Chapter 12.3, Problem 32PPS

a.

To determine

To describe: The faces of solid that remains after the top portion has been cut by a plane parallel to the base.

Answer to Problem 32PPS

The frustum is made of removing an upper pyramid of slant height 8.18 in. and base edge of 6 in from a bigger pyramid of slant height 23.73 in. and base edge of 17 in.The faces of solid are squares of edges 6 in. and 17 in.

Explanation of Solution

Given:

The ferret tent is shown in Figure-1 here.

Concept used:

The ratio of sides of two similar triangles are equal.

Calculations:

A frustum can be visualized as the frustum of a pyramid, the complete pyramid is shown in Figure-2:

The edge of square base of the pyramid is 17 in. From Figure-2, clearly triangles ZAB and ZCD are similar triangles. Therefore,

  $\Rightarrow \frac{ZN}{ZE}=\frac{AB}{CD}$

  $\Rightarrow \frac{15+ZE}{ZE}=\frac{17}{6}$

  $\Rightarrow \frac{15}{ZE}+1=\frac{17}{6}$

  $\Rightarrow \frac{15}{ZE}=\frac{17}{6}-1=\frac{11}{6}$

  $\Rightarrow ZE=\frac{15}{11}\times 6=8.18$ in.

Thus, the frustum is made of removing an upper pyramid of slant height 8.18in. and base edge of 6 in from a bigger pyramid of slant height 23.77 in. and base edge of 17 in.The faces of solid are squares of edges 6 in. and 17 in.

Conclusion:

Thefrustum is made of removing an upper pyramid of slant height 8.18 in. and base edge of 6 in from a bigger pyramid of slant height 23.18in. and base edge of 17 in.The faces of solid are squares of edges 6 in. and 17 in.

b.

To determine

To find: The lateral area and the surface area of the frustum.

Answer to Problem 32PPS

The lateral area and surface area of frustum are 788.12 inch² and 1113.13 inch² respectively.

Explanation of Solution

Given:

The frustum in part (a)

Concept used:

The lateral area of a pyramid of perimeter of base P and slant height l is given by $L=\frac{1}{2}Pl$ and surface area $S=\frac{1}{2}Pl\text{+Base}\text{area}$ .

Calculations:

The lateral area of the complete tent (pyramid) is

  $L=\frac{1}{2}\left(17+17+17+17\right)\times 23.18=788.12$ inch².

The surface area of frustum $S=L\text{+area of both ends}$

  $\Rightarrow S=788.12\text{+}{\left(\text{6}\right)}^{\text{2}}\text{+}{\left(\text{17}\right)}^2=1113.12$ inch².

Conclusion:

The lateral area and surface area of frustum are 788.12 inch² and 1113.12 inch² respectively.

c.

To determine

To find: The surface area of the frustum.

Answer to Problem 32PPS

The surface area of frustum is 3128.25 inch².

Explanation of Solution

Given:

The another tent is made by cutting the top off of a pyramid with a height of 12 cm slant height of 20 cm and square base of 32 cm.

Concept used:

The lateral area of a pyramid of perimeter of base P and slant height l is given by $L=\frac{1}{2}Pl$ and surface area $S=\frac{1}{2}Pl\text{+Base}\text{area}$ .

Calculations:

As from the figure-2 part (b)

  $\Rightarrow \frac{ZN}{ZE}=\frac{AB}{CD}$

  $\Rightarrow \frac{20+12}{12}=\frac{20}{CD}$

  $\Rightarrow CD=\frac{20\times 12}{32}=7.5$

The lateral area of the complete tent (pyramid) is

  $L=\frac{1}{2}\left(32+32+32+32\right)\times 32=2048$ inch².

The surface area of frustum $S=L\text{+area of both ends}$

  $\Rightarrow S=2048\text{+}{\left(\text{7}\text{.5}\right)}^{\text{2}}\text{+}{\left(\text{32}\right)}^2=3128.25$ inch².

Conclusion:

The surface area of frustum is3128.25inch².