A 100 inch piece of string is cut into 3 pieces. One piece forms a circle, one forms a square and 1 forms an equilateral triangle. If the perimeter of the triangle is equal in length to the perimeter of the square how long should each piece of string be to minimize the total area? What is the diameter of the circle? How long is each side of the square? How long is each side of the triangle?
let each side of the triangle be 4x (you will see why ...)
then each side of the square is 3x (ahh, how clever)
height of the triangle is 2√3x , by Pythagorus
let the radius of the circle be r
2πr + 24x = 100
πr = 50 - 12x
r = (50-12x)/π
total area (T) = πr^2 + 9x^2 + (1/2)(4x)(2√3x)
= π(50-12x)^2/π^2 + 4√3x^2
= (2500 - 1200x + 144x^2)/π + 9x^2 + 4√3x^2
d(T)/dx = -1200/π + 288x/π + 18x + 8√3x
= 0 for a min of T
timex π
-1200 + 288x + 18πx + 8√3π x=0
x(288 + 18π + 8√3π) = 1200
x = 3.092
r = 4.104
so diameter of circle is 8.21
perimeter of rectangle is 12x = 37.106
perimeter of square is 37.106
check:
2πr + 12x + 12x = 100
To minimize the total area, we need to find the dimensions of each shape that would minimize their respective areas.
Let's start with finding the length of each piece of string.
Since the perimeter of the triangle is equal to the perimeter of the square, we can set up an equation:
3x = 4y (where x is the length of each side of the equilateral triangle, and y is the length of each side of the square)
To solve for x and y, we need another equation relating all three sides.
The perimeter of the triangle is given by P = 3x. The perimeter of the square is given by P = 4y. And the perimeter of the circle is given by P = πd, where d is the diameter.
We know that the total length of the string is 100 inches, so we can set up another equation:
3x + 4y + πd = 100
Now, we have two equations and two unknowns. With some algebraic manipulation, we can solve for x, y, and d.
Substituting 3x for 4y in the second equation:
3x + 3x + πd = 100
6x + πd = 100
Now we have one equation in terms of x and d. We can rearrange this equation to solve for d:
πd = 100 - 6x
d = (100 - 6x)/π
We can substitute this value of d into the equation 3x + 4y + πd = 100 to solve for y:
3x + 4y + π((100 - 6x)/π) = 100
3x + 4y + 100 - 6x = 100
150 - 3x = 4y
y = (150 - 3x)/4
Now, we have expressions for y and d in terms of x. To minimize the total area, we need to minimize the area of each shape.
For the circle, the area is given by A = πr². Since the diameter is d, the radius is half of the diameter, so r = d/2. Thus, the area of the circle is:
A(circle) = π(d/2)² = π(d²/4)
For the square, the area is given by A = s², where s is the length of each side. Thus, the area of the square is:
A(square) = y² = (150 - 3x)²/16
For the equilateral triangle, the area is given by A = (√3/4) * s², where s is the length of each side. Thus, the area of the triangle is:
A(triangle) = (√3/4) * x²
To minimize the total area, we need to minimize the sum of the areas of the three shapes, i.e., find the minimum of:
Total Area = A(circle) + A(square) + A(triangle)
= π(d²/4) + (150 - 3x)²/16 + (√3/4) * x²
To find the values of x, y, and d that minimize the total area, you can differentiate the Total Area equation with respect to x, set the derivative equal to zero, and solve for x. This will give you the critical points where the total area is minimized. Then you can substitute these values into the equations for y and d to find the corresponding dimensions.