HP 15c Scientific 000–, 002–, 001–42,21,3

Section 13: Finding the Roots of an Equation 189

If you have some knowledge of the behavior of the function f(x) as it varies with different values of x, you are in a position to specify initial estimates in the general vicinity of a zero of the function. You can also avoid the more troublesome ranges of x such as those producing a relatively constant function value or a minimum of the function's magnitude.

Example: Using a rectangular piece of sheet metal 4 decimeters by 8 decimeters, an open-top box having a volume of 7.5 cubic decimeters is to be formed. How should the metal be folded? (A taller box is preferred to a shorter one.)

Solution: You need to find the height of the box (that is, the amount to be folded up along each of the four sides) that gives the specified volume. If x is the height (or amount folded up), the

length of the box is (8 – 2x) and the width is (4 – 2x). The volume V is given by

V= (8 – 2x)(4 – 2x) x.

By expanding the expression and then using Horner's method (page 79), this equation can be rewritten as

V = 4 ((x – 6) x + 8) x.

To get V= 7.5, find the values of x for which

f(x) = 4 ((x – 6) x + 8) x – 7.5 = 0.

The following subroutine calculates f(x):

Keystrokes	Display
¥	000–		Program mode.
´b3	001–42,21, 3		Label.
6	002–	6	Assumes stack loaded with x.