Question

PLEASE HELP!! Need to finish my quiz!!

Juan wants to build a greenhouse in his back yard. He needs to maximize the ratio of floor area to structure volume in order to provide the most room for plants while minimizing the area to be heated.
Find the description of the structure that best meets Juan’s criteria.
A.) A hemisphere with a diameter of 12 feet
B.) A cube with a side length of 12 feet
C.) A cone with a diameter of 12 feet and a height of 10 feet
D.) A square pyramid with a base side length of 12 feet and a height of 9 feet

Answer 1

Square

Explanation:

Given are 3 options with same dimension of 12 feet. The condition is base area should be minimum for maximum volume

A) A hemisphere has

`V=(2)/(3) \pi ((12)/(2) )^3\\=452.57\\Base area = \pi ((12)/(2) )^2\\=113.04\\Ratio=4.00`

B) A cube with side length 12 ft

`V=6^3 =216\\Base area = 36\\Ratio = 6`

C) A cone with diameter 12 ft and height 10 ft

`V=(1)/(3) \pi(6^2)10\\=377.14\\Base area =113.04\\Ratio =3.34\\`

D) A square pyramid

`V=(lwh)/(3) \\=(144(9))/(3) \\=432\\Base area = 144\\Ratio =3`

To maxmize the ratio of floor area to volume, we have to maximise the ratio of volume to floor area

Hence of these 4 figures square is the best one that meets the criteria