Chris cuts half a rectangular lawn, 40m by 30m, by mowing strips of equal width around the perimeter. Jerri cuts the small rectangle in the middle that's left. How wide a strip does Chris cut so that they share the work equally?
Solve by using quadratic formula.
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Let the width of the cut done by Chris be x m
That leaves a rectangle of 40-2x by 30-2x to be cut by Jerri.
(40-2x)(30-2x) = (1/2)(40)(30)
4x^2 - 140x + 1200 = 600
x^2 - 35x + 150 = 0
(x-5)(x-30) = 0
x = 5 or x = 30 , but x = 30 makes no sense
x = 5
The strip should be 5 m wide.
To solve this problem, let's first assume the width of the strip that Chris cuts is "x" meters.
If Chris cuts a strip of width "x", it would reduce both dimensions of the rectangular lawn by 2x. So, the dimensions of the remaining rectangle after Chris cuts his strips would be (40-2x) meters by (30-2x) meters.
The area of the remaining rectangle would be given by:
Area = (40-2x) * (30-2x)
And the area that Jerri cuts would be equal to half of the original area of the rectangular lawn minus the area that Chris cuts. So we have:
(0.5 * 40 * 30) - [(40-2x) * (30-2x)]
Let's set these two expressions equal to each other since Chris and Jerri share the work equally:
(0.5 * 40 * 30) - [(40-2x) * (30-2x)] = [(40 * 30) - (0.5 * 40 * 30)] / 2
To simplify, let's first multiply out the terms:
600 - [(40-2x) * (30-2x)] = [(1200) - (600)] / 2
600 -[(1200 - 40*2x - 30*2x + 4x^2)] = [1200 - 600] / 2
Now let's further simplify:
600 - (1200 - 80x - 60x + 4x^2) = 600
600 -1200 + 140x - 4x^2 = 600
Rearranging and combining like terms, we get:
4x^2 - 140x = 0
Now we can solve this quadratic equation by using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 4, b = -140, and c = 0. Plugging these values into the formula, we get:
x = (-(-140) ± √((-140)^2 - 4*4*0))/(2*4)
Simplifying further:
x = (140 ± √(19600))/(8)
x = (140 ± 140)/(8)
x = (140 + 140)/(8) or x = (140 - 140)/(8)
x = 280/8 or x = 0/8
x = 35 or x = 0
Since the width of the strip can't be zero, we can conclude that Chris cuts a strip of width 35 meters so that they share the work equally.
To find the width of the strip that Chris cuts, we can set up an equation based on the given information.
Let's denote the width of the strip as 'x'.
The length and width of the large rectangle are 40m and 30m respectively. Chris cuts strips of equal width around the perimeter, so the length and width of the small rectangle left after Chris cuts the strips would be (40 - 2x) and (30 - 2x) respectively.
The area of the small rectangle cut by Jerri would be (40 - 2x) * (30 - 2x).
Now, let's set up the equation based on the given condition that they share the work equally. The total area of the rectangular lawn is 40m * 30m, so the area cut by Chris should be half of that:
(40 * 30) / 2 = (40 - 2x) * (30 - 2x)
Now, we can solve this quadratic equation using the quadratic formula.
The quadratic equation is in the form ax^2 + bx + c = 0, where:
a = (40 - 2x) * (30 - 2x) = 4x^2 - 140x + 1200
b = 0
c = -600
Using the quadratic formula, x can be calculated as:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
In this case, with b = 0:
x = ± sqrt((-4)(-600) / (2)(4))
Simplifying further:
x = ± sqrt(600) / 4
x = ± (10 sqrt(6)) / 4
x = ± (5 sqrt(6)) / 2
Therefore, the width of the strip that Chris cuts so that they share the work equally is (5 sqrt(6)) / 2 or approximately 5.477 m.