Problem 103 on page 605 of the textbook; Elementary and Intermediate Algebra, shows us the following problem:

Sailboat stability. To be considered safe for ocean sailing, the capsize screening value C should be less than 2. For a boat with a beam (or width) b in feet and displacement d in pounds, C is determined by the function C=4d-13 b. (Dugopolski, 2012).

It instructs us to solve the problem in 3 different ways using the following figure:

a) Find the capsize screening value for the Tartan 4100, which has a displacement of 23,245 pounds and a beam of 13.5 feet.

b) Solve this formula for d.

c) The accompanying graph shows C in terms of d for the Tartan 4100 (b = 13.5). For what displacement is the Tartan 4100 safe for ocean sailing? (Dugopolski, 2012).

A. Using the equation C = 4d-1/3b,

Determine C, given that b = 13.5 and d = 23245.

C = 4 x 23245-1/3 x 13.5

Since negative exponents are the same as the reciprocal value of the expression, the equation can be re-written as:

C = (4 x 13.5) / 232451/3

Fractional exponents can be written in radical form as x1/y is equivalent to y√x, the expression now becomes:

C = 54 / 3√23245

Taking the cube root of the denominator leaves:

C = 54 / 28.539

C = 1.892

B. Solving for d:

Starting with the equation C = 4d-1/3b, place the exponent expression d1/3 into the denominator since d-1/3 is the equivalent of 1/ d1/3 which is the same as the radical expression 1/ 3√d. Now the expression is:

C = 4b/3√d

Solving for d by multiplying both sides by the radical 3√d and the variable 1/C gives the following:

C3√d = 4b

3√d = 4b/C

Raising expressions on by sides by an exponent of 3 will remove d from the cube root radical expression and leaves the final result:

d = (4b/C) 3

C. The graphical display of C for the sail boat crosses the 2.0 value at a displacement of 20,000 pounds. Any displacement greater than 20,000 will give a C value of less than 2.0 and would meet the capsizing criteria....