Radical Formulas in Sailboat Stability: A Real-World Application

Introduction and Problem Context

This paper addresses Problem 103 from page 605 of an algebra textbook, which applies radical formulas and negative exponents to a practical maritime engineering scenario: determining sailboat stability. The problem, drawn from Elementary and Intermediate Algebra (Dugopolski, 2012; McGraw-Hill, 2011), centers on the capsize screening formula, which naval architects use to evaluate whether a sailboat is safe for ocean sailing.

The capsize screening value C is determined by the formula:

C = 4d^−1/3b

where d is displacement in pounds and b is the beam (width) in feet. For a boat to be considered safe for ocean sailing, C must be less than 2. The problem requires solving three connected parts:

This problem exemplifies how negative and fractional exponents appear in real-world design and safety calculations.

To find the capsize value for the Tartan 4100, we substitute d = 23,245 pounds and b = 13.5 feet into the formula C = 4d^−1/3b.

Step 1: Substitute known values

C = 4(23,245)^−1/3(13.5)

Step 2: Convert 13.5 to a fraction

Part A: Calculating Capsize Screening Value

C = 4(23,245)^−1/3 × (135/10)

Step 3: Find prime factorization

Breaking down the components:

Step 4: Apply the negative exponent rule

Convert d^−1/3 to 1/d^1/3:

C = 4 × (1/(5 × 4,649)^1/3) × (3³ × 5)/(2 × 5)

Step 5: Simplify the denominator

Multiply the denominator: (5^1/3 × 4,649^1/3) ≈ (1.710 × 16.689) ≈ 28.539

Step 6: Combine numerator

The numerator simplifies to: 4 × 27/2 = 54

Step 7: Divide to get final result

C = 54/28.539 ≈ 1.89

Since C ≈ 1.89 is less than 2, the Tartan 4100 meets the safety criterion for ocean sailing.

To find a general formula for displacement d in terms of capsize value C and beam b, we rearrange the original formula algebraically.

Starting formula:

C = 4d^−1/3b

Step 1: Rewrite the negative exponent

C = 4b/d^1/3

Step 2: Multiply both sides by d^1/3

Part B: Solving for Displacement

C × d^1/3 = 4b

Step 3: Isolate d^1/3

d^1/3 = 4b/C

Step 4: Cube both sides to eliminate the fractional exponent

(d^1/3)³ = (4b/C)³

Step 5: Simplify using exponent rules

d^{1/3 × 3} = 4³ × b³/C³

d^3/3 = 64b³/C³

Final result:

d = 64b³/C³

This formula allows naval architects and boat manufacturers to determine the displacement needed to achieve a target capsize value, given a specified beam width.

To find the displacement range that ensures safety (C < 2) for the Tartan 4100 with b = 13.5 feet, we set up and solve an inequality using the formula from Part A.

Set up the inequality:

For safety, C < 2. Substituting the formula:

4d^−1/3(13.5) < 2

Part C: Finding Safe Displacement Range

Which simplifies to:

54/d^1/3 < 2

Step 1: Divide both sides by 2

54/2 < d^1/3

27 < d^1/3

Step 2: Recognize that 27 = 3³

3³ < d^1/3

Step 3: Cube both sides

(3³)³ < (d^1/3)³

3⁹ < d

Step 4: Calculate 3⁹

3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 = 19,683

Final result:

d > 19,683 pounds

Therefore, for the Tartan 4100 to be safe for ocean sailing with a beam of 13.5 feet, its displacement must exceed 19,683 pounds. This threshold represents a critical design parameter that boat builders must respect when constructing vessels of this size and width.