Problem 117 - Algebra and Trigonometry

Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Liv notices a blue jay in a tree. Initially she must look up 5 degrees from eye level to see the jay, but after moving 6 feet closer she must look up 7 degrees from eye level. How high is the jay in the tree if you add 5.5 feet to account for Liv's height? Round to the nearest tenth.

Step-by-Step Solution

Answer

The jay is approximately 7.4 feet high in the tree.

Step 1: Draw a diagram

Draw a right triangle to represent the situation. Label the height of the tree as 'h' (the vertical distance from Liv's eyes to the jay). Let the distance from Liv's initial position to the base of the tree be 'd'.

Step 2: Establish the trigonometric relationships

From the initial position, the angle of elevation is 5 degrees. Therefore, use the tangent function: \[ \tan(5^\circ) = \frac{h}{d} \]

Step 3: Create an equation for the second observation

After moving 6 feet closer, the new distance to the base of the tree is (d - 6) feet, and the angle of elevation is 7 degrees. Use the tangent function again: \[ \tan(7^\circ) = \frac{h}{d - 6} \]

Step 4: Solve for 'h'

Set up the equations: \[ \tan(5^\circ) = \frac{h}{d} \implies h = d \cdot \tan(5^\circ) \] \[ \tan(7^\circ) = \frac{h}{d-6} \implies h = (d-6) \cdot \tan(7^\circ) \] Since both expressions equal 'h', set them equal to each other: \[ d \cdot \tan(5^\circ) = (d-6) \cdot \tan(7^\circ) \]

Step 5: Simplify the equation

Distribute and solve for 'd': \[ d \cdot \tan(5^\circ) = d \cdot \tan(7^\circ) - 6 \cdot \tan(7^\circ) \] \[ d \cdot (\tan(5^\circ) - \tan(7^\circ)) = -6 \cdot \tan(7^\circ) \] \[ d = \frac{6 \cdot \tan(7^\circ)}{\tan(7^\circ) - \tan(5^\circ)} \]

Step 6: Calculate 'd'

Substitute the values of \( \tan(5^\circ) \) and \( \tan(7^\circ) \): \[ \tan(5^\circ) \approx 0.08749 \] \[ \tan(7^\circ) \approx 0.12278 \] \[ d = \frac{6 \cdot 0.12278}{0.12278 - 0.08749} \approx 21.3 \text{ feet} \]

Step 7: Compute 'h'

Now substitute 'd' back into the equation for 'h': \[ h = d \cdot \tan(5^\circ) = 21.3 \cdot 0.08749 \approx 1.9 \text{ feet} \]

Step 8: Add Liv's height

Add Liv's height to the height of the jay above her eye level: \[ h_{total} = h + 5.5 = 1.9 + 5.5 = 7.4 \text{ feet} \]

Key Concepts

tangent function

The tangent function, often abbreviated as 'tan', is a trigonometric function that relates the angles of a right triangle to the ratio of the lengths of the opposite side over the adjacent side. Specifically, for an angle θ in a right triangle, the tangent of θ is defined as: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]In our exercise, Liv uses the tangent function to find the height of the blue jay relative to her eye level. She notes the angles of elevation at different distances, forming the basis of two tangent equations. These equations later help us solve for the height of the bird in the tree. By understanding how to set up the tangent function in this context, you can tackle various real-world problems involving angles and distances.

right triangle

The right triangle plays a crucial role in solving trigonometric problems. A right triangle is a triangle with one angle exactly equal to 90 degrees. The sides of a right triangle include:

• The hypotenuse: the longest side opposite the right angle.
• The opposite side: the side opposite the angle you are examining.
• The adjacent side: the side next to the angle you are examining, excluding the hypotenuse.

In this exercise, we visualize the problem by drawing two right triangles. One for Liv’s initial position and one after she moves 6 feet closer to the tree. The vertical side represents the height of the blue jay, while the horizontal sides represent the initial and new distances from the tree. Drawing these triangles helps in setting up and understanding the trigonometric equations used to find the bird's height.

angle of elevation

The angle of elevation is the angle between the horizontal line from the observer's eye to the top of an object being viewed. It is measured upward from the horizontal line. In Liv's scenario, she first spots the blue jay at an angle of elevation of 5 degrees and then at 7 degrees after moving closer.To work with angles of elevation in trigonometry:

• Identify the horizontal distance from the observer to the object.
• Use the angles provided to set up the tangent functions, as they help link the distances to the object's height.

By noting these angles, Liv can form expressions involving the tangent of these angles relative to the known distances. This allows us to determine the height of the blue jay through trigonometric relationships.

solving equations

Solving trigonometric equations typically involves isolating the unknown variable. In this exercise, we set up two equations based on tangent functions at different angles and distances. The steps are as follows:

• Write down the equations based on the tangent function for each angle of elevation.
• Express both equations in terms of the variable representing the height, 'h'.
• Set the two expressions for 'h' equal to one another since they represent the same height.
• Simplify the resultant equation to isolate the variable 'd', representing the initial distance from the tree.
• Substitute back to find the height 'h'.

Using these steps, we calculate 'd' first, then use it to find 'h'. Then finally, add Liv's height to find the total height of the blue jay. This process exemplifies how trigonometry and algebra work together to solve real-world problems.