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NANA2007
13.10.2020 •
Mathematics
Can someone please help me?
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Ответ:
a) A(r) = ( 1/2) * (750*x - 5*x²)
b) Dimensions
x = 75 ft
y = 187,5 ft
A (max) = 14062,5 ft²
Step-by-step explanation:
Fencing material available 750 ft
Rectangula area A(r)
Let x and y dimensions of rectangular area, and x the small side of the rectangle, then
The perimeter of the rectangle is
P(r) = 2*x + 2*y (1)
To get the four pens we have to place three more fence in between the two x sides of the rectangle, in such way that the total fence is
P(r) + 3*x = 750
So 2*x + 2*y + 3*x = 750 ⇒ 5*x + 2*y = 750 ⇒ y = (750- 5x)/2
Plugging that value in (1)
A(r) = x* y
A(r) = x* ( 750 - 5*x)/ 2 ⇒ A(r) = ( 1/2) * (750*x - 5*x²)
Taking derivatives in both sides of the equation we get:
A´(r) = (1/2) * ( 750 - 5*x) ⇒ A´(r) = 0 ⇒ (1/2) * ( 750 - 10*x) = 0
750 - 10*x = 0 ⇒ x = 750/10 ⇒ x = 75 ft
And y would be
y = (750- 5x)/2 ⇒ y = (750 - 5*75) / 2 ⇒ y = 375 / 2
y = 187,5 ft
A(max) = 187,5*75
A(max) = 14062,5 ft²