Problem 41
Question
The tallest television transmitting tower in the world is in North Dakota. From a point on level ground 5280 feet \((1 \text { mile })\) from the base of the tower, the angle of elevation is \(21.3^{\circ} .\) Approximate the height of the tower to the nearest foot.
Step-by-Step Solution
Verified Answer
The height of the tower after rounding to the nearest foot is approximately 2000 feet.
1Step 1: Understand the Problem
We need to find the height of the tower. The angle of elevation and the distance from the tower are given. Knowing that tangent of an angle is the ratio of opposite side (in this case, the height of tower) to the adjacent side (in this case, the distance from the tower), we can write the equation as \(\tan(21.3^{\circ}) = \frac{{\text{{Height of Tower}}}}{5280}.\)
2Step 2: Solve the Equation for the Height of the Tower
Solve the equation for the height of the tower to get \(\text{{Height of Tower}} = \tan(21.3^{\circ}) \times 5280.\)
3Step 3: Calculate the Height of the Tower
Plug \(21.3^{\circ}\) into the equation and calculate the result to estimate the height of the tower.
4Step 4: Present the Final Approximate Value
The exact height can be given in decimal form, but the problem asks to approximate it to the nearest foot. Therefore, we will need to round our answer accordingly.
Key Concepts
Angle of ElevationTangent RatioSolving Right TrianglesTrigonometric Equations
Angle of Elevation
The angle of elevation is a measure of how high something is relative to the observer's position. Imagine you're standing on level ground and you look up to see the top of a structure; the angle between the line of sight and the horizontal ground is the angle of elevation. In the context of our exercise, the angle of elevation to the top of the television tower is given as 21.3 degrees.
Understanding the angle of elevation is crucial in trigonometry problem solving as it can be used to find heights or distances of objects when one or more quantities are known. When combined with the right trigonometric ratio, the angle of elevation can help solve right triangles effectively.
Understanding the angle of elevation is crucial in trigonometry problem solving as it can be used to find heights or distances of objects when one or more quantities are known. When combined with the right trigonometric ratio, the angle of elevation can help solve right triangles effectively.
Tangent Ratio
In trigonometry, the tangent ratio is one of the three fundamental ratios, along with sine and cosine. It is defined for acute angles in a right triangle as the ratio of the opposite side to the adjacent side. Mathematically, it is expressed as:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
In the context of our exercise problem, the tangent ratio is used with the angle of elevation to find the height of the tower. The formula described relates the height of the tower (the opposite side) to the horizontal distance from the tower (the adjacent side) using the known angle.
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
In the context of our exercise problem, the tangent ratio is used with the angle of elevation to find the height of the tower. The formula described relates the height of the tower (the opposite side) to the horizontal distance from the tower (the adjacent side) using the known angle.
Solving Right Triangles
Solving right triangles involves finding the missing lengths of sides or measures of angles in a right-angled triangle, given some initial information. This often utilizes the Pythagorean theorem in conjunction with trigonometric ratios such as sine, cosine, and tangent.
Using these steps, we can systematically approach and solve a variety of problems, including the one in our exercise by utilizing the tangent ratio.
Steps for Solving Right Triangles:
- Identify the known values (sides and angles).
- Choose the appropriate trigonometric ratio based on the given information.
- Set up the equation using the trigonometric ratio.
- Isolate the unknown variable and solve for it.
- Check the solution within the context of the problem.
Using these steps, we can systematically approach and solve a variety of problems, including the one in our exercise by utilizing the tangent ratio.
Trigonometric Equations
A trigonometric equation is an equation that involves trigonometric functions of an unknown variable - commonly an angle. Trigonometric equations often require finding all the angles that satisfy the equation, but they can also be used for practical problems like the one in our exercise, where we need to find a specific measure (in this case, the height of a tower).
To solve trigonometric equations:
To solve trigonometric equations:
- Ensure the equation is in the correct form, with one side isolated if solving for a variable.
- If necessary, use trigonometric identities to simplify the equation.
- Solve for the unknown angle or side length.
- Consider the range of possible angles if it's an angle you need to determine.
- Adapt the solution to the context of the problem, including any constraints provided.
Other exercises in this chapter
Problem 41
Find the exact value of each expression, if possible. Do not use a calculator. $$ \tan ^{-1}\left(\tan \frac{2 \pi}{3}\right) $$
View solution Problem 41
In Exercises \(33-42,\) let $$ \sin t=a, \cos t=b, \text { and } \tan t=c $$ Write each expression in terms of \(a, b,\) and \(c .\) $$ \begin{aligned} \cos t+\
View solution Problem 41
Determine the amplitude and period of each function. Then graph one period of the function. $$y=-\frac{1}{2} \cos \frac{\pi}{3} x$$
View solution Problem 41
Use a calculator to find the value of the trigonometric function to four decimal places. $$ \tan 32.7^{\circ} $$
View solution