Minimising the perimeter of a tunnel with a fixed area

[henryc09]

Homework Statement

A tunnel cross-section is to have the shape of a rectangle surmounted by a semicircular roof. The total cross-sectional area must be A, but the perimeter minimised to save building costs. Find its dimensions

Homework Equations

The Attempt at a Solution

I have that the perimeter would equal pi*r + 4r + 2h (where r is the radius of semi circle, h is height of rectangle) and that A=0.5pi*r^2 + 2rh, but am unsure as to how to get started. Any help would be appreciated.

 

[cepheid]

The fact that area must be equal to 'A' has given you a second equation. From it, you can determine the required relationship between r and h...meaning that the problem is reduced down to one variable (either r or h). Finding the value of that one variable that minimizes the perimeter is then just a differential calculus problem.

 

[tomcue]

I would approach this problem by first defining the objective of minimizing the perimeter. This means that we want to find the dimensions of the tunnel that will result in the smallest possible perimeter while still maintaining the required cross-sectional area.

Next, I would consider the equations provided for the perimeter and the cross-sectional area. In order to minimize the perimeter, we need to find the critical points of the perimeter equation. This can be done by taking the derivative of the equation with respect to r and setting it equal to zero. This will give us the value of r that results in the minimum perimeter.

Once we have the value of r, we can plug it back into the equation for the perimeter to find the corresponding value of h. This will give us the dimensions of the tunnel that will minimize the perimeter while still maintaining the required cross-sectional area.

It is also important to consider any practical constraints or limitations that may affect the design of the tunnel, such as the materials and construction methods that will be used. These factors may need to be taken into account in order to find a feasible solution that not only minimizes the perimeter, but also meets all necessary requirements.