Question

I dont understand the ?

Square pyramids A and B are similar. In pyramid​ A, each base edge is 9 cm. In pyramid​ B, each base edge is 3 cm and the volume is 12 cmcubed.
a. Find the volume of pyramid A.
b. Find the ratio of the surface area of A to the surface area of B.
c. Find the surface area of each pyramid

Answer 1

(a) Volume of pyramid A is `324cm^3`
(b) Ratio is `9:1`
(c) Surface area of pyramid A is `34.632` `cm^2` and that of pyramid A is `311.688` `cm^2`.

Explanation:

In similar geometrical bodies if sides are in the ratio of `a:1`, surface areas are in the ratio `a^2:1` and volumes are in the ratio `a^3:1`. As here the ratio of base edges is `3:1`, surface areas must be inn the ratio of `9:1` and volumes in the ratio `27:1`.

In a square pyramid, if `a` is the side of base square and `h` is the height,

its volume is `1/3a^2h`, lateral surface area is `asqrt(4h^2+a^2)` and total surface area is `a^2+asqrt(4h^2+a^2)`.

Now in similar geometric bodies all sides are proportuonal. So if base of pyramid A is `3`-times that of pyramid B, so is the height of pyramid A is `3`-times that of pyramid B's height.

Now let us solve the questions in reverse order.

(c) As volume of pyramid B is `12cm^3`, we have `1/3xx3^2xxh=12` or `h=4cm.` and its total surface area is `3^2+3sqrt(4*4^2+3^2)=9+3sqrt73=34.632` `cm^2`.

The height of pyramid is `4xx3=12`, hence its total surface area is `9^2+9sqrt(4*12^2+9^2)`
= `81+9sqrt(576+81)=81+9sqrt657=81+27sqrt73=311.688` `cm^2`.

(b) Ratio of total surface areas of pyramids is hence

`(81+27sqrt73)/(9+3sqrt73)=9`

(a) Volume of pyramid A is `1/3xx9^2xx12=27xx12=324` and hence ratio of volumes of pyramids is `(27xx12)/12=27:1`