Mat 222 Week 3 Assignment

Real World Radical Formulas
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....