Chapter 12.3, Problem 35PPS

a.

To determine

To sketch: The square pyramid with a base edge 3 units.

Explanation of Solution

Given:

The edge of square based pyramid is 3 units.

Concept used:

The square based pyramid is a figure with slant height $l=\sqrt{h^2+{\left(\frac{a}{2}\right)}^2}$ units where “ a” is the length of edge of square.

Sketch:

The sketch of square based parameter of edge 3 units is shown in figure here.

b.

To determine

To write:A tableshowing the lateral area of the pyramid for slant heights 3 units and 9 units.

Answer to Problem 35PPS

The lateral areas of the pyramid for slant heights 3 units and 9 units are 18 units and 54 units.

Explanation of Solution

Given:

The base of the pyramid is square of edge 3 units and slant heights 3 units and 9 units.

Formula/ concept used:

The lateral area of the pyramid of base perimeter P and slant height l is given by $L=\frac{1}{2}Pl$

Tabulation:

The table showing the lateral areas of the pyramid for slant heights 3 units and 9 units is given below:

Edge of base( square)	Perimeter	L for l = 3 units	L for l = 9 units
3 units	$4\times 3=12$	$\frac{1}{2}Pl=\frac{1}{2}\cdot 12\cdot 3=18$ units	$\frac{1}{2}Pl=\frac{1}{2}\cdot 12\cdot 9=54$ units

Conclusion:

The lateral areas of the pyramid for slant heights 3 units and 9 units are 18 units and 54 units.

c.

To determine

To describe:The effect on the lateral area of the pyramid when slant height tripled.

Answer to Problem 35PPS

When slant height of pyramid is tripledthe lateral area is also tripled.

Explanation of Solution

Given:

The base of the pyramid is square of edge 3 units and slant height is tripled

Formula/ concept used:

The lateral area of the pyramid of base perimeter P and slant height l is given by $L=\frac{1}{2}Pl$

Let the initial lateral area of the pyramid is $L=\frac{1}{2}Pl$ . Since, base of the pyramid remains as it is only slant height is tripled so new lateral area $L^{\prime }=\frac{1}{2}P\left(3l\right)=3\left(\frac{1}{2}Pl\right)=3L$ .

Thus,when slant height of pyramid is tripled the lateral area is also tripled.

Conclusion:

When slant height of pyramid is tripled the lateral area is also tripled.

d.

To determine

To make:A conjecture about the lateral area of a square pyramid.

Answer to Problem 35PPS

The conjecture about the square pyramid is:

The lateral area ( L ) of a square pyramid is directly proportional to the edge ( a ) of base square and the slant height ( l ), i.e., $L\propto a$ , and $L\propto l$ .

Explanation of Solution

Given:

The slant height and base edge of squarepyramid are tripled

Formula/ concept used:

The lateral area of the pyramid of base perimeter P and slant height l is given by $L=\frac{1}{2}Pl$

Let the initial lateral area of the pyramid is $L=\frac{1}{2}Pl$ . The perimeter of the square pyramid of base edge “ a ” is 4a . Therefore, the lateral area of square pyramid is given by

  $L=\frac{1}{2}Pl=\frac{1}{2}\times 4a\times l=2al$

$\Rightarrow L\propto a$ , and $L\propto l$ as 2 is a constant.

Thus, the conjecture about the square pyramid is:

The lateral area ( L ) of a square pyramid is directly proportional to the edge ( a ) of base square and the slant height ( l ), i.e., $L\propto a$ , and $L\propto l$ .

Conclusion:

The conjecture about the square pyramid is:

The lateral area ( L ) of a square pyramid is directly proportional to the edge ( a ) of base square and the slant height ( l ), i.e., $L\propto a$ , and $L\propto l$ .