Problem 69 - Algebra and Trigonometry

Question

The speed that a sailboat is capable of sailing is determined by three factors: its total length \(L,\) the surface area \(A\) of its sails, and its displacement \(V\) (the volume of water it displaces), as shown in the sketch. In general, a sailboat is capable of greater speed if it is longer, has a larger sail area, or displaces less water. To make sailing races fair, only boats in the same "class" can qualify to race together. For a certain race a boat is considered to qualify if $$0.30 L+0.38 A^{1 / 2}-3 V^{1 / 3} \leq 16$$ where \(L\) is measured in feet, \(A\) in square feet, and \(V\) in cubic feet. Use this inequality to answer the following questions. (a) A sailboat has length 60 \(\mathrm{ft}\) , sail area 3400 \(\mathrm{ft}^{2}\) , and dis- placement 650 \(\mathrm{ft}^{3} .\) Does this boat qualify for the race? (b) A sailboat has length 65 \(\mathrm{ft}\) and displaces 600 \(\mathrm{ft}^{3} .\) What is the largest possible sail area that could be used and still allow the boat to qualify for this race?

Step-by-Step Solution

Answer

(a) Yes, it qualifies; (b) Maximum sail area is about 3292.82 sq ft.

Step 1: Evaluate the inequality for the first sailboat (Part a)

Given the sailboat has length \(L = 60\) feet, sail area \(A = 3400\) square feet, and displacement \(V = 650\) cubic feet. Substitute these values into the inequality:\[0.30 \times 60 + 0.38 \times \sqrt{3400} - 3 \times 650^{1/3} \leq 16\].

Step 2: Calculate each term of the inequality (Part a)

1. Calculate \(0.30 \times 60\): the result is 18.2. Calculate \(\sqrt{3400}\) and then \(0.38 \times \sqrt{3400}\): \(\sqrt{3400} \approx 58.31\), so \(0.38 \times 58.31 \approx 22.16\).3. Calculate \(650^{1/3}\) and then \(3 \times 650^{1/3}\): \(650^{1/3} \approx 8.673\), so \(3 \times 8.673 \approx 26.019\).

Step 3: Evaluate the inequality (Part a)

Substitute the calculated terms into the inequality:\[18 + 22.16 - 26.019 = 14.141\].This result satisfies the condition as \(14.141 \leq 16\). Therefore, the sailboat qualifies for the race.

Step 4: Establish the inequality for sail area (Part b)

Given \(L = 65\) feet and \(V = 600\) cubic feet, find the largest possible sail area \(A\) that satisfies the inequality. First, substitute known values into the inequality:\[0.30 \times 65 + 0.38 \times A^{1/2} - 3 \times 600^{1/3} \leq 16\].

Step 5: Solve for maximum sail area (Part b)

1. Calculate \(0.30 \times 65\): the result is 19.5.2. Calculate \(600^{1/3}\) and then \(3 \times 600^{1/3}\): \(600^{1/3} \approx 8.434\), so \(3 \times 8.434 \approx 25.302\).3. Set up the equation for \(A^{1/2}\):\[19.5 + 0.38 \times A^{1/2} - 25.302 \leq 16\].

Step 6: Isolate and solve for \(A^{1/2}\) (Part b)

Simplify the inequality:\[0.38 \times A^{1/2} \leq 16 - 19.5 + 25.302\].This becomes:\[0.38 \times A^{1/2} \leq 21.802\].Solving for \(A^{1/2}\) gives:\[A^{1/2} \leq \frac{21.802}{0.38}\approx 57.374\].

Step 7: Calculate the maximum possible sail area (Part b)

Square both sides to solve for \(A\):\[A \leq (57.374)^2 \approx 3292.82\].Therefore, the largest possible sail area is approximately 3292.82 square feet.

Key Concepts

Inequalities

Inequalities are mathematical expressions that show the relationship between two values, indicating one is either greater than, less than, or equal to another. They are important in many areas of mathematics and are used to represent conditions or constraints. In the context of our sailboat racing problem, inequalities ensure fair competition by creating strict qualification criteria that boats must meet to race together.

In our example, the inequality is given as: \[ 0.30L + 0.38A^{1/2} - 3V^{1/3} \leq 16 \] This measures whether a sailboat's characteristics (length, sail area, and displacement) fit within the range considered acceptable for a race. By substituting the boat's specifications into the inequality, we can determine if the boat qualifies.

Solving inequalities generally involves finding values for variables that make the inequality true. In this exercise, we calculated unknown expressions, like the maximum sail area for a certain boat length and displacement, which help meet the qualification criteria.

Sailboat Racing Qualification

Sailboat racing qualification ensures that participating boats race under fair and competitive conditions. The criteria for qualification are set so similar-sized boats with similar capabilities compete against each other. This allows for a well-matched event where skill and strategy, rather than the technical advantages of a better design, often determine the outcome.

For the race in question, a specific criterion is given that involves calculations from the boat's physical attributes. These include its length, sail area, and the water displacement. Each attribute contributes uniquely to the boat's potential speed and maneuverability.

The inequality formula used to determine qualification takes into account these attributes, ensuring each boat fits within a 'class' of boats having comparable features. It places mathematical boundaries on these physical characteristics, ensuring no sailboat has excessive advantages based solely on engineering.

Mathematical Modeling

Mathematical modeling in this context involves using mathematical expressions to simulate real-world scenarios and solve practical problems. For our sailboat racing problem, a model is established through the inequality that describes a boat's speed potential relative to its length, sail area, and displacement.

This involves representing the complex interaction of these factors with a formula, allowing for predictions and decisions based on the variables involved. Mathematical models simplify real-world phenomena into manageable equations, which can then be analyzed to support decision-making.

In this case, the inequality serves as a model to determine if a sailboat qualifies for a race based on its physical measurements. It takes into account each contributing factor by applying weights (coefficients like 0.30, 0.38, and -3) to balance their effects, providing an objective way to assess sailboat potential and ensure fair competition.

Square Roots and Cube Roots

Square roots and cube roots are fundamental concepts in mathematics, and they help in breaking down complex calculations into simpler forms. They are essential in many real-world applications, such as calculating areas and volumes.

In the sailboat racing inequality, both square and cube roots are used to adjust for the sail area and water displacement respectively:

• Square root (\(A^{1/2}\) ) is used because sail area affects speed, but its impact isn't linear. The square root moderates its effect in the inequality.
• Cube root (\(V^{1/3}\) ) adjusts for water displacement, reflecting its influence on buoyancy and resistance.

These operations allow us to implement more scaled and realistic measurements for sail area and volume, key in gauging the boat's qualifications in a way that fits competitive settings.

Understanding how to compute these roots and applying them to real-life problems is crucial for accurate and meaningful modeling. They provide a way to scale inputs such as area and volume to a common standard, important for fair comparisons and evaluations.