Problem 47 - Precalculus : Building Concepts and Connections

Question

The distance from the top of a utility pole to a certain point \(P\) on the surrounding level ground is 20 feet, and the angle of elevation of the top of the pole with respect to point \(P\) is \(38^{\circ} .\) What is the distance from the top of the pole to a point on the ground that is 10 feet farther away from the base of the pole than \(P ?\)

Step-by-Step Solution

Answer

The final answer to the problem is the calculated value for \(x\) from Step 4.

Step 1: Identify known values

We know that the distance from the top of the utility pole to point \(P\) on the ground is 20 feet. The angle of elevation of the pole with respect to \(P\) is \(38^{\circ}\). We need to find the distance from the top of the pole to a point on the ground that is 10 feet farther away from the base of the pole than \(P\). Let's call this unknown distance \(x\). Since we have a right triangle and know one angle other than the right angle, we can use the tangent function.

Step 2: Apply the tangent function

The tangent of an angle in a right triangle is the length of the opposite side divided by the length of the adjacent side. Here, the length of the side opposite to the angle 38 degree (height of the utility pole) can be denoted by \(h\), and the length of the adjacent side (distance from the base of the pole to point \(P\)) can be denoted by \(a\). We can write the equation as: \(\tan(38^{\circ}) = \frac{h}{a}\). Solve this equation for \(h\). This gives us \(h = a*\tan(38^{\circ})\). Substituting \(a = 20\), we get the height \(h\) of the pole.

Step 3: Find the new distance

We now have to find the distance from the top of the pole to a point on the ground that is 10 feet farther away from the base of the pole than \(P\). This is the hypotenuse of a right triangle where the adjacent side (distance from the base of the pole to the new point) is \(a + 10\). We can use the Pythagorean theorem for this: \(x = \sqrt{{h^2 + (a + 10)^2}}\)

Step 4: Calculate the distance

Substituting the known values into the equation of Step 3, we can now calculate the unknown distance \(x\), which is the final answer to the problem. Make sure to follow the BODMAS rule (also known as the order of operations) where brackets are calculated first, followed by orders (powers and square roots), then division and multiplication, and finally addition and subtraction.

Key Concepts

Right Triangle

In the world of trigonometry, the right triangle is a fundamental shape that sets the stage for many concepts and calculations. A right triangle is a type of triangle where one of the angles is exactly 90 degrees. This right angle divides the triangle into two shorter sides known as the legs, and a longer side known as the hypotenuse, which is opposite the right angle.

When dealing with a right triangle in real-world problems, it's essential to identify the hypotenuse and the legs correctly, as these are used in various formulas. For example:

• The hypotenuse is always the side opposite the right angle, and it is the longest side.
• The legs are the two sides that form the right angle.

Understanding these basic components is crucial for applying trigonometric functions, which often involve using the relationships between these sides to solve problems involving sizes and angles.

In the context of the exercise, we are working with a right triangle where the height of the pole is perpendicular to the ground, forming the right angle.

Tangent Function

The tangent function is an essential part of trigonometry, especially when dealing with right triangles. It’s one of the three primary trigonometric ratios, alongside sine and cosine. The tangent of an angle in a right triangle is calculated as:

\[\tan(\theta) = \frac{\text{Opposite Side}}{\text{Adjacent Side}}\]

This function effectively tells us how steep a slope is if you know just one angle besides the right angle and the sides involved. In our exercise, the tangent function helps us find a missing side length (the height of the pole) given the angle of elevation and the distance on the ground.

Steps to apply the tangent function here:

• Identify the angle, which is the angle of elevation, 38 degrees in our case.
• Recognize the opposite side corresponding to this angle, which is the height of the pole.
• Determine the adjacent side, the known distance on the ground from the base of the pole to point \( P \).

By rearranging the tangent formula, we can solve for the unknown side as we've done to find the height of the pole.

Angle of Elevation

The angle of elevation is an angle formed between the horizontal line of sight and a line above that horizontal line. It is a common concept in trigonometry when determining distances or heights that aren’t directly measurable.

Consider looking up at the top of a pole or building; the angle your line of sight makes with the ground is the angle of elevation:

• It is measured from the horizontal plane - think of it like looking straight in front of you and then tilting your head up.
• This angle helps in establishing relationships between distances and heights in problems involving right triangles.

In our original exercise, you measure the angle of elevation to be 38 degrees from point \( P \) on the ground to the top of the utility pole. Understanding the angle of elevation is crucial because it directly influences how we use trigonometric functions like tangent to find unknown side lengths, as you can see by using it to identify the steepness or slope represented by the height over the distance from the pole.