Chapter 12.7, Problem 22PPS

Solution

(a)

To find: the surface area of a square pyramid.

The complete table is

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

Given: Consider the table

Side Length	1	2	3	4
Slant height	10	20	30	40
Surface area

Formulas used:

  $L=\frac{1}{2}Pl$

The surface area $S$ of a square pyramid is given by the formula

  $S=L+B$

Calculation:

The objective is to find the surface area of a square pyramid with side length $1$ and slant height $10$ .

Let $x$ be the side of the base of the square pyramid.

On combing the two formulas the surface area of a square pyramid is given by

  $\begin{array}{l}S=L+B\\ \text{ =}\frac{1}{2}Pl+B\\ \text{ =}\frac{1}{2}.4xl.+x^2\\ \text{ =2xl+}{x}^2\end{array}$

Put $\text{x=1}$ and $\text{l=10}$ in the combine formula for surface area, the surface area ${\text{S}}_1$ of the square pyramid is

  $\begin{array}{l}\text{x=1}\\ \text{l=10}\\ {\text{S}}_1=\text{2xl+}{x}^2\\ \text{ =2}\text{.1}{\text{.10+1}}^2\\ \text{ =20+1}\\ \text{ =21}\end{array}$

Therefore, the surface area of the square pyramid with side length of and a slant height of $\text{10}$ is $\text{21 square units}\text{.}$

Put $\text{x=2}$ and $\text{l=20}$ in the combine formula for surface area, the surface area ${\text{S}}_2$ of the square pyramid is

  $\begin{array}{l}{\text{S}}_2=\text{2xl+}{x}^2\\ \text{ =2}\text{.2}{\text{.20+2}}^2\\ \text{ =80+4}\\ \text{ =84}\end{array}$

Put $\text{x=3}$ and $\text{l=30}$ in the combine formula for surface area, the surface area ${\text{S}}_3$ of the square pyramid is

  $\begin{array}{l}{\text{S}}_3=\text{2xl+}{x}^2\\ \text{ =2}\text{.3}{\text{.30+3}}^2\\ \text{ =180+9}\\ \text{ =189}\end{array}$

Put $\text{x=4}$ and $\text{l=40}$ in the combine formula for surface area, the surface area ${\text{S}}_4$ of the square pyramid is

  $\begin{array}{l}{\text{S}}_4=\text{2xl+}{x}^2\\ \text{ =2}\text{.4}{\text{.40+4}}^2\\ \text{ =320+16}\\ \text{ =336}\end{array}$

Therefore, the complete table is

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

(b)

To explain: the effect to the surface area to the given conditions.

Slant height are double the surface area increased by $\frac{84}{21}$ times that is $\text{4}$

Slant height are double the surface area increased by $\frac{84}{21}$ times that is $9$

Slant height are double the surface area increased by $\frac{84}{21}$ times that is $16$

Explanation:

The objective of state the change in surface area if the base length and slant height are doubled, tripled and multiplied by $\text{4}$

From the table you can say that when the base length and slant height are double the surface area increased by $\frac{84}{21}$ times that is $\text{4}$

From the table you can say that when the base length and slant height are double the surface area increased by $\frac{189}{21}$ times that is $9$

From the table you can say that when the base length and slant height are double the surface area increased by $\frac{336}{21}$ times that is $16$ .

(c)

To predict: surface area of the pyramid.

The surface area of the square pyramid is $525$

Calculation:

The objective is to predict the surface area of the pyramid with the side length of $5$ and slant height of $50$ .

The surface area of the square pyramid is

  $\begin{array}{l}{5}^2.21=25.21\\ \text{ =525}\end{array}$

Conclusion:

Therefore, the surface area of the square pyramid is $525$