An architect created a plan for a four story building with a central courtyard.
The building measures 96 feet on each side. The courtyard measure 40 feet on a side. His clients asked him to work out an alternative design with the same volume but without the courtyard. Find the side length of such a building if it is designed as a four-story square prism.
Any help would be wonderful thanks.
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Hi! :) Well, let's get the volume of the hollow prism first. Let h be the height of the building. The rectangular prism without the courtyard would has a volume of (96 ft)(96 ft)(h ft)=9216h ft^3. Now, with the courtyard, imagine cutting out the center of a sytrofoam box. The volume would then be the original minus the removed center. Therefore, the volume of the building WITH the courtyard would have to be VOL w/o courtyard - VOL of courtyard = VOL of initial plan of architect. The volume of the courtyard is (40 ft)(40 ft))(h ft)=1600h ft^3. VOL of initial plan, therefore, is 7616h ft^3.
From there, we can answer the problem. The side length is asked. Without any courtyard, we want to achieve the same volume as before.
Let s be the side length.
7616h ft^3 = (s ft)(s ft)(h ft)
7616h ft^3 = (s^2)(h) ft^3
7616 = s^2
s= sqrt (7616) ft