A 24-inch piece of string is cut into two pieces. One piece is used to form a circle and the other is used to form a square. How should the string be cut so that the sum of the areas is a minimum?
Your string is cut into X and (24-X). Suppose the X part is used to form the circle, and the (24-X) part is used to form the square. Then since the circumference X = 2ðr, the radius r of the circle is X/(2ð), and so the area is ðr² = ðX²/(4ð²) = X²/(4ð).
The area of the square is just the square of one side, and one side is (24-X)/4 = 6 - X/4, so the area is (6 - X/4)².
So the sum of the areas is just X²/(4ð) + (6 - X/4)². You want to minimize this, so differentiate it with respect to X, and set the derivative to zero:
f'(X) = X/(2ð) - (6 - X/4)/2 = 0, so X/ð - 6 + X/4 = 0, so X = 6 / (1/4 + 1/ð).
Note that we don't know if this is a minimum or a maximum yet, so you'll need to work out what the area would have been if all the string had been used to create the circle, and then work out what the area would have been if all the string had been used to create the square. You should find that the above answer is less than either of them. However I strongly recommend you also check my math above: if I've got it right this is rather a messy one.
(That funny symbol ð should be a pi, by the way: it isn't being interpreted correctly.)
thank you David Q
To find the lengths of the string that should be used to form the circle and square such that the sum of their areas is minimized, we can use calculus and the concepts of optimization.
Let's represent the lengths of the string used for the circle and square as x and (24 - x) respectively.
The circumference of a circle is given by the formula C = πd or C = 2πr, where d is the diameter and r is the radius. We know that the string used for the circle will form its circumference, so we can set up the equation:
x = 2πr
To find the radius r in terms of x, we divide both sides of the equation by 2π:
r = x / (2π)
The perimeter of a square is given by the formula P = 4s, where s is the length of a side. We know that the string used for the square will form its perimeter, so we can set up the equation:
24 - x = 4s
To find the length of a side s in terms of x, we divide both sides of the equation by 4:
s = (24 - x) / 4
The area of a circle is given by the formula A = πr^2, and the area of a square is given by the formula A = s^2. We can substitute the values we found for r and s into these formulas and add them together:
A = π((x / (2π))^2) + ((24 - x) / 4)^2
A = (x^2 / (4π)) + ((24 - x)^2 / 16)
To minimize the sum of the areas, we can take the derivative of A with respect to x and set it equal to zero:
dA/dx = (2x / (4π)) - (2(24 - x) / 16) = 0
Simplifying this equation, we get:
x / (2π) = (24 - x) / 8
Solving for x, we find:
x = 16π / 5
To find the corresponding length for the square, we substitute this value of x into the equation we set up earlier:
s = (24 - x) / 4
s = (24 - 16π / 5) / 4
Now we have the lengths of the string that should be used to form the circle and square such that the sum of their areas is minimized.