Chapter 12.2, Problem 8PSB

a.

To determine

To calculate: The lateral area and total area of pyramid PABCD if each side of base has a length of 14 and altitude PQ is 24.

Answer to Problem 8PSB

The lateral area is $700$ and total area is $896$ .

Explanation of Solution

Given information:

Side AB=BC=CD=DA = 14,

Side PQ = 24.

Formula used:

The below theorem is used:

Pythagoras theorem states that “In a right angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides”.

  

In right angle triangle,

  $a^2+b^2\text{= }{c}^2$

Area of triangle: $A=\frac{1}{2}\times b\times h$

b = base of triangle

h = height of triangle Area of square: $A=s^2$

s = side of square

Calculation:

  

  $RQ=\frac{1}{2}CB=\frac{1}{2}\times 14=7$

Side PR can be calculated by applying Pythagoras Theorem.

In right angle triangle PQR , we get

  $\begin{array}{l}{\left(PQ\right)}^2+{\left(QR\right)}^2={\left(PR\right)}^2\\ {\left(24\right)}^2+{\left(7\right)}^2={\left(PR\right)}^2\\ 576+49={\left(PR\right)}^2\\ {\left(PR\right)}^2=625\\ \left(PR\right)=\sqrt{625}\\ PR=25\end{array}$

Area of triangle:

  $\begin{array}{l}A=\frac{1}{2}\times DC\times PR\\ A=\frac{1}{2}\times 14\times 25\\ A=175\end{array}$

Lateral Area = 4 $\times$ Area of triangle

Lateral Area $=4\times 175=700$

Area of base ABCD $={\left(14\right)}^2=196$

Total Area = Lateral Area + Area of base ABCD

Total Area $=700+196$

Total Area $=896$

b.

To determine

To find: The lateral area and total area of pyramid PABCD if each slant height is 17 and altitude PQ is 15.

Answer to Problem 8PSB

The lateral area is $700$ and total area is $896$ .

Explanation of Solution

Given information:

Side PA=PB=PC=PD = 17,

Side PQ = 15.

Formula used:

The below theorem is used:

Pythagoras theorem states that “In a right angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides”.

  

In right angle triangle,

  $a^2+b^2\text{= }{c}^2$

Area of triangle: $A=\frac{1}{2}\times b\times h$

b = base of triangle

h = height of triangle Area of square: $A=s^2$

s = side of square

Calculation:

  

  $PS=PA=17$

Side QS can be calculated by applying Pythagoras Theorem.

In right angle triangle PQS , we get

  $\begin{array}{l}{\left(PQ\right)}^2+{\left(QS\right)}^2={\left(PS\right)}^2\\ {\left(15\right)}^2+{\left(QS\right)}^2={\left(17\right)}^2\\ {\left(QS\right)}^2=289-225\\ {\left(QS\right)}^2=64\\ \left(QS\right)=\sqrt{64}\\ QS=8\\ \overline{RQ}=\overline{QS}=8\\ \overline{CB}=2\overline{RQ}=16\\ \overline{CB}\cong \overline{DC}=16\end{array}$

Area of triangle:

  $\begin{array}{l}A=\frac{1}{2}\times CB\times PS\\ A=\frac{1}{2}\times 16\times 17\\ A=136\end{array}$

There are four triangles.

Lateral Area = 4 $\times$ Area of triangle

Lateral Area $=4\times 136=544$

Area of base ABCD $={\left(16\right)}^2=256$

Total Area = Lateral Area + Area of base ABCD

Total Area $=544+256$

Total Area $=800$