Problem 91 - Introductory Algebra for College Students

Question

The formula $$p=15+\frac{5 d}{11}$$ describes the pressure of sea water, \(p,\) in pounds per square foot, at a depth of d feet below the surface. Use the formula to solve Exercises. The record depth for breath-held diving, by Francisco Ferreras (Cuba) off Grand Bahama Island, on November \(14,1993,\) involved pressure of 201 pounds per square foot. To what depth did Ferreras descend on this illadvised venture? (He was underwater for 2 minutes and \(9 \text { seconds! })\)

Step-by-Step Solution

Answer

Francisco Ferreras descended to a depth of approximately 409.2 feet.

Step 1: Substitute the given pressure into the formula

Replace \(p\) in the formula with 201, which is the pressure in pounds per square foot endured by Francisco Ferreras during his dive.

Step 2: Solve for depth

Rearrange the equation to solve for \(d\). Start by subtracting 15 from both sides, and then multiplying the result by 11 and dividing by 5 to get \(d\). That is: \(d = \frac{(p - 15) * 11}{5}\). After substituting \(p = 201\), the equation becomes: \(d = \frac{(201 - 15) * 11}{5}\). Solve this equation to find the value of \(d\).

Step 3: Simplify to get the answer

Solving the equation \(d = \frac{(201 - 15) * 11}{5}\), you get \(d = 409.2\). Therefore, Ferreras descended to a depth of approximately 409.2 feet on his venture.

Key Concepts

Pressure Calculations

In the realm of introductory algebra, pressure calculations often involve using a formula to determine the pressure exerted by a fluid at a given depth. The pressure of sea water can be determined using the formula:\[p = 15 + \frac{5d}{11}\]where \(p\) is the pressure in pounds per square foot and \(d\) is the depth in feet below the surface. To solve problems using this formula, we need to understand that:

• The formula helps us calculate how pressure increases with depth under water.
• As the depth \(d\) increases, the fraction \(\frac{5d}{11}\) increases, leading to a corresponding increase in pressure \(p\).
• This formula is specific to sea water and might differ for other fluids or conditions.

Considering these points, pressure calculations are crucial for anyone interested in understanding how different environmental factors, like depth, affect underwater experiences.

Formula Manipulation

Understanding formula manipulation is essential in solving algebraic problems, particularly when faced with the task of isolating a specific variable. To find the depth \(d\) using the given pressure \(p\), we must adjust the formula. Here’s how it's done:

• First, start by isolating the fraction: Subtract 15 from both sides, resulting in \(p - 15 = \frac{5d}{11}\).
• Next, clear the fraction: multiply both sides by 11 to obtain \(11(p - 15) = 5d\).
• Finally, divide both sides by 5 to solve for \(d\): \(d = \frac{11(p - 15)}{5}\).

This series of steps allows us to manipulate the formula efficiently to find any unknown variable. It highlights the importance of performing the inverse operations in reverse order to undo what has been applied to the variable we are solving for.

Problem-Solving Steps

Solving algebraic problems requires strategic thinking and clear steps to reach the correct solution. Here's a breakdown of the problem-solving steps used in this scenario:

• Identify known values: In this exercise, we know the pressure \(p = 201\) pounds per square foot from the dive.
• Substitute into the formula: Replace \(p\) with 201 in the formula to begin isolating \(d\).
• Apply arithmetic operations: As detailed in formula manipulation, subtract 15, solve the resulting equation, and follow through till \(d\) is isolated.
• Simplification: Conduct arithmetic to simplify calculations: \(d = \frac{(201 - 15) \times 11}{5}\).
• Verification: Ensure the calculations are correct and the units are appropriate.

These problem-solving steps are crucial for systematically approaching and solving algebraic equations efficiently, ensuring that we find a meaningful solution.

Breath-Held Diving

Breath-held diving is an extreme yet fascinating activity that challenges the physical limits of human endurance. In the case of Francisco Ferreras, the dive involved a depth of approximately 409.2 feet, where he experienced a pressure of 201 pounds per square foot. Here’s why this information is remarkable:

• Physics of the dive: Underwater pressure increases significantly with depth, demanding exceptional lung capacity and body resistance to pressure change from divers.
• Safety considerations: Such dives require careful planning and an understanding of one's limits, as extreme depths can pose substantial risks.
• Historical context: Ferreras's dive remains one of the significant achievements in the field of freediving, showcasing human capability and the challenge of breaking limits.

Breath-held diving highlights how understanding concepts like pressure calculations and formula manipulation can have practical applications, especially in fields pushing human boundaries.