this is my last question from applications of derivatives!
A 216m^2 rectangular pea patch is to be enclosed by a fence and divided into two equal parts by another fence parallel to one of the sides. What dimensions for the outer rectangle will require the smallest total length of fence? How much fence will be needed.
Thank you all for helping.
Let's suppose your dividing fence is vertical and has a length of x. Let the length of the other side be y.
Since the area must be 216, y = 216/x
Perimeter P = 3x + 2y which after substituting is equal to 3x + 432/x
Find derivative: dP/dx = 3 - 432/x^2. Set it to zero and solve for x.
3x^2 = 432
x^2 = 144
We can accept the positive solution only. Therefore, x = 12.
y = 18 (sub 12 for x into y = 216/x)
The dimensions are 18m and 12m
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The total length of fence needed is 3x + 2y = 3(12) + 2(18) = 78m.
To find the dimensions for the outer rectangle that will require the smallest total length of fence, we can use calculus.
First, let's label the length of the dividing fence as x and the length of the other side as y.
Since we know the area of the rectangular pea patch is 216m^2, we can set up the equation y = 216/x (since the area of a rectangle is the product of its length and width).
Now, let's find the perimeter of the rectangular pea patch. The perimeter is given by the equation P = 2x + 2y. However, since we are dividing the patch into two equal parts with a dividing fence, we only need to calculate the length of the outer perimeter. So, the length of the outer perimeter will be P = 3x + 2y.
Now, let's substitute y with 216/x in the equation for the perimeter: P = 3x + 2(216/x). Simplifying this equation, we get P = 3x + 432/x.
To find the dimensions that require the smallest total length of fence, we can consider the derivative of the perimeter function with respect to x (dP/dx). Let's find the derivative:
dP/dx = 3 - 432/x^2
Next, we set the derivative equal to zero and solve for x to find the critical points:
3 - 432/x^2 = 0
Simplifying this equation, we get:
3x^2 = 432
Now, divide both sides of the equation by 3:
x^2 = 144
Taking the square root of both sides, we get:
x = ±12
Since the dimensions cannot be negative, we can accept the positive solution: x = 12.
Now, substitute x = 12 into the equation y = 216/x to find the value of y:
y = 216/12
y = 18
Therefore, the dimensions for the outer rectangle that require the smallest total length of fence are 18m and 12m.
To find out how much fence will be needed, we can substitute these dimensions into the equation for the perimeter:
P = 3x + 2y
P = 3(12) + 2(18)
P = 36 + 36
P = 72
So, the length of the dividing fence needed will be 72 meters.
To find the dimensions for the outer rectangle that will require the smallest total length of fence, we can use the given area of 216m^2.
Let's label the length of the rectangle as y and the width as x. Since the area must be 216m^2, we have the equation:
xy = 216
Solving for y, we find:
y = 216/x
The total length of the fence is given by the perimeter of the rectangular pea patch. The perimeter P is given by:
P = 2x + 2y
Substituting y = 216/x, we get:
P = 2x + 2(216/x)
Simplifying the equation, we have:
P = 2x + 432/x
To find the dimensions that will require the smallest total length of fence, we need to minimize the function P with respect to x. We can do this by finding the critical points of P.
To find the critical points, we take the derivative of P with respect to x:
dP/dx = 2 - 432/x^2
Setting dP/dx equal to zero and solving for x, we have:
2 - 432/x^2 = 0
Rearranging the equation, we get:
2 = 432/x^2
Simplifying further, we have:
x^2 = 432/2
x^2 = 216
Taking the square root of both sides, we find:
x = sqrt(216)
Since we're dealing with a length, we can accept the positive square root. Therefore, x = sqrt(216) = 12.
Substituting this value of x back into the equation y = 216/x, we find:
y = 216/12 = 18
So the dimensions for the outer rectangle that will require the smallest total length of fence are 18m (length) and 12m (width).
To find the amount of fence needed, we calculate the perimeter P with these dimensions:
P = 2(18) + 2(12)
P = 36 + 24
P = 60
Therefore, the total length of fence needed is 60m.