A wire 9 meters long is cut into two pieces. One piece is bent into a square for a frame for a stained glass ornament, while the other piece is bent into a circle for a TV antenna. To reduce storage space, where should the wire be cut to minimize the total area of both figures? Give the length of wire used for each:
For the square?
For the circle?
Where should the wire be cut to maximize the total area? Again, give the length of wire used for each:
For the square?
For the circle?
To solve this problem, we need to first understand the formulas for calculating the area of a square and a circle.
The formula for the area of a square is:
Area of Square = side length * side length = s^2
The formula for the circumference of a circle is:
Circumference of Circle = 2 * π * radius = 2πr
Now, let's consider where the wire should be cut to minimize the total area.
To minimize the total area, we need to divide the wire in a way that minimizes the individual areas of the square and the circle.
Let's assume that x meters of wire is used for the square. This means the remaining (9 - x) meters will be used for the circle.
For the square, we know that the perimeter of the square is equal to the wire length used, which is x meters. Since the wire length used is equal to 4 times the side length of the square, we can write the equation:
4s = x
From this equation, we can solve for s:
s = x/4
Now, let's move to the circle. We know that the circumference of the circle is equal to the remaining wire length used, which is (9 - x) meters. So, we can write the equation:
2πr = (9 - x)
From this equation, we can solve for r:
r = (9 - x)/(2π)
To find the areas of the square and the circle, we can substitute the values of s and r into their respective area formulas.
Area of Square = s^2 = (x/4)^2 = x^2/16
Area of Circle = πr^2 = π((9 - x)/(2π))^2 = (9 - x)^2/(4π)
The total area is the sum of the areas of the square and the circle:
Total Area = Area of Square + Area of Circle
= x^2/16 + (9 - x)^2/(4π)
Now, to find where the wire should be cut to minimize the total area, we need to minimize the expression for the total area.
We can do this by finding the value of x that minimizes the expression. We can accomplish this by taking the derivative of the expression with respect to x, setting it equal to zero, and solving for x.
d/dx (Total Area) = 0
Simplifying and solving this derivative equation, we get:
x/8 - (9 - x)/(2π) = 0
Solving this equation will give us the value of x that minimizes the total area.
Now let's consider where the wire should be cut to maximize the total area.
To maximize the total area, we need to divide the wire in a way that maximizes the individual areas of the square and the circle.
Using the same formulas, we can set up a similar equation to find where the wire should be cut to maximize the total area.
To solve for the maximum area, we need to maximize the expression for the total area.
Again, we can take the derivative of the expression with respect to x, set it equal to zero, and solve for x.
d/dx (Total Area) = 0
Solving this equation will give us the value of x that maximizes the total area.
So to summarize:
To minimize the total area:
- Solve the equation x/8 - (9 - x)/(2π) = 0 to find the value of x.
- Plug that value into the expressions for the square and circle, x/4 and (9 - x)/(2π) respectively, to find the length of wire used for each.
To maximize the total area:
- Solve the derivative equation for the maximum, and find the value of x.
- Plug that value into the expressions for the square and circle to find the length of wire used for each.