Hi pls help me i dunno what to do show me complete solution please please Thank you.
a piece of wire 10ft. long is cut into two pieces , one piece is bent into the shape of a circle and the other into the shape of a square. How should the wire be cut so that:
a.)The combined area of the two figures is as small as possible?
b.)The combined area of the two figures is as large as possible?
Got Answer from my notes but no solution:
a.) Radius of circle = 5/(pi+4);
Length of side of square = 10/(pi+4)
b.) Radius of circle = 5/pi and there is no square
let the radius of circle be r
let the side of the square be x
4x + 2πr = 10
2x + πr = 5
x = (5 - πr)/2
total area = πr^2 + x^2
replace the x with (5 - πr)/2
careful with the squaring.
a) Differentiate, set that equal to zero and solve for r
sub back into the expression for x to find the side of the square.
b) for a max, since a circle is the largest shape possible for a given perimeter, use all of the wire for the circle.
radius of circle = r
side of square = s
10 = 4 s + 2 pi r
so s = 2.5 - .5 pi r
A = pi r^2 + s^2
or
A = pi r^2 + 6.25 - 2.5 pi r + .25 pi^2 r^2
A = (.25 pi +1)pi r^2 -2.5 pi r + 6.25
I am going to use calculus, you can complete the square to find vertex of parabola
dA/dr = 0 for max or min in domain so
0 = 2(pi)(1+.25 pi)r -2.5 pi
0 = (1+.25 pi)r - 1.25
r = 1.25/(1 + .25 pi)
or
r = 5/(4 + pi) agreed
To find the solution, we can use some basic geometry principles. Let's break it down step by step:
a) To minimize the combined area, we need to minimize the areas of both the circle and square.
- Let's start with the circle: The circumference of the circle is equal to the length of the wire given (10ft). So, the circumference of the circle should be 10ft, which means the length of the wire used for the circle is fully utilized.
- Now, the circumference of a circle is given by the formula: C = 2 * π * r, where r is the radius of the circle.
- From the above formula, we can find that the radius of the circle should be 10 / (2 * π) = 5 / π.
- Next, let's move on to the square: The remaining wire after cutting for the circle will be used to form the square.
- Since the length of the wire used for the square needs to be maximized, we should use all the remaining wire for the square, which is 10ft - (circumference of circle) = 10 - (2 * π * r) = 10 - (2 * π * (5 / π)) = 10 - 10 = 0ft.
- With 0ft of wire remaining, it means we cannot form a square. Hence, there is no square in this case.
b) To maximize the combined area, we need to maximize the areas of both the circle and square.
- Let's start again with the circle: We want to maximize the area of the circle this time.
- The formula to calculate the area of a circle is A = π * r^2, where r is the radius.
- We know from case a) that the radius should be 5 / π. So, the area of the circle would be π * (5 / π)^2 = 25 / π.
- Now, let's move on to the square: Since we want to maximize the total area, we should utilize all of the wire to form the square.
- In this case, all 10ft of the wire is used to form the square.
- The formula to calculate the area of a square is A = s^2, where s is the length of a side.
- So, the area of the square would be (10 / (π + 4))^2.
Therefore, the solution for maximizing the combined area is:
- Radius of circle = 5 / π.
- No square can be formed in this case, so there is no length for the side of the square.
I hope this explanation helps you understand how to find the solutions for both parts of the question.