Respuesta :
Answer:
The dimensions are x = 23 and y = 23
The maximum area of the rectangle A = 529
Step-by-step explanation:
Step(i):-
Let given function area of the rectangle
A = x y ....(i)
Given Perimeter of rectangle
2 ( x + y ) = 92
(x +y ) = 46
y = 46 - x ....(ii)
Substitute y = 46 - x in equation (i)
A = x (46 - x)
A = 46 x - x² ....(iii)
Differentiating equation (iii) with respective to 'x'
[tex]\frac{dA}{dx} = 46(1) - 2 x[/tex] ...(iv)
Step(ii):-
[tex]\frac{dA}{dx} = 46(1) - 2 x = 0[/tex]
46 - 2 x = 0
46 = 2 x
x = 23
Again Differentiating equation (iv) with respective to 'x'
[tex]\frac{d^{2} A}{dx^{2} } = - 2 (1) = -2 < 0[/tex]
The maximum value at x = 23
Step(iii):-
From(ii)
y = 46 - x
y = 46 - 23
y = 2 3
The dimensions are x = 23 and y = 23
Final answer :-
The dimensions are x = 23 and y = 23
The maximum area of the rectangle
A = x y
A = 23 ×23
A = 529
The maximum area of the rectangle = 529
The dimension of the rectangle with perimeter 92 m whose area is as large as possible is 23m by 23m
If the perimeter of a rectangle is 92m, hence;
2(x+y) = 92
x is the length
y is the width
x + y = 46 ....... 1
If the area is as large as possible, hence;
xy = maximum .........2
From equation 1, x = 46 - y
Substitute into the second equation
(46-y)y = P(y)
P(y) = 46y - y²
If the product is at the maximum, hence;
dP/dy = 46 - 2y = 0
46 - 2y = 0]
2y = 46
y = 23
Since x + y = 46
x = 46 - y
x = 46 - 23
x = 23
Hence the dimension of the rectangle with perimeter 92 m whose area is as large as possible is 23m by 23m
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