This week I’m asked to solve the following word problem in relation to a real world radical formula. Problem 103 on pages 605-606 states: To be considered safe for ocean sailing, the capsize screening value C should be less than 2 (www.sailing.com). For a boat with a beam (or width) b in feet and displacement d in pounds, C is determined by the function: C=〖4d〗^(-1/3) b.
Find the capsize screening value for the Tartan 4100, which has a displacement of 23,245 pounds and a beam of 13.5 feet.
Solve this formula for d?
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) The first part of the problem requires that I substitute the variables with their given values. I need to find the value of C, which represents the capsize screening value. To do so, I need to replace d, the displacement value in pounds, with 23,245; and, also replace b, the beam’s width in feet, with 13.5. I do not need to convert the inches to feet using a decimal value because that was already done. By following the order of operations I first need to solve for the exponent before multiplying across. The radical exponent of -1/3 means that I have to apply the reciprocal of the cubed root of d and use that value within my multiplication. C=4d^(-1/3) bCapsize formula
C=4(23245)^(-1/3) (13.5)Replace variables with given values C=4(1/〖23245〗^(1/3) )(13.5)Convert the reciprocal of the negative radical exponent C=4(1/28.539)(13.5)Factor the radical exponent, then the rational number (computed with a calculator and then rounded to thousandths place) C=4(0.035)(13.5)
Multiply all terms
C=1.89Capsize screening value is less than 2; this boat is safe to sail. b) The second part of the problem asks that I solve the formula for d, the displacement value in pounds. Since I will use the same capsize formula, I will not replace any of the variables. I just need to convert the formula to solve for d.