farmer wishes to fence a rectangular area along the river bank. No fence is required on the side adjacent to the river. The material for the fence costs P16.00 per meter for the side parallel to the river, P12.00 per meter for the side perpendicular to the river. The farmer has a budget of P12 000.00. Find the dimension of the lot that will have the largest possible area.
? :O i'm not sure how to solve this. :(
Assuming farmer uses up his budger of 12000, and let
L=length along river (costs 16/m)
w=width (costs 12/m)
L=(12000-12w*2)/16
Area,
A(w)=w(12000-24w)/16
=750w-(3/2)w²
Differentiate with respect to w and equate to zero for maximum:
A'(w)=750-3w=0
w=250
L=(12000-24w)/16=375
L= 60m
W= 30m
No worries! I'll help you break down the problem step by step.
To find the dimensions of the lot that will have the largest possible area within the given budget, we need to consider the costs and constraints mentioned in the question.
Let's denote the length of the side parallel to the river as L and the length of the side perpendicular to the river as W. The area of the rectangular lot is given by A = L * W.
Based on the given information, the cost of fencing is P16.00 per meter for the side parallel to the river and P12.00 per meter for the side perpendicular to the river. Therefore, the total cost C can be calculated as:
C = 2L * P16.00 + 2W * P12.00
Since the farmer has a budget of P12,000.00, we can write the equation:
C = 2L * P16.00 + 2W * P12.00 = P12,000.00
Now, we need to express the total cost C in terms of one variable. Let's solve the equation for W:
2L * P16.00 + 2W * P12.00 = P12,000.00
2L * P16.00 = P12,000.00 - 2W * P12.00
2L * P16.00 = P12,000.00 - 24W
L = (P12,000.00 - 24W) / (2 * P16.00)
Next, substitute this expression for L in the area equation A = L * W:
A = [(P12,000.00 - 24W) / (2 * P16.00)] * W
Now we have the area A in terms of the variable W. To find the dimensions of the lot that will have the largest possible area, we need to find the value of W that maximizes the area A.
To do this, we can take the derivative of the area A with respect to W, set it equal to zero, and solve for W:
dA/dW = [(P12,000.00 - 24W)(2 * P16.00) - W(2P12.00)] / (2 * P16.00)^2 = 0
Solving this equation will give us the value of W that maximizes the area. Once we have W, we can substitute it back into the expression for L to find the corresponding value of L.
After finding the values of W and L, you will have the dimensions of the rectangular lot that will have the largest possible area within the given budget.