Problem 89 - Precalculus : Building Concepts and Connections

Question

In this set of exercises, you will use right triangle trigonometry to study real-world problems. Unless otherwise indicated, round answers to four decimal places. The angle of depression of a certain point on the surface of the water in a swimming pool with respect to the end of the diving board is \(25^{\circ} .\) That point is at a horizontal distance of 15 feet from the end of the diving board. A 6 -foot-tall swimmer aims for the point in question and makes a straight-line dive, head first. If the top of his head makes contact with the water at that point, find the distance \(d\) traversed by his feet between the time they left the end of the diving board and the time his head hit the water. (IMAGES CANNOT COPY)

Step-by-Step Solution

Answer

The distance \(d\) traversed by his feet between the time they left the end of the diving board and the time his head hit the water can be calculated by first finding the total height from the head to the feet of the diver, and then using this value with the known horizontal distance to solve for \(d\) using the Pythagorean theorem. Finally, round the answer to four decimal places.

Step 1: Compute the height of the triangle using the given angle and distance

We know that the tangent of an angle in a right triangle is equal to the opposite side divided by the adjacent side (tangent = opposite / adjacent). Let's denote the height of the swimming pool as \(h\). Given the angle of depression \(25^{\circ}\) and the horizontal distance of \(15\) feet, we can setup the equation using the tangent of the angle of depression:\[\tan(25^{\circ}) = \frac{h}{15}\]Then, solve this equation for \(h\): \[h = 15 \times \tan(25^{\circ})\]

Step 2: Compute the height from diver's head to feet

The height from the diver's head to his feet is the sum of the height of the swimmer (6 feet) and the height of the swimming pool. So,\[h_{\text{total}} = h + 6\]where \(h_{\text{total}}\) is the distance from diver's head to his feet

Step 3: Compute the distance \(d\) using the Pythagorean theorem

Now that we have both the horizontal distance from the end of the diving board to the point of contact with the water (15 feet) and the height from the head to feet of the diver (\(h_{\text{total}}\)), we can solve for \(d\) using the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. In our case, \[d^2 = {h_{\text{total}}}^2 + 15^2\] Then, solve this equation for \(d\):\[d = \sqrt{{h_{\text{total}}}^2 + 15^2}\]

Step 4: Round to the nearest four decimal places

Finally, round the final answer you get in step 3 to four decimal places to comply with the problem's instructions.

Key Concepts

Angle of Depression

The concept of the angle of depression is commonly used in right triangle trigonometry, especially in real-world applications such as these exercises. To visualize it, imagine standing on a diving board and looking down at a point in the swimming pool. The angle between this sight line and the horizontal line, parallel to the ground, is the angle of depression. This concept is useful when determining dimensions in various geometric setups.
The angle of depression is numerically equal to the angle of elevation from the point being observed if you switch perspectives. However, these two angles are measured from different reference points: one from above and the other from below. This is important when setting up trigonometric equations to solve problems such as the height of a triangle, where we use this angle as part of calculating distances in right triangles.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry used to solve for the lengths of sides in a right triangle. This theorem 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 written as:

• \[ c^2 = a^2 + b^2 \]

where:

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

In the context of our problem, the hypotenuse is the path taken by the diver's body in the water, whose length we need to find. Using the heights derived from the previous calculations, along with the known horizontal distance from the diving board to the water, the Pythagorean theorem helps to compute this hypotenuse length efficiently.

Tangent Function

The tangent function is one of the key trigonometric functions used to solve problems involving right triangles, often in relation to angles of depression or elevation. The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side:

• \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]

In our exercise, the tangent function is employed to find the vertical height of the triangle formed by the diving angle and the distance. Using the given angle of depression and the horizontal distance to the point on the pool's surface, we can easily calculate the height using this function. By setting up the equation with the tangent function, we express the height as a direct function of the known angle and adjacent side, providing us with a clear path to finding the necessary distances to solve the overall problem.