Answer to Question #148341 in Calculus for Angelo

Question #148341

A one-story building having a rectangular floor space of 13,200 𝑓𝑑^2 is to be constructed

where a 22 ft walkway is required in the front and back, and a 15 ft walkway is required on each side. Find the dimensions of the lot having the least area on which this building can be located.


1
Expert's answer
2020-12-10T11:19:58-0500

Consider the rectangular floor of length "x" and width "y".


Given, "xy=13200" , so "y=\\frac{13200}{x}"


The area of the lot on which the building is to be constructed is,


"A=(x+30)(y+44)"


Substitute "y=\\frac{13200}{x}" to express the area in a single variable "x" as,


"A(x)=(x+30)(\\frac{13200}{x}+44)"


"=13200+44x+\\frac{396000}{x}+1320"


"=14520+44x+\\frac{396000}{x}"


Differentiate "A(x)" with respect to "x" and set it equal to zero in order to find the critical value as,


"A'(x)=0"


"44-\\frac{396000}{x^2}=0"


"x^2=9000"


"x=30\\sqrt{10}"


Here, "A"(x)=\\frac{792000}{x^3}>0" for "x=30\\sqrt{10}" , so "A" is minimum at "x=30\\sqrt{10}"


Therefore, the dimensions of the floor are:"x=30\\sqrt{10}\\approx94.87" ft"y=\\frac{13200}{30\\sqrt{10}}=44\\sqrt{10}\\approx139.14" ftThe dimensions of the lot are:
"x+30=30\\sqrt{10}+30\\approx124.87 ft"
"y+44=44\\sqrt{10}+44\\approx 183.14 ft"

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