Mr. Snooty Curmudgeon donated a nice sum of $9109360.96 for construction of a ve
ID: 2867672 • Letter: M
Question
Mr. Snooty Curmudgeon donated a nice sum of $9109360.96 for construction of a very boxy business building. As part of the donation, Mr. H. demands a square base, and that we use exactly the materials he desires. Naturally, the university wants the largest possible volume for the building. The foundation costs $40 per square foot, the siding costs $20 per square foot, and the roof costs $60 per square foot.
Give a formula for the total cost of a boxy business building if the length is x feet and the height is h feet: ______ dollars.
What is the formula for the volume only in terms of x?_______
What are the dimensions of the best boxy business building? Length is____ feet, width is _____feet, and the height is____feet.
What is the maximum volume that his money ($9109360.96) can buy?_____ cubic feet.
PLEASE SHOW ALL WORK
Explanation / Answer
Mr. Snooty wants a cuboidal building, the buildings base will have a square base with side = x feet
the sides will be rectangular in shape with length = x feet and width = h feet
the roof will be square shaped as well, with sides = x feet
Area of the base = x^2 square feet
Area of the side = x*h square feet
Area of the roof = x^2 square feet
the foundation costs = $ 40 per square foot
therefore cost for x^2 square feet = $ 40x^2
the sides cost = $ 20 per square foot
there will be 4 sides
therefore the cost of the sides = 4*20*xh = $ 80*x*h
the roof costs = $ 60 per square foot
therefore the cost of the roof = $ 60*x^2
the total cost of the Boxy building is = $ [40x^2 + 80xh + 60x^2]
total cost = $ [100x^2 + 80xh]
Volume of the building is V = x^2*h
We know the total cost is = $ 9109360.96
therefore, 100x^2 + 80xh = 9109360.96
h= (9109360.96 - 100x^2)/(80x)
thus, Volume = x^2*(9109360.96 - 100x^2)/(80x)
V = x*(9109360.96 - 100x^2)/80 <==== Volume in terms of x
For the best diamensions we'll have to differentiate the above oquation
V '(x) = -3.75(x^2-30364.5)
V '(x) = 0
width = 174. 254 ft
height = 437.637 ft
plug the value of x in equation
h= (9109360.96 - 100x^2)/(80x)
the diamenstions for the best possible building will be length
x = 174. 254 ft
width = 174. 254 ft
height = 437.637 ft
maximum Volume is = $ 13227900
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