What are the dimensions of the garden with the largest area she can enclose?
The dimensions of the garden with the largest area she can enclose are length for cedar side = 7 foot and width for other side = 10 foot
Let L = length of the fence and W = width of the fence.
Also, let L be the length of cedar fencing.
Since we require one side of cedar fencing and the other 3 sides of cheaper metal fencing, and the cedar fencing costs $13/foot and the cheaper material costs $7/foot, the total cost of fencing is C = 13L + 7L + 2W × 7
= 13L + 7L + 14W
= 20L + 14W.
Since the total amount my friend wants to spend on fencing is $280, C = 280.
So, 20L + 14W = 280
10L + 7W = 140 (1)
Also, the area of the rectangular area is A = LW
From (1), L = (140 - 7W)/10
Substituting L into A, we have
A = LW
A = (140 - 7W)W/10
A = 14W - 0.7W²
To find the value of W that makes A maximum, we differentiate A with respect to W.
So, dA/dW = d(14W - 0.7W²)/dt
dA/dW = 14 - 1.4W
Equating it to zero, we have
14 - 1.4W = 0
14 = 1.4W
W = 14/1.4
W = 10 feet
To determine if this value maximizes A, we differentiate dA/dW with respect to W.
So, d²A/dW² = d(14 - 1.4W)/dW = -1.4
Since d²A/dW² = -1.4 < 0, W = 10 feet is a maximum point.
Since L = (140 - 7W)/10,
substituting W = 10 into the equation, we have
L = (140 - 7W)/10
L = (140 - 7 × 10)/10
L = (140 - 70)/10
L = 70/10
L = 7 feet
So, the dimensions of the garden with the largest area she can enclose are length for cedar side = 7 foot and width for other side = 10 foot
What is the largest area that can be enclosed?
The largest area that can be enclosed is area = 70 foot-squared
Since the area A = LW
Substituting the values of L and W into the e quation, we have
A = LW
A = 7 × 10
A = 70 ft²
So, the largest area that can be enclosed is area = 70 foot-squared
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