Question

The angle of elevation to the top, $\mathrm{B}$, of a vertical tower $\mathrm{AB}$ is ${19}^{\circ }{3}^{\prime }$ when measured from a point, $\mathrm{C},27.3\mathrm{m}$ from the base of the tower. Calculate (a) the height of the tower (b) the distance BC.

Short answer

Answer: (a) The height of the tower is approximately 9.99 meters. (b) The distance between point B and C is approximately 29.1 meters.

Step-by-step solution

We are given an angle of elevation, $\mathrm{\angle }B={19}^{\circ }{3}^{\prime }$, and the horizontal distance from the base of the tower to the point where angle was measured, $AC=27.3$ m. We need to find the height of the tower (the opposite side) and the distance BC (the hypotenuse). #Step 2: Convert the angle in degrees and minutes to degrees only

The given angle is in degrees and minutes, so we have to convert it to a decimal value in degrees. Minutes are a fraction of degrees (60 minutes = 1 degree), so we'll divide the given minutes by 60 to get the decimal portion of the angle in degrees. $\mathrm{\angle }B={19}^{\circ }+\frac{3^{\prime }}{60}={19}^{\circ }+{0.05}^{\circ }={19.05}^{\circ }$. #Step 3: Use the tangent function to find the height of the tower (AB)

We can use the tangent function to find the height of the tower, where $\tan (\mathrm{\angle }B)=\frac{AB}{AC}$. Rearrange this formula to solve for the height AB: $AB=AC\times \tan (\mathrm{\angle }B)$. Now plug in the known values: $AB=27.3\mathrm{m}\times \tan ({19.05}^{\circ })\approx 9.99\mathrm{m}$. The height of the tower is approximately $9.99$ meters. #Step 4: Use the Pythagorean theorem to find the distance BC

Now that we know the height of the tower AB, we can use the Pythagorean theorem to find the distance BC: $B{C}^2=A{B}^2+A{C}^2$. Plugging in the known values, we have $B{C}^2=(9.99\mathrm{m}{)}^2+(27.3\mathrm{m}{)}^2\approx 846.24{\mathrm{m}}^2$. Taking the square root of both sides, we get that $BC\approx 29.1\mathrm{m}$. The distance BC is approximately $29.1$ meters. Answers: (a) The height of the tower (AB) is approximately $9.99$ meters. (b) The distance BC is approximately $29.1$ meters.

Key concepts

Angle of Elevation

When trying to find the height of a vertical object like a tower, engineers and surveyors often rely on the concept known as the 'angle of elevation'. This angle is the angle above horizontal ground from an observer's line of sight to the top of an object. Imagine standing a certain distance away from a tower and looking up to see the top; the angle your line of sight makes with an imaginary horizontal line is the angle of elevation.

To measure this angle, professionals might use tools like a theodolite or a transit. In real-world applications, knowing the angle of elevation is crucial because it helps determine the height of the object without needing to measure it directly, which can be particularly useful when the object is very tall or difficult to access. To calculate this height, the tangent function in trigonometry is commonly used.

Tangent Function

In trigonometry, the tangent function is extremely important for dealing with right triangles. The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. The formula can be expressed as $\tan (\theta )=\frac{\text{opposite}}{\text{adjacent}}$.

In the context of the given exercise, the tangent of the angle of elevation ($\theta$) is used to determine the height of the tower (opposite side), when the distance from the observer to the base of the tower (adjacent side) is known. After converting the angle to a decimal using standard degree measure, the tangent function can accurately provide the necessary height measurement. This function is widely used in engineering fields, such as in the design of roads, construction of buildings, and when working with slopes and inclines.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry, particularly relevant to right-angled triangles. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Expressed in a formula, it's $c^2=a^2+b^2$, where $c$ represents the length of the hypotenuse, while $a$ and $b$ are the lengths of the triangle's other two sides.

In engineering, the Pythagorean theorem allows for the determination of distances or heights that are otherwise difficult to measure. In our exercise example, after finding the height of the tower using the tangent function, we applied the Pythagorean theorem to calculate the distance from point C to the top of the tower, which is crucial in various engineering applications such as determining the line of sight, signal strength for communication towers, or even for laying out construction projects.

• Easy to measure base length, $a$
• Calculated vertical height, $b$
• Sought distance or hypotenuse, $c$