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In the context of the given problem, the Pythagorean theorem helps to find the length of the speed zone, represented as a right triangle. Imagine the patrol car, the two ends of the zone, and the length of the zone itself forming this triangle. The theorem states that in a right triangle, the sum of the squares of the two shorter sides equals the square of the hypotenuse (the longest side).

Given the patrol car's distances from both ends – 200 feet and 150 feet – these become the two shorter sides of your triangle. You can express this relationship mathematically as:
\[ a^2 + b^2 = c^2 \]where:

• \( a = 200 \)
• \( b = 150 \)

Plugging these into the formula yields:\[ 200^2 + 150^2 = c^2 \]Calculating, we get:\[ 40000 + 22500 = c^2 \]\[ c^2 = 62500 \]\[ c = \sqrt{62500} \]\[ c = 250 \]
Thus, the length of the zone, or hypotenuse, is 250 feet. The Pythagorean theorem offers a straightforward way to find side lengths when dealing with right triangle scenarios like this one.

To calculate the angles within the triangle formed by the patrol car's position and the speed zone, we employ basic trigonometric functions.
This involves using the tangent function and knowledge about the sum of angles in a triangle.

Let's start with angle \( A \), which is opposite the side measuring 150 feet and adjacent to the side measuring 200 feet. The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side, expressed as:

• \( \tan(A) = \frac{b}{a} = \frac{150}{200} \)

To find angle \( A \), use the inverse tangent function:\[ A = \arctan(\frac{150}{200}) \]By calculating, angle \( A \) comes out to be approximately 36.87 degrees.

Next, for finding angle \( B \), remember that the sum of angles in any triangle is 180 degrees. Since one angle is already the right angle (90 degrees), angle \( B \) is:\[ B = 180 - 90 - A \]Substituting the known values gives:\[ B = 90 - 36.87 = 53.13 \]
Therefore, angle \( B \) is approximately 53.13 degrees. Understanding how to leverage trigonometric functions and properties of triangles enables us to pinpoint angle measures with confidence.

In physics, the relationship between speed, distance, and time is crucial. It allows us to determine how long a vehicle takes to traverse a specified distance at a certain speed. The formula to remember is:

• \( \text{speed} = \frac{\text{distance}}{\text{time}} \)

Rearranging it to solve for time gives:\[ \text{time} = \frac{\text{distance}}{\text{speed}} \]From the problem, we know:

• The length of the speed zone, or distance, is 250 feet.
• The speed limit posted is 35 miles per hour.

The key here is to keep the units consistent. Since the speed is initially in miles per hour, and we want to find the time in seconds, a conversion is necessary.
Convert 35 miles per hour to feet per second using the factor 5280 feet per mile and 3600 seconds per hour:\[ 35 \text{ miles/hour} = 35 \times \frac{5280}{3600} \approx 51.33 \text{ ft/sec} \]Now use the formula to find the time it takes to cross the zone:\[ \text{time} = \frac{250}{51.33} \approx 4.87 \text{ seconds} \]Thus, the minimum time a vehicle can take to pass through the enforcement zone, without breaching the speed limit, is approximately 4.87 seconds. By understanding the units and carefully applying the formula, you can easily tackle similar speed-distance-time problems.