Respuesta :
Answer:
(a). The length of the field is 1,300 - 2·x
(b) The area of the rectangular field is 1,300·x - 2·x²
(c) x leading to the maximum area is 325 meters
(d) The maximum area is 211,250 m²
Step-by-step explanation:
The given parameters are;
The length of fencing the compound owner has = 1,300 m
The shape of the field he wants to enclose = Rectangular field
The width of the field he wants to enclose = x
The fence on the side along the river = No fencing along the river
(a). Let 'l' represent the length of the field, we, have;
2·x + l = 1.300
∴ l = 1,300 - 2·x
The length of the field, l = 1,300 - 2·x
(b) The area of a rectangular field, A = Length of the field, l × Width of the field, w
Therefore;
The area of the rectangular field, A = x × (1,300 - 2·x) = 1,300·x - 2·x²
The area of the rectangular field = 1,300·x - 2·x²
(c) The coefficient of x² is negative, therefore, the graph of the area has a maximum point
The value of 'x' leading to maximum area is given by differentiating the function representing the area, equating the result to 0, and fining the value of 'x' as follows;
d(A)/dx = d(1,300·x - 2·x²)/dx = 1,300 - 4·x
d(A)/dx = 0 at the maximum point
∴ 1,300 - 4·x = 0
4·x = 1,300
x = 1,300/4 = 325
x leading to the maximum area = 325 meters
(d) The maximum area, [tex]A_{max}[/tex] is given by A = 1,300·x - 2·x², when x = 325
∴ [tex]A_{max}[/tex] = 1,300 m × 325 m - 2 × (325 m)² = 211,250 m²
The maximum area, [tex]A_{max}[/tex] = 211,250 m²
