Alfonse cuts half a rectangular lawn, 50m by 20m, by mowing strips of equal width around the perimeter. Bart cuts the small rectangle in the middle that's left. How wide a strip does Alfonse cut so that they share the work equally?
50 * 20 = 1000
strip width = s
interior dimensions
(50-2s)(20-2s) = half original = 500
1000 - 140 s + 4 s^2 = 500
so
4 s^2 - 140 s +500 = 0
s^2 - 35 s + 125 = 0
s = [35 +/- sqrt (1225-500) ]/2
s = [35+/- 26.92 ] /2
s = 30.96 or 4.04
must be 4.04 because strips of 31 each end will not work
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check
50 - 2(4.04) =41.92
20 - 2(4.04) = 11.92
11.92 * 41.92 = 499.6 fine
gustave is a really demanding student who finishes his work in half the time of other students if he isnt kept busy he takes off his shoes and dances on the desks. His teach assigns him to add the first 5000 odd numbers. come up with the answer and illustrate how gustave can do this fast and get back to his dancing.
To find out how wide a strip Alfonse cuts, we first need to determine the area of the rectangular lawn and the area of the small rectangle.
The area of the rectangular lawn is calculated by multiplying its length by its width: 50m * 20m = 1000 square meters.
Since Alfonse cuts a strip around the perimeter of the rectangular lawn, the length and width of the remaining small rectangle will each be reduced by twice the width of the strip.
Let's assume the width of the strip is 'x' meters. So, the length and width of the small rectangle become (50m - 2x) and (20m - 2x) meters, respectively.
The area of the small rectangle is the product of its length and width: (50m - 2x) * (20m - 2x) = (1000 - 140x + 4x^2) square meters.
Since Alfonse cuts half of the rectangular lawn, the area of the small rectangle left for Bart will be half of the total area of the lawn.
So, the equation becomes: (1000 - 140x + 4x^2) / 2 = 1000
To solve the equation, we'll simplify it and bring all terms to one side:
4x^2 - 140x + 1000 - 2000 = 0
4x^2 - 140x - 1000 = 0
Now we can either use the quadratic formula or factor the equation. Let's factor it:
4x^2 - 140x - 1000 = (x - 25)(4x + 40) = 0
From the factored equation, we have two possibilities:
1) x - 25 = 0, which gives us x = 25
2) 4x + 40 = 0, which gives us x = -10
Since the width of a strip cannot be negative, the answer is x = 25 meters.
Therefore, Alfonse should cut a strip of width 25 meters so that they share the work equally.