Download Optimization Word Problems (page 20), Solutions

Optimization Word Problems (page 20), Solutions
1. The product of two positive real numbers, x and y, is 24.
(a) Find the minimal sum of these two numbers.
(b) Find the minimal value of the expression 3x + 2y.
Solution:
(a) Let S be the sum, S = x + y. But xy = 24, so y = 24x−1 , and so
S = x + 24x−1
so
S 0 = 1 − 24x−2 = 0 =⇒ x2 = 24
√
√
√
√
√
√
so x = 24. Also, y = 24 · x−1 = 24/ 24 = 24. So the sum is S = 24 + 24 = 2 24.
(b) Let S be the expression S = 3x + 2y. But xy = 24, so y = 24x−1 , and so
S = 3x + 48x−1
so
S 0 = 3 − 48x−2 = 0 =⇒ x2 = 16
so x = 4. Also, y = 24 · x−1 = 24/4 = 6. So the minimal expression is S = 3 · 4 + 2 · 6 = 24.
√
2. Find the point on the curve y = x which is closest to the point (3, 0). (To get started, you may
find √
it helpful to find a formula for the distance between (3, 0) and a general point (x, y) on the curve
y = x.)
Solution:
Let
q
√
D = (x − 3)2 + ( x − 0)2 ,
√
the distance between the point (x, x) and (3, 0). The distance and the square of the distance will be
minimized at the same point, so we just need to minimize the function
√
F (x) = (x − 3)2 + ( x − 0)2 = x2 − 6x + 9 + x = x2 − 5x + 9
But F 0 (x) = 2x − 5 = 0 implies x = 5/2. The point on the curve is therefore
r !
5
5
,
2
2
3. Find the dimensions of the largest rectangle which has one side on the positive x-axis, one side
on the positive y-axis, one vertex at the origin, and the opposite vertex on the curve y = e−x . (To get
started, you may find it helpful to draw a picture.)
Solution:
The base of the rectangle has length x and the height of the rectangle has length e−x , so the area is
A = x e−x
But
A0 = (x)0 e−x + x (e−x )0 = e−x − x e−x = (1 − x) e−x
so A0 = 0 implies that (1 − x) = 0, and therefore x = 1. Thus, the base has legth 1 and the height is
therefore e−1 .
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