Page 134 - How To Solve Word Problems In Calculus

A (x) = 4x − 2916x −2
2916
0 = 4x −
x 2
2916
= 4x
x 2
3
4x = 2916
3
x = 729
x = 9

486 486
The height of the box, determined from y = ,is = 6. The
x 2 81
dimensions for minimum surface area are 9 cm, 18 cm, and 6 cm.
We confirm that this yields a minimum area (optional):
5832

A (x) = 4 + 5832x −3 = 4 +
x 3
Since x = 9 is a positive number, A (9) > 0, and this value

corresponds to a relative minimum. Since it is the only relative
extremum on the interval (0, ∞), it yields the absolute minimum.
13.

(0, 1)

d (x, y)

By symmetry, there appear to be two points closest to (0, 1). We
shall find the point in the first quadrant.
Let (x, y) be an arbitrary point on the parabola. The distance
between (x, y) and (0, 1) is

2
d = (x − 0) + (y − 1) 2
2
Since y = x for all points on the parabola, the distance may be
expressed as a function of x:
121