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Please help me solve this area and perimeter word problem.

Please help me solve this area and perimeter word problem.-example-1

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Explanation:

x = length

y = width

let's assume the side with the adjoining road is a length. we will see that given the properties of a rectangle it does not matter. we will get the same result either way.

the perimeter is

2x + 2y = perimeter

the area is

x × y = area

we know that

2×8×y + 1×8×x + 1×12×x = 1200

16y + 20x <= 1200 (but can safely take this as "=" to get the maximum area based on the maximum possible perimeter)

20x = 1200 - 16y

x = (1200 - 16y)/20 = 60 - 16y/20 = 60 - 4y/5

we use this in the area equation :

area = f(y) = (60 - 4y/5) × y = 60y - 4y²/5

this function has an extreme value at the zero of the first derivative :

f'(y) = 60 - 8y/5 = 0

-8y/5 = -60

8y = 300

y = 300/8 = 75/2 = 37.5 ft

x = 60 - 4y/5 = 60 - 4×37.5/5 = 30 ft

the second derivative tells us, if it is a max. or a min.

if it is positive, the point is a relative minimum, and if it is negative, the point is a relative maximum.

f''(y) = -8/5 negative for y = 37.5, so it is a max.

now, if we assume that the side with the adjoining road is a width, we only need to switch x and y for each other in all the equations above. we get f(x) instead of f(y), but that does not matter, we get the same equations with the same results : one type of side is 30 ft, the other is 37.5 ft long. the side with the adjoining road is of the 30 ft type.

perimeter = 2×30 + 2×37.5 = 60 + 75 = 135 ft

area = 30 × 37.5 = 1,125 ft²

the dimensions that form the maximum area are

30 ft × 37.5 ft.

one of the 30 ft sides is along the adjoining road.

this max. area is therefore 1,125 ft²

User Fogedi
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