Suppose that 430 ft of fencing is used to enclose a corral in the shape of a rectangle with a semicircle whose diameter is a side of the rectangle. (In the figure below, the blue outline represents the fencing.) Find the dimensions of the corral with maximum area.
x=?
y=?
i tried couple of ways but got the wrong answer, i know i have to take the derivative and find the minimum value for x but i don't know how to set it up.
i did get that and i got farther than that but then got completely lost
Since we can't see your diagram, your problem is not clear to me.
I will assume you have a rectangle with a semicircle mounted on top of the rectangle.
let the base of the rectangle be 2x, (that way I can use x as the radius of the semicircle)
let the height of the rectangle be y
so first equation:
2x + 2y + (1/2)2π(x) = 430
2x + πx + 2y= 430
y = (430-2x-πx)/2
Area = 2xy + (1/2)πx^2
= 2x (430-2x-πx)/2 + (1/2)πx^2
= 430x - 2x^2 - πx^2 + (1/2)πx^2
= 430x - 2x^2 - (1/2)πx^2
d(area)/dx = 430 - 4x - πx = 0
4x + πx = 430
x = 430/(4+π) = 60.21
sub back into y = ... and state your conclusion
To find the dimensions of the corral with the maximum area, we'll need to set up an equation and then differentiate it to find the maximum value.
Let's start by defining the variables:
- Let x represent the width of the rectangle (the base of the corral).
- Let y represent the length of the rectangle (the height of the corral).
Now, we can set up the equation for the perimeter:
Perimeter = Length of the rectangle + 2 * Width of the rectangle + Circumference of the semicircle
The length of the rectangle is y, the width is x, and the circumference of the semicircle can be calculated using its formula:
Circumference = π * (diameter) / 2
Since the diameter is equal to the width of the rectangle (x), the formula becomes:
Circumference = π * x / 2
Now, we can write the equation for the perimeter:
430 = y + 2x + πx/2
We can then isolate y in terms of x:
y = 430 - 2x - πx/2
Next, we need to find the equation for the area of the corral. It is the sum of the area of the rectangle and the area of the semicircle:
Area = Length of the rectangle * Width of the rectangle + Area of the semicircle
The area of the rectangle is simply:
Area(Rectangle) = y * x
The area of the semicircle can be calculated using its formula:
Area(Semicircle) = π * (radius^2) / 2
Since the radius is equal to half of the diameter (which is x), the formula becomes:
Area(Semicircle) = π * (x^2) / 8
Now, we can write the equation for the area of the corral in terms of x and y:
Area = y * x + π * x^2 / 8
To find the dimensions of the corral with maximum area, we need to find the critical points of the area function. We can do this by taking the derivative of the area equation with respect to x.
Differentiating the equation:
d(Area)/dx = d(y * x)/dx + d(π * x^2 / 8)/dx
d(Area)/dx = y + π * x / 4
To find the maximum area, we need to find the values of x and y where the derivative is equal to zero. So, we set d(Area)/dx = 0 and solve for x:
y + π * x / 4 = 0
y = -π * x / 4
Now, we substitute this y value back into the equation for the perimeter: y = 430 - 2x - πx/2
-π * x / 4 = 430 - 2x - πx/2
Now, we can solve for x using algebraic methods. Once we have the value of x, we can substitute it back into the equation for y to find the dimensions of the corral with maximum area.