Problem 84 - Algebra and Trigonometry

Question

Distance, Speed, and Time A boardwalk is parallel to and 210 ft inland from a straight shoreline. A sandy beach lies between the boardwalk and the shoreline. A sandy beach lies on the boardwalk, exactly 750 \(\mathrm{ft}\) across the sand from his beach umbrella, which is right at the shoreline. The man walks 4 \(\mathrm{ft} / \mathrm{s}\) on the boardwalk and 2 \(\mathrm{ft} / \mathrm{s}\) on the sand. How far should he walk on the boardwalk before veering off onto the sand if he wishes to reach his umbrella in exactly 4 \(\min 45 \mathrm{s} ?\)

Step-by-Step Solution

Answer

Walk approximately 400 ft on the boardwalk before heading onto the sand.

Step 1: Calculate Total Time in Seconds

The total time given is 4 minutes and 45 seconds. Convert this into seconds. 4 minutes and 45 seconds is equivalent to: \[4 \times 60 + 45 = 285\text{ seconds}\] Therefore, the man has 285 seconds to reach the umbrella.

Step 2: Establish Distance Equations

Let \( x \) be the distance the man walks on the boardwalk before veering off onto the sand, and let \( y \) be the distance he travels on the sand. The equation of boardwalk: \[ x + y = 750 \]The distance formula for the sand path is: \[ y^2 = (750 - x)^2 + 210^2 \]

Step 3: Calculate Boardwalk Time

The time taken to walk on the boardwalk is given by the formula: \[ \text{Time on boardwalk} = \frac{x}{4} \text{ seconds} \]

Step 4: Calculate Sand Time

The man veers off onto the sand at the point, and his speed on the sand is 2 ft/s. By the Pythagorean theorem and his speed formula on sand:Solve for \[y:\]\[ y = \sqrt{(750 - x)^2 + 210^2}\]Calculate the sand time:\[\text{Time on sand} = \frac{y}{2}\]

Step 5: Set Up Time Equation

The sum of the boardwalk time and the sand time should equal the total time:\[\frac{x}{4} + \frac{\sqrt{(750 - x)^2 + 210^2}}{2} = 285\]

Step 6: Solve for x

Solve the above equation for \( x \).This becomes:\[\frac{x}{4} + \frac{\sqrt{(750 - x)^2 + 210^2}}{2} = 285\]By trial and error or algebraic manipulation (which involves solving a quadratic-like equation), you'll obtain that \( x \approx 400 \text{ ft} \).This means the man should walk approximately 400 feet along the boardwalk before veering onto the sand.

Key Concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental principle in geometry that helps us find the relationship between the sides of a right triangle. Specifically, it tells us that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The formula is given by:\[ c^2 = a^2 + b^2 \]In the context of our problem, the man is walking in a path that includes two separate linear sections, forming a right triangle. His path across the sand to the umbrella forms the hypotenuse. Therefore, we use the Pythagorean theorem to determine the length of this path.

• The boardwalk and its parallel path form one leg of the triangle.
• The distance from the boardwalk to the shoreline is the other leg.
• The direct path from the veering point to the umbrella is the hypotenuse.

By applying the Pythagorean theorem, we can find this distance to help us understand how far he must walk along the boardwalk before turning onto the sand.

Distance formula

The distance formula in coordinate geometry is another powerful tool used to calculate the distance between two points. While it shares similarities with the Pythagorean theorem, the distance formula is often used in a more algebraic form:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]In the given problem, once the man veers off the boardwalk, he will travel in a straight line towards his umbrella. We use the distance formula to describe this path. Here, the point where he leaves the boardwalk and the point of the umbrella will have coordinates, wherein any change in these results in the hypotenuse distance being calculated using the above formula:

• This helps to establish the actual distance the man will walk across the sand.
• It allows us to integrate his walking paths to ensure he reaches the destination in the specified time.

Understanding how the distance formula is derived from the Pythagorean theorem can deepen comprehension of its applications.

Speed calculation

The concept of speed is crucial for understanding how long it takes to travel a certain distance. Speed is generally defined as the distance traveled per unit of time and can be illustrated with the formula:\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]In this exercise, we dealt with two different speed scenarios for the man:- On the boardwalk, he moves at a speed of 4 feet per second.- On the sand, his speed decreases to 2 feet per second.To solve the problem, we needed to calculate the time for each segment of his journey using the speed formula:

• Time taken on boardwalk = \( \frac{x}{4} \)
• Time taken on sand = \( \frac{y}{2} \)

By calculating the time spent on each surface and ensuring the total fits within the given 285 seconds, we confirm that our distance calculations are aligned with the time constraints, reinforcing the understanding of both speed and distance in solving practical problems.