Chapter 12.7, Problem 24HP

Solution

To explain: whether pyramid has greater surface area or cone has greater surface area.

Surface area of cone is greater than the surface area of the Square pyramid.

Given:

Cone:diameter is x units and slant height is $l$ units.

Square pyramid: length is $x$ units and a slant height $l$ units

Calculation:

The objective is to state which figure has greater surface area.

The lateral area $L$ of a cone is given by the formula

  $L=\pi rl$

Where $r$ is the radius of the circle and $l$ is the slant height the cone

The surface are $S$ of a cone is given by the formula

  $S=L+B$

Where $L$ the lateral is surface area and $B$ is the area of the base that is circle.

The lateral area $L$ of a Square pyramid is given by the formula

  $L=\frac{1}{2}Pl$

Where $P$ is the perimeter of the base and $l$ is the slant height the Square pyramid.

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

  $S=L+B$

Where $L$ the lateral is surface area and $B$ is the sera of the base.

First find the lateral are of the cone.

Put $r=x$ and $l=l$ lateral area formula to find the lateral area of the cone.

Thus,

  $L=\pi xl$

The area $B$ of the cone is given by

  $\begin{array}{l}B=\pi r^2\\ \text{ =}\pi {(}^x\\ \text{ =}\pi x^2\end{array}$

Put $B\text{=}\pi x^2$ and $L\text{=}\pi xl$ in the surface are formula to find the surface area of the cone.

  $\begin{array}{l}S=L+B\\ \text{ =}\pi xl+\text{=}\pi x^2\\ \text{ =}\pi (xl+x^2)\\ \approx 3.14(xl+x^2)\\ =3.14xl+3.14x^2\end{array}$ [Since $\pi =3.14$ ]

Next find the lateral are of the Square pyramid.

The side of the square is $x$ units.

The perimeter $P$ of the square is

  $P=4(side)$

Put $side=x$ in the perimeter formula to find the perimeter of the square.

Thus,

  $\begin{array}{l}P=4.x\\ \text{ =4x}\end{array}$

The slant height of the Square pyramid is $l$ units.

Put $P=4x$ and $\text{l=l}$ in the lateral surface area to find the lateral surface area of the Square pyramid.

  $\begin{array}{l}L=\frac{1}{2}Pl\\ \text{ =}\frac{1}{2}\mathrm{.4.}x.l\\ \text{ =2xl}\end{array}$

The area $\text{B}$ of the base is given by

  $\begin{array}{l}{\text{B=(side)}}^2\\ {\text{ =x}}^2\end{array}$

Put $B=x^2$ and $L=2xl$ in the surface area formula to find the surface area of the Square pyramid.

  $\begin{array}{l}S=L+B\\ \text{ =2xl+}{x}^2\end{array}$

Since the value of $\pi$ is greater than $2$ . So $\pi (\text{xl+}{x}^2)>2\text{xl+}{x}^2$

Therefore, surface area of cone is greater than the surface area of the Square pyramid.