91Ó°ÊÓ

The Pythagorean theorem is a fundamental principle in mathematics, particularly involving right triangles. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This can be expressed in the formula: \[ c^2 = a^2 + b^2 \]where:

• \( c \) is the length of the hypotenuse,
• \( a \) and \( b \) are the lengths of the triangle's other two sides.

In the given problem, the person walks towards the beach forming a right triangle with the shoreline and the boardwalk. Here, the hypotenuse is 750 ft, and one of the sides is 210 ft. By applying the Pythagorean theorem, you can calculate the other side (\( y \)) which is the direct path to the umbrella across the sand. This is done as follows:\[ y = \sqrt{750^2 - 210^2} \] By finding \( y \), you determine the actual distance that needs to be traveled across the sand to reach the destination.

A right triangle is a type of triangle that has one angle equal to 90 degrees. The longest side opposite this angle is known as the hypotenuse, while the other two sides are referred to as the legs. In practical applications, right triangles are very useful for calculating distances and understanding spatial relationships. They allow us to use the Pythagorean theorem to determine unknown lengths or distances that are otherwise not straightforward to measure. In this specific optimization problem, the right triangle is formed between the shore, the distance along the sand, and the perpendicular distance from the boardwalk to the shoreline. This can visually be imagined as the man making a straight line path across the sand after walking straight along the boardwalk. The goal was to optimize his path to ensure he spends exactly 4 minutes and 45 seconds walking. Knowing the properties of a right triangle allows us to compute the lateral path he can safely take without exceeding or falling short of this time requirement.

Time-distance problems involve calculating how long it takes an object or person to travel a particular distance given a certain speed. These problems can also require determining distances or speeds when other variables are known. This exercise involves such a problem where different speeds are covered in separate parts of the journey:

• The first part of the journey is along the boardwalk at a speed of 4 ft/s.
• The second part is across the sand at a slower speed of 2 ft/s.

To find out how far the man walks on the boardwalk before switching to the sandy terrain, we need to sum the time for each part and equate it to the total desired time.By breaking down the equation based on these segments and their respective speeds:\[ \frac{x}{4} + \frac{y}{2} = 285 \text{ seconds} \]you can solve for \( x \), understanding fully that each change in speed affects how much distance can be covered within the given time period.

Speed is determined by dividing the distance traveled by the time it takes to travel that distance. In mathematical terms: \[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]In this problem, this fundamental concept is applied to both parts of the man's journey:

• Along the boardwalk: speed is 4 ft/s.
• Across the sand: speed is 2 ft/s.

Therefore, to achieve the entire trip in a set time limit (4 minutes and 45 seconds which equals 285 seconds), it is crucial to balance the distance and speed.Using the equation from the time-distance problem:\[ \frac{x}{4} + \frac{720}{2} = 285 \]We realize the calculation involves ensuring the speed at each segment allows for reaching the umbrella just at or before the computed time. Solving correctly for \( x \) finds the exact distance walked on the boardwalk first, satisfying the prescribed condition of the total time constraint.