Chapter 12.10, Problem 22IP

a)

To determine

To calculate: Surface area of square pyramid with base length 1 and slant height 10.

Answer to Problem 22IP

21 square units.

Explanation of Solution

Given information:

The given information is, A pyramid with vase length 1 and slant height 10 is given.

Formula used:

Surface area $=4\times areaofeachface+areaofbase$

Heron’s formula: semi perimeter $s=\frac{a+b+c}{2}$

Area of triangle $=\sqrt{5(5-a)(5-b)(5-c)}$

Calculation:

  $\begin{array}{l}s=\frac{1+10+10}{2}=10.5\\ \therefore surfacearea=4\sqrt{(10.5)(9.5)(0.5)(0.5)}+1\times 1\\ =21squareunits.\end{array}$

b)

To determine

To calculate: To complete the given table.

Answer to Problem 22IP

Side length	1	2	3	4
Slant height	10	20	30	40
Surface area	21	84	189	336

Explanation of Solution

Given information:

The given information is, Pyramids with square base are given as shown in the table.

Formula used:

Surface area $=4\times areaofeachface+areaofbase$

Heron’s formula: semi perimeter $s=\frac{a+b+c}{2}$

Area of triangle $=\sqrt{5(5-a)(5-b)(5-c)}$

Calculation:

  $\begin{array}{l}a)s=\frac{2+20+20}{2}=21\\ surfacearea=4\sqrt{21\times 19\times 1\times 1}+2\times 2\\ =84\\ b)s=\frac{3+30+30}{2}=31.5\\ surfacearea=4\sqrt{31.5\times 28.5\times 1.5\times 1.5}+3\times 3\\ =189\\ c)s=\frac{4+40+40}{2}=42\\ surfacearea=4\sqrt{42\times 38\times 2\times 2}+4\times 4\\ =226\end{array}$

c)

To determine

To calculate: To predict surface area of pyramid with side length 5 and slant height 5, then check prediction.

Answer to Problem 22IP

Prediction: 525

Acmol value: 525

Explanation of Solution

Given information:

The given information is, A pyramid with base length 5 and slant height 50 is given.

Formula used:

Surface area $=4\times areaofeachface+areaofbase$

Heron’s formula: semi perimeter $s=\frac{a+b+c}{2}$

Area of triangle $=\sqrt{5(5-a)(5-b)(5-c)}$

Calculation:

  $\begin{array}{l}prediction:surfaceareashouldbe21\times 25=525\\ s=\frac{5+50+50}{2}=52.5\\ surfacearea=4\sqrt{52.5\times 47.5\times 2.5\times 2.5}+5\times 5\\ =525\end{array}$