Page 122 - How To Solve Word Problems In Calculus

40y − 2xy = 400 − 40x

20y − xy = 200 − 20x
Finally we factor y from each term on the left and divide by its
coefficient.
(20 − x)y = 200 − 20x

200 − 20x
y =
20 − x
The smallest allowable value of x, leading to a degenerate triangle, is
x = 0; the largest value occurs when y = 0. Since

2
2
x + y + x + y = 20, when y = 0, 2x = 20 or x = 10. Hence
0 ≤ x ≤ 10.
To compute the critical values, we first express the area in
terms of x.
1
A = xy
2

1 200 − 20x
= x
2 20 − x
100x − 10x 2
A(x) =
20 − x
The derivative is calculated using the quotient rule. Once obtained,
we set it to 0.
2
(20 − x)(100 − 20x) − (100x − 10x )(−1)

A (x) =
(20 − x) 2
2
2000 − 500x + 20x + 100x − 10x 2
=
(20 − x) 2
2
10x − 400x + 2000
=
(20 − x) 2
2
10x − 400x + 2000
0 =
(20 − x) 2
2
0 = 10x − 400x + 2000
2
0 = x − 40x + 200
√
2
−b ± b − 4ac
To find x, we use the quadratic formula: x = .
2a
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