Answer:
a.
35.2792 cm from one end (The square)
And 27.7208 cm from the other end (The circle)
b. See (b) explanation below
Step-by-step explanation:
Given
Length of Wire ,= 63cm
Let L be the length of one side of the square
Circumference of a circle = 2πr
Perimeter of a square = 4L
a. To minimise
4L + 2πr = 63 ----- make r the subject of formula
2πr = 63 - 4L
r = (63 - 4L)/2π
r = (31.5 - 2L)/π
Let X = Area of the Square. + Area of the circle
X = L² + πr²
Substitute (31.5 - 2L)/π for r
So,
X² = L² + π((31.5 - 2L)/π)²
X² = L² + π(31.5 - 2L)²/π²
X² = L² + (31.5 - 2L)²/π
X² = L² + (992.25 - 126L + 4L²)/π
X² = L² + 992.25/π - 126L/π +4L²/π ------ Collect Like Terms
X² = 992.25/π - 126L/π + 4L²/π + L²
X² = 992.25/π - 126L/π (4/π + 1)L² ---- Arrange in descending order of power
X² = (4/π + 1)L² - 126L/π + 992.25/π
The coefficient of L² is positive so this represents a parabola that opens upward, so its vertex will be at a minimum
To find the x-cordinate of the vertex, we use the vertex formula
i.e
L = -b/2a
L = - (-126/π) / (2 * (4/π + 1)
L = (126/π) / ( 2 * (4 + π)/π)
L = (126/π) /( (8 + 2π)/π)
L = 126/π * π/(8 + 2π)
L = (126)/(8 + 2π)
L = 63/(4 + π)
So, for the minimum area, the side of a square will be 63/(4 + π)
= 8.8198 cm ---- Approximated
We will need to cut the wire at 4 times the side of the square. (i.e. the four sides of the square)
I.e.
4 * (63/(4 + π)) cm
Or
35.2792 cm from one end.
Subtract this result from 63, we'll get the other end.
i.e. 63 - 35.2792
= 27.7208 cm from the other end
b. To maximize
Now for the maximum area.
The problem is only defined for 0 ≤ L ≤ 63/4 which gives
0 ≤ L ≤ 15.75
When L=0, the square shrinks to 0 and the whole 63 cm wire is made into a circle.
Similarly, when L =15.75 cm, the whole 63 cm wire is made into a square, the circle shrinks to 0.
Since the parabola opens upward, the maximum value is at one endpoint of the interval, either when
L=0 or when L = 15.75.
It is well known that if a piece of wire is bent into a circle or a square, the circle will have more area, so we will assume that the maximum area would be when we "cut" the wire 0, or no, centimeters from the
end, and bend the whole wire into a circle. That is we don't cut the wire at
all.
