The speed that a sailboat is capable of sailing is...

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?

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?

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Key Concepts

Inequality Constraints

This concept involves understanding that the performance and eligibility of a sailboat are determined by an inequality that sets a boundary condition on a combination of physical parameters. It requires recognizing that certain combinations of the boat's length, sail area, and displacement must satisfy a specific inequality for the boat to qualify, reflecting trade?offs in design characteristics.

Dimensional Analysis

This key concept involves the proper handling of physical quantities with their respective units in an equation. It includes understanding that length, area, and volume have different dimensions, and that operations like taking square roots and cube roots correctly adjust these units to maintain a consistent relationship. This is important in verifying that the terms in the inequality are dimensionally compatible.

Algebraic Manipulation of Inequalities

This concept covers the skills necessary to rearrange the given inequality to isolate and solve for a particular variable. When certain parameters are fixed, the inequality often needs to be manipulated to determine the range or maximum value of the remaining variable that still satisfies the qualifying condition. This is crucial for solving optimization problems within constrained systems.

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