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Chapter 20: Problem 9

Peter is measuring the height of a church steeple. He stands on level ground 500 feet from the base of the church and determines that the angle of elevation from the ground to the base of the steeple is \(23^{\circ}\). From the same spot he measures the angle of elevation to the highest point of the steeple and finds it is \(29^{\circ}\). (a) How high is the church, from the base of the church at ground level to the tip of the steeple? Give an exact answer and then give a numerical approximation. (b) How high is the steeple? Give an exact answer and then give a numerical approximation.

Short Answer

Expert verified
Therefore, the total height of the church is given by \[ 500 \times \tan(29^{\circ}) \] and the height of the steeple is given by \[ 500 \times \tan(29^{\circ}) - 500 \times \tan(23^{\circ}) \]

Step by step solution

01

Calculate total height of the church (base to steeple top)

The total height of the church can be found by applying the tangent formula: \[ \tan(\theta) = \frac{opposite side}{adjacent side} \] This gives: \[ \tan(29^{\circ}) = \frac{total height}{500} \], solving for total height, we will have: \[ total height = 500 \times \tan(29^{\circ}) \]
02

Calculate the height of the church to the base of the steeple

This can also be found using the tangent formula: \[ \tan(23^{\circ}) = \frac{church height}{500} \], solving for church height we will have: \[ church height = 500 \times \tan(23^{\circ}) \]
03

Calculate the height of the steeple

The steeple height can be obtained by subtracting the height of the church to the base of the steeple from the total height of the church, thus: \[ steeple height = total height - church height = 500 \times \tan(29^{\circ}) - 500 \times \tan(23^{\circ}) \] The numerical approximation can be found by evaluating each expression in a calculator and rounding off to the nearest whole number.
04

Verification of Results

The results obtained in steps 1 to 3 can be verified by checking that the sum of the church height and the steeple height is equal to the total height of the church. In other words: \[ church height + steeple height = total height \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tangent Formula
At the heart of many trigonometry and calculus problems lies the tangent formula. It鈥檚 a fundamental trigonometric function that relates an angle in a right-angled triangle to the ratio of the opposite side over the adjacent side. Formally, if \( \theta \) is an angle in a right-angled triangle, then the tangent of \( \theta \) is given by:

\[ \tan(\theta) = \frac{opposite side}{adjacent side} \
\] This formula is incredibly useful when you know either the length of the sides of the triangle or the angle and need to find the missing measure. For instance, within the context of the church steeple problem, Peter can use the tangent formula to calculate the height of the church and steeple using angles of elevation and a known horizontal distance.
Angle of Elevation
Imagine looking upwards from the ground at the tip of a tall structure; the angle your line of sight makes with the horizontal is known as the angle of elevation. It is a common scenario in trigonometry problems related to real-world applications, such as determining the height of a building, a tree, or, in our example, a church steeple. It is calculated using the tangent formula in combination with the known horizontal distance from the observer to the object.

In Peter's case, he uses the angle of elevation to establish a relationship between the known horizontal distance (500 feet from the church's base) and the unknown heights he is trying to find. Measuring the angle of elevation is the starting point from which he applies the tangent formula to solve for the heights.
Height Calculation Using Trigonometry
Trigonometry offers a straightforward method for height calculation without physically measuring the object. By forming a right-angled triangle with the height of the object, the distance to the object, and the angle of elevation, one can use trigonometric ratios to find the unknown height. Here's how it鈥檚 done in the context of our exercise:

Firstly, the total height of the church, from its base to the tip of the steeple, is found using the angle to the steeple's tip, \(29^\circ\), and the distance from the base (500 feet). Similarly, the height from the ground to the base of the steeple is found using the \(23^\circ\) elevation angle. Lastly, the height of the steeple alone is calculated by subtracting the church height from the total height.

Let's break down the process:
  • For the total height, apply the tangent formula using \(29^\circ\).
  • For the church height upto the base of the steeple, apply the tangent formula using \(23^\circ\).
  • The steeple's height is found by subtracting the height of the church (obtained in the previous step) from the total height.

These step-by-step calculations allow anyone to compute the height of tall objects without the need for complex equipment, providing a practical and efficient solution in various fields such as architecture, navigation, and even photography.

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Most popular questions from this chapter

When designing the one-third-of-a-mile-long Georgia World Congress Center, the building that housed nearly one-fifth of the events of the 1996 Olympics, engineers had to take into account the curvature of the earth (Sports Illustrated, August 5,1996 ). Assuming a constant curvature of the earth, how many feet would it curve in one-third of a mile? In other words, assume a cross-section of the earth is a perfect circle and draw a tangent line to the curve of this circle at one end of the building. How far away would the tangent line be from the circle itself at the other end of the building? (Use 3960 miles as the radius of the earth.)

Due to the scampering of a goat, a rock has been dislodged from a mountain and s sliding down an incline making a \(70^{\circ}\) angle with level ground. The weight of the rock exerts a downward force of 3 pounds. What is the component of this force in the direction of motion of the rock?

The function \(f(x)=\tan x\) has an inverse function when its domain is restricted to \((-\pi / 2, \pi / 2)\) (a) Graph \(y=\tan ^{-1}(x)\). Where is the derivative positive? Negative? (b) Is \(\tan ^{-1}(x)\) an even function, an odd function, or neither? (c) Is the derivative of \(\tan ^{-1}(x)\) even, odd, or neither?

Force \(A\) has a horizontal component of 3 pounds and a vertical component of 4 pounds. Force \(B\) has a horizontal component of 5 pounds and a vertical component of 12 pounds. (a) What is the strength of force \(A\) ? What angle does this force vector make with the horizontal? (Give a numerical approximation in degrees.) (b) What is the strength of force \(B\) ? What angle does this force vector make with the horizontal? (Give a numerical approximation in degrees.) (c) What is the component of force \(A\) in the direction of force \(B\) ?

Find exact values for each of the following. (No calculator-or use it only to check your answers.) (a) \(\cos (\pi / 4)\) (b) \(\cos (5 \pi / 4)\) (c) \(\cos (-3 \pi / 4)\) (d) \(\sin (5 \pi / 6)\) (e) \(\sin (-13 \pi / 6)\) (f) \(\cos (-2 \pi / 3)\)
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