000}{18}}}{V_\text{large}}=\bigg(\dfrac{{\color{#0000FF}{3}}}{{\color{#009600}{4}}}\bigg)^3

▼

Solve for V_\text{large}

(a/b)^m=a^m/b^m

\dfrac{18}{V_\text{large}}=\dfrac{3^3}{4^3}

Calculate power

\dfrac{18}{V_\text{large}}=\dfrac{27}{64}

\text{LHS} \cdot V_\text{large}=\text{RHS}\cdot V_\text{large}

18=\dfrac{27}{64}(V_\text{large})

LHS * 64=RHS* 64

18 \cdot 64=27 (V_\text{large})

.LHS /27.=.RHS /27.

\dfrac{18 \cdot 64}{27}= V_\text{large}

Rearrange equation

V_\text{large}=\dfrac{18 \cdot 64}{27}

Use a calculator

V_\text{large}=42.666666\ldots

Round to 1 decimal place(s)

V_\text{large}\approx 42.7 \text{ in}^3

The volume of the larger cone is about 42.7 cubic inches.

Extra

Why Do We Raise the Ratio to the Third Power?

The ratio between volumes of two similar solids is equal to the ratio of any linear measures raised to the third power. For example, the height h. \begin{gathered} \dfrac{V_\text{A}}{V_\text{B}}=\bigg(\dfrac{h_\text{A}}{h_\text{B}}\bigg)^3 \end{gathered} Why the third power? The answer is surprisingly simple, and makes for a great way of remembering this formula. Linear measures are one-dimensional and volumes are three-dimensional! To get something three-dimensional, we need to multiply three one-dimensional measures.

Subchapter links

Exercises