Respuesta :
Answer:
y = 70 yd
A(max) = 9800 yd²
Step-by-step explanation:
Let call " y " sides of the rectangular area running perpendicular to the river, then:
Perimeter of the area (without one side is )
P = x + 2*y and we have 280 yards of fencing material
280 = x + 2*y ⇒ y = ( 280 - x ) / 2
And rectangular area is:
A = x*y
Area as a function of x is:
A(x) = x * ( 280 - x ) / 2
A(x) = 280*x /2 - x²/2
Taking derivatives on both sides of the equation we get:
A´(x) = 140 - x = 0
140 - x = 0
x = 140 yd
And y
y = ( 280 - x ) / 2
y = 140 /2
y = 70 yd
And
A( max ) = 140*70
A(max) = 9800 yd²
The length of each side running perpendicular to the river is 70 yards.
The length of the side running parallel to the river is 140 yards(recall its just one side)
The largest total area that can be enclosed is 140 × 70 = 9800 yards²
let
the length of each side running perpendicular to the river = x
the length of the side running parallel to the river = y
perimeter of the fence = 2x + y (because of the river)
280 = 2x + y
y = 280 - 2x
Area = xy
Therefore,
Area = x(280 - 2x)
Area = 280 - 2x²
Area = -2x² + 280
This is a parabola facing downward because the leading coefficient is less than zero.
The maximum point are (h, k).
h = - b / 2a (this gives the maximizing point)
h = - 280 / 2 × - 2
h = -280 / -4
h = 70
Therefore,
x = 70 yards
- The length of each side running perpendicular to the river is 70 yards.
- The length of the side running parallel to the river is 140 yards(recall its just one side)
- The largest total area that can be enclosed is 140 × 70 = 9800 yards²
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