A home gardener plans to enclose two rectangular gardens with fencing. The dimensions of the garden: x by 12-x, y by 12-x-y
a. Find the values of x and y that maximize the total area enclosed.
b. What is the maximum total area enclosed?
c. How many meters of fencing are needed?
The total area A=A(x,y).
A=x(12-x)+y(12-x-y)=12x-x^2+12y-xy-y^2
Partial derivatives
A'x=12-2x-y
A'y=12-x-2y
Solve the equations A'x=A'y=0 => a. x=y=4
b. Amax=32+16=48
c. 24+16=40
hey thanks! but where did you get 16 in b?
y(12-x-y)=4(12-4-4)=16
To find the values of x and y that maximize the total area enclosed, we need to consider the area function and apply calculus.
a. The area function in this case is given by A = xy. Since we have two rectangular gardens, the total area enclosed is the sum of the areas of the two gardens: A_total = A1 + A2 = x(12-x) + y(12-x-y).
To find the maximum total area, we need to take the derivative of A_total with respect to both x and y, set them equal to zero, and solve for x and y. The critical points obtained will give us the maximum values.
Let's start by taking the derivative with respect to x:
dA_total/dx = 12 - 2x - y
Setting this derivative equal to zero, we have:
12 - 2x - y = 0
Next, let's take the derivative with respect to y:
dA_total/dy = 12 - 2y - x
Setting this derivative equal to zero, we have:
12 - 2y - x = 0
Now we have a system of equations to solve.
Solving the equations simultaneously will give us the values of x and y that maximize the total area enclosed.
b. To find the maximum total area enclosed, substitute the optimal values of x and y (obtained from part a) into the area function A_total = x(12-x) + y(12-x-y). This will give us the maximum total area.
c. To find the amount of fencing needed, we need to calculate the perimeter of the two rectangular gardens.
The perimeter of the first garden is given by P1 = 2x + 2(12-x) = 24 - 2x.
The perimeter of the second garden is given by P2 = 2y + 2(12-x-y) = 24 - 2x - 2y.
To find the total amount of fencing needed, add the perimeters of the two gardens together:
Total fencing needed = P1 + P2 = (24 - 2x) + (24 - 2x - 2y).
Simplify the expression to get the total amount of fencing needed.