I want to fence in the largest patch of land that I can for a skatepark that I am building. I only have 34 miles of fence. Which dimensions should I use?
The largest area of a region with a fixed perimeter is a circle, so ....
Can someone help plz
epic
To find the dimensions that will enclose the largest patch of land with 34 miles of fence, we can use the concept of perimeter and area.
First, let's set up an equation for the perimeter of the fence. The perimeter of a rectangular shape is given by the formula: P = 2(l + w), where l is the length and w is the width of the rectangular shape.
Since we have 34 miles of fence, we can write the equation as: 34 = 2(l + w).
Next, let's solve the equation for one variable in terms of the other. For example, solving for l gives us l = (34 - 2w) / 2.
Now, we want to find the dimensions that maximize the enclosed area. The area of a rectangular shape is given by the formula: A = l * w.
Substituting the expression for l into the area equation, we have A = (34 - 2w) / 2 * w.
To maximize the area, we can differentiate the equation and find the critical points. Taking the derivative of A with respect to w, we get: dA/dw = (34 - 4w) / 2.
Setting the derivative equal to zero and solving for w, we find: 34 - 4w = 0. Solving for w gives us w = 8.5.
Substituting w = 8.5 back into the equation for l, we find: l = (34 - 2*8.5) / 2 = 8.5.
Therefore, the dimensions that will enclose the largest area with 34 miles of fence are 8.5 miles for both the length and width of the rectangular shape.