The designers of a water park are creating a new slide and...

The designers of a water park are creating a new slide and have sketched some preliminary drawings. The length of the ladder is 30 feet, and its angle of elevation is $60^{\circ}$ (see figure). (a) Find the height $h$ of the slide. (b) Find the angle of depression $\theta$ from the top of the slide to the end of the slide at the ground in terms of the horizontal distance $d$ the rider travels. (c) The angle of depression of the ride is bounded by safety restrictions to be no less than $25^{\circ}$ and not more than $30^{\circ} .$ Find an interval for how far the rider travels horizontally.

The designers of a water park are creating a new slide and have sketched some preliminary drawings. The length of the ladder is 30 feet, and its angle of elevation is $60^{\circ}$ (see figure).
(a) Find the height $h$ of the slide.
(b) Find the angle of depression $\theta$ from the top of the slide to the end of the slide at the ground in terms of the horizontal distance $d$ the rider travels.
(c) The angle of depression of the ride is bounded by safety restrictions to be no less than $25^{\circ}$ and not more than $30^{\circ} .$ Find an interval for how far the rider travels horizontally.

Key Concepts

Inverse Trigonometric Functions

These functions are used to determine angle measures from known ratios of side lengths. They are essential when reversing a trigonometric calculation, such as finding an angle from given height and horizontal distance.

Angle of Elevation and Depression

These angles are defined relative to a horizontal line; the angle of elevation is measured upward from the horizontal, while the angle of depression is measured downward. They are crucial for determining heights or distances in real-world applications where the observer’s line of sight is angled.

Inequalities and Interval Constraints in Trigonometry

This concept deals with applying restrictions to the range of possible values for angles or distances, ensuring that computed measurements meet specific safety or design criteria. It often involves solving inequalities that use trigonometric expressions.

Right Triangle Trigonometry

This concept involves analyzing right triangles by relating the lengths of their sides with the measures of their angles. It typically uses trigonometric functions such as sine, cosine, and tangent to solve for unknown sides or angles, making it fundamental in problems combining angles and distances.

Trigonometric Ratios

These are formulas that relate the angles of a right triangle to the ratios of its sides. For example, sine, cosine, and tangent are used to compute unknown dimensions such as height or horizontal distance when one angle and one side are known.

Transcript

00:01 So in part a of this problem, they want us to find the height of the slide.

00:04 Well, if you look at that first triangle on the left, you're told that that bottom angle is 60 degrees, and you know the hypotenuse.

00:11 So we can use our sign function to help us.

00:14 So we're going to do the sign of 60 degrees is equal to the opposite side, which is age, divided by the hypotenuse, which is 30.

00:22 So now to solve for h, we simply just need to multiply both sides of our equation by 30.

00:27 Well, 30 times the sign of 60 degrees is approximately 25 .98, so 25 .98 feet.

00:37 So this will represent the height of the water slide.

00:41 Okay, so now let's take a look at part b.

00:45 In part b, we're told that we're asked to find the angle of depression, theta, going from the top of the slide to the end of the slide at the ground in terms of the horizontal distance d.

00:55 Well, remember, the angle of depression would be this angle right here.

01:01 But it also is the same as this angle down at the bottom because they're alternate interior angles.

01:06 So now in this separate triangle, we have theta.

01:10 We have our d, that's our distance.

01:12 And we also know the height of the slide.

01:14 We just found that in part a to be 25 .98 feet.

01:18 So now we can solve for theta using our tangent function.

01:22 So the tangent of theta is going to equal the opposite.

01:26 Side, which is our height, which is 25 .98 feet, divided by the adjacent side, which is d.

01:33 So if we need to solve for theta, we just have to take the inverse tangent of both sides.

01:38 So what we'd find is that theta is equal to the inverse tangent of 25 .98 feet divided by d.

01:46 So this would be our expression to find theta...

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