A rancher has 296 feet of fencing to enclose two adjacent rectangular corrals. What dimensions will produce the largest total area?

What is the maximum total area?

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To find the dimensions that will produce the largest total area, we can use the concept of optimization. In this case, we need to maximize the total area of the two adjacent rectangular corrals.

Let's assume the length of one corral to be 'x' feet. Since there are two adjacent corrals, the length of the other corral will also be 'x'. The width of each corral will be denoted by 'y'.

To find the total area, we multiply the length and width of each corral and sum them up. The total area (A) can be calculated as:

A = 2xy

Now, we know that the total amount of fencing available is 296 feet. The perimeter of each corral consists of the sides parallel to the length (2x) and the sides parallel to the width (2y). So, the total amount of fencing required is:

4x + 4y = 296

Simplifying the equation, we can express 'y' in terms of 'x':

y = (296 - 4x) / 4
y = (74 - x) / 1

We can substitute this value of 'y' in the equation for the area (A):

A = 2xy
A = 2x(74 - x)

To find the values of 'x' and 'y' that maximize the area, we can use calculus. Taking the derivative of A with respect to x:

dA/dx = 2(74 - 2x)

Setting the derivative equal to zero:

2(74 - 2x) = 0
148 - 4x = 0
4x = 148
x = 37

Substituting this value of 'x' back into the equation for 'y':

y = (74 - x) / 1
y = (74 - 37) / 1
y = 37

So, the dimensions that will produce the largest total area are 37 feet by 37 feet for each corral.

To find the maximum total area, substitute the values of 'x' and 'y' into the area equation:

A = 2xy
A = 2(37)(37)
A = 2(1369)
A = 2738 square feet

Therefore, the maximum total area is 2738 square feet.

A=xy

3x+2y=296
2y=-3x+296
y=-1.5x+148

296=x(-1.5x+148)
= -1.5x^2+148x-296