a farmer wants to enclose a rectangle garden using the sides of her barn as one side of the rectangle. what is the maximum area which she can enclose with 60ft of fence? what should the dimensions of the garden be to give this area?
Max area which she can enclose with 60ft of fence in sq.ft?
dimensions to give area is 30ft by___ft?
2 w + L = 60 so L =60-2w
A = w L
A = w (60-2w)
A = -2w^2+60 w
2 w^2 -60 w = -A
w^2 - 30 w = -A/2
w^2 - 30 w + 225 = -A/2 + 450/2
(w-15)^2 =-(1/2)(A-450)
w = 15
L = 60-30 = 30
A = 450
If you know any calculus, you can do this much faster
A = -2w^2+60 w
dA/dw = 0 at extreme = -4 w + 60
so w = 60/4 = 15 at max
I can verify that answer as correct
A batter hits a baseball. The ball's height (in feet) h(t) after t seconds is given by h(t)=-16t^2+160+5
A batter hits a baseball. The ball's height (in feet) h(t) after t seconds is given by h(t)=-16t^2+160+5
To find the maximum area the farmer can enclose with 60 feet of fence, we need to determine the dimensions of the rectangle. Let's break down the problem step by step:
1. Recognize that the fence will consist of three sides of the rectangle, while the barn will serve as the fourth side.
2. Let's call the length of the rectangle (parallel to the barn) "L" and the width (perpendicular to the barn) "W".
3. Based on the information given, we know that 2L + W = 60 ft. This is because we have two lengths and one width, and the total fence length is 60 ft.
4. We can rewrite the equation as 2L = 60 ft - W.
5. Now, let's express the area of the rectangle in terms of L and W. The formula for the area of a rectangle is A = L * W.
6. By substituting the value of 2L from step 4 into the area formula, we get A = (60 ft - W) * W.
7. Expand the equation to get A = 60W - W^2.
8. To find the maximum area, we need to determine the value of W that will maximize the area. This can be done using calculus or by recognizing that the area will be maximized when W is equal to half the fence length (30 ft).
9. Plug in the value W = 30 ft into the equation A = 60W - W^2.
10. Calculate the area: A = 60 * 30 - 30^2 = 1800 - 900 = 900 square feet.
Therefore, the maximum area the farmer can enclose with 60 ft of fence is 900 square feet. The dimensions to achieve this maximum area are 30 ft by 30 ft.