What is the angle at which a surfer approaches the steepest part of a wave if he travels at a constant speed of 3.5 meters/second for 45 seconds and covers a horizontal displacement of 150 meters?

To find the angle at which the surfer approaches the steepest part of the wave, we need to use trigonometry and the given information about the surfer's speed and displacement.

First, let's understand the situation. The surfer's path can be visualized as a right triangle, where the horizontal displacement is the base of the triangle, and the surfer's path is the hypotenuse. The surfer's speed is the rate at which he covers the hypotenuse.

In this case, the horizontal displacement is 150 meters, and the time taken is 45 seconds. Therefore, we can calculate the length of the hypotenuse (the surfer's path) using the formula d = rt, where d is the displacement, r is the rate or speed, and t is the time.

Hence, the length of the surfer's path (hypotenuse) is 3.5 meters/second * 45 seconds = 157.5 meters.

Now, we have two sides of the right triangle: the base (horizontal displacement) and the hypotenuse (surfer's path). To find the angle at which the surfer approaches the steepest part of the wave, we can use trigonometric functions.

The sine function relates the angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse, which in this case is the ratio of the vertical displacement to the surfer's path.

So, we need to find the vertical displacement to use the sine function. Since we only have the horizontal displacement and the surfer's path length, we need to calculate the vertical displacement.

Considering the right triangle we have, to find the vertical displacement, we can use the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Using the Pythagorean theorem:
(displacement)^2 = (horizontal displacement)^2 + (vertical displacement)^2
(157.5 m)^2 = (150 m)^2 + (vertical displacement)^2

To solve this equation, we can subtract (150 m)^2 from both sides:
(vertical displacement)^2 = (157.5 m)^2 - (150 m)^2

Now, calculate:
(vertical displacement)^2 = 2475 m^2

To find the vertical displacement, we take the square root of both sides:
vertical displacement = √(2475 m^2)

Thus, the vertical displacement is approximately 49.75 meters.

Now that we have the vertical displacement and the surfer's path length, we can find the angle using the sine function.
sine (angle) = vertical displacement / surfer's path length
sine (angle) = 49.75 m / 157.5 m

To find the angle, we need to take the inverse sine (arcsine) of this ratio:
angle = arcsine(49.75 m / 157.5 m)

Calculating this using a calculator or appropriate software will give you the value of the angle, which is approximately 18.19 degrees.

Therefore, the angle at which the surfer approaches the steepest part of the wave is approximately 18.19 degrees.