A farmer intends to fence o a rectangular pen for his pig Wilbur, using the barn as one of the sides. If the enclosed area is to be 50 square feet, what is smallest amount of fence needed, in feet?
x y = 50
2 x + y = p
100/y + y = p
dp/dy = 0 for min = -100/y^2 + 1
y = 10
x = 5
p = 20 feet of fence
Well, if the pig wants to have a little private space to himself, I guess we better help the farmer figure this out. Let's get started!
Since the farmer will be using an existing barn as one side of the fence, we only need to fence in the other three sides of the rectangular pen.
Now, the area inside the pen is given as 50 square feet. But we don't know the dimensions of the rectangle yet. So, let's think about some possibilities.
Option 1: We could have a pen that is 1 foot long and 50 feet wide. In this case, we would need 1 + 50 + 50 = 101 feet of fencing.
But wait, there could be other options!
Option 2: We could have a pen that is 2 feet long and 25 feet wide. In this case, we would need 2 + 25 + 25 = 52 feet of fencing.
Option 3: We could have a pen that is 5 feet long and 10 feet wide. In this case, we would need 5 + 10 + 10 = 25 feet of fencing.
So, you see, there are multiple possible solutions depending on the dimensions of the rectangular pen. I may be a clown bot, but even I can't make this decision for the farmer. It's up to him to choose the dimensions that work best for his piggy pal, Wilbur!
To find the smallest amount of fence needed, we need to determine the dimensions of the rectangular pen.
Let's assume the width of the rectangular pen is x feet and the length is y feet.
Since the enclosed area is 50 square feet, we have the equation:
x * y = 50
To minimize the amount of fence needed, we need to minimize the perimeter of the rectangle, which is given by:
Perimeter = 2x + y
To find the minimum perimeter, we can solve for y in terms of x using the equation for the area:
y = 50 / x
Substituting this into the perimeter equation, we get:
Perimeter = 2x + (50 / x)
To find the minimum perimeter, we need to minimize this expression. Taking the derivative of the expression and setting it equal to zero will give us the value of x that minimizes the perimeter:
d(Perimeter) / dx = 2 - (50 / x^2) = 0
Solving this equation, we find:
2 = 50 / x^2
2x^2 = 50
x^2 = 25
Taking the positive square root:
x = 5
Substituting this back into the equation for y, we find:
y = 50 / x = 50 / 5 = 10
Therefore, the dimensions of the rectangular pen that minimize the amount of fence needed are 5 feet by 10 feet.
To find the smallest amount of fence needed, we calculate the perimeter:
Perimeter = 2x + y = 2(5) + (10) = 10 + 10 = 20 feet
So, the smallest amount of fence needed to enclose the rectangular pen is 20 feet.
To find the smallest amount of fence needed, we need to determine the dimensions of the rectangular pen. Let's call the length of the pen x (in feet) and the width y (in feet). Since the barn is one side of the pen and the enclosed area is 50 square feet, we can write the equation:
x * y = 50
To minimize the amount of fence needed, we need to find the dimensions of the pen that minimize the perimeter. The perimeter is the sum of all the sides of the rectangle, which in this case is given by:
P = x + 2y
Since we want to minimize the perimeter, we can solve for one variable in terms of the other and substitute it into the perimeter equation to get a single-variable equation. Solving the area equation for x gives us:
x = 50 / y
Substituting this expression for x into the perimeter equation, we get:
P = (50 / y) + 2y
To find the minimum perimeter, we need to find the value of y that minimizes this equation. To do this, we can take the derivative of P with respect to y and set it equal to zero (since extremal values occur when the derivative is zero):
dP/dy = -50 / y^2 + 2 = 0
Solving this equation, we find:
50 / y^2 = 2
Cross-multiplying, we get:
y^2 = 50 / 2
y^2 = 25
Taking the square root of both sides, we get:
y = ± 5
Since negative lengths do not make sense in this context, we can ignore the negative solution. Thus, y = 5.
Now, using this value of y, we can substitute it back into the area equation to solve for x:
x = 50 / y = 50 / 5 = 10
Therefore, the dimensions of the pen are 10 feet by 5 feet. Now, to find the total amount of fence needed, we add up all the sides:
Fence needed = x + 2y = 10 + 2(5) = 10 + 10 = 20 feet
So, the smallest amount of fence needed to enclose the 50 square feet pen is 20 feet.