Calculus 1 Question-Find the Dimensions

Question

The sum of the perimeters of an equilateral triangle and a square is 20. Find the dimensions of the triangle and the square that produce a minimum total area.

Answer 1

Let x = the sides of the equilateral triangle

Let y = the sides of the square

 

We need to find the height , h, of the triangle using Pythagorean theorem.

 

h = √((x² - (x/2)²)

h = √(x² - x²/4)

h = √((4x² - x²) / 4)

h = (1/2)x√(3)

 

 

Using these variables, we can create equations that represent the perimeter and area.

 

3x + 4y = 20                eq1    --->  sum of perimeters

 

[(1/2)x²√(3) *(1/2)] + y²               eq2     ----->  total area 

 

Total Area = (1/4)x²√(3) + y²                     

 

 

Next, you want to substitute eq1 into the total area function, so that the function is in terms of only x.  Then once you have done that, take the derivative of the area function with respect to x and set it equal to zero.  That is

 

dA/dx = 0

 

Solve for x from this equation.  The x values will be your possible critical points.

 

In other words, you will perform the first derivative test.

 

 

When you find your critical points, evaluate the derivative of the area around those critical points.  If the derivative changes from negative to positive, then you found your critical point that you need to use.  That x value that you found should be positive, since your dimensions can only be positive.

 

Then once you have your x value, plug it into eq1 to find y.