Question

It says to find the volume of the figure and round to the nearest hundredth where necessary. Please help!

Answer 1

Given:

A frustrum with a rectangular top and base.

To find the volume of the figure, we make the sketch as shown below.

The volume of the frustrum is given by;

`Volume\text{ of frustum}=volume\text{ of big pyramid}-volume\text{ of small pyramid}`

The volume of a pyramid is given by;

`\begin{gathered} V=(1)/(3)Ah \\ where; \\ A\text{ is the base area} \\ h\text{ is the height} \end{gathered}`

Hence, we get the heights of both pyramids by similarity;

`\begin{gathered} height\text{ of big pyramid}=25.7+x \\ height\text{ of small pyramid}=x \\ base\text{ of small pyramid}=15 \\ base\text{ of big pyramid}=37 \\ \\ Hence,\text{ by similarity,} \\ (h_1)/(b_1)=(h_2)/(b_2) \\ (25.7+x)/(37)=(x)/(15) \\ \\ Cross\text{ multiplying,} \\ 15\left(25.7+x\right)=37* x \\ 385.5+15x=37x \\ Collecting\text{ the like terms,} \\ 385.5=37x-15x \\ 385.5=22x \\ Dividing\text{ both sides by 22;} \\ (385.5)/(22)=x \\ x=17.52cm \\ \\ Thus, \\ he\imaginaryI ght\text{ of b}\imaginaryI\text{g pyram}\imaginaryI\text{d}=25.7+x=25.7+17.52 \\ he\mathrm{i}ght\text{ of b}\mathrm{i}\text{g pyram}\mathrm{i}\text{d}=43.22cm \\ \\ he\mathrm{i}ght\text{ of small pyram}\mathrm{i}\text{d}=x \\ he\mathrm{i}ght\text{ of small pyram}\mathrm{i}\text{d}=17.52cm \end{gathered}`

Hence, the volume of the pyramids is calculated;

`\begin{gathered} V=(1)/(3)Ah \\ Volume\text{ of the big pyramid}=(1)/(3)* l* b* h \\ Volume\text{ of the big pyramid}=(1)/(3)*37*20*43.22=31,982.8cm^3 \\ \\ Volume\text{ of the small pyram}\imaginaryI\text{d}=(1)/(3)* l* b* h \\ Volume\text{ of the small pyram}\imaginaryI\text{d}=(1)/(3)*15*20*17.52=5256cm^3 \\ \\ \\ Volume\text{ of the figure}=31982.8-5256 \\ Volume\text{ of the figure}=26726.80cm^3 \end{gathered}`

Therefore, the volume of the figure to the nearest hundredth is 26,726.80 cubic centimeters.