Problem 36 - Contemporary Precalculus

Question

A pole tilts at an angle \(9^{\circ}\) from the vertical, away from the sun, and casts a shadow 24 feet long. The angle of elevation from the end of the pole's shadow to the top of the pole is \(53^{\circ} .\) How long is the pole?

Step-by-Step Solution

Answer

Answer: The length of the pole is approximately 27.61 feet.

Step 1: Identify the triangles

First, let's identify the triangles that we will work with. We have a right triangle with the vertical line (the part of the pole that is straight up), the shadow on the ground, and the tilted pole itself as its sides. We also have another right triangle with the shadow, the horizontal distance from the end of the shadow to the base of the pole, and the vertical distance from the base of the pole to the top of the pole as its sides.

Step 2: Label the sides and angles of the triangles

Let's label the first triangle's sides as follows: - Let x be the length of the vertical line (the part of the pole that is straight up). - Let 24 feet be the length of the shadow. - Let y be the length of the second horizontal segment (the distance from the end of the shadow to the base of the pole). - Let z be the length of the tilted pole (the hypotenuse of the first triangle). - Let h be the length of the second triangle's vertical side. - Given angles are \(9^{\circ}\) for the angle between the tilted pole and the vertical line, and \(53^{\circ}\) for the angle of elevation from the end of the shadow to the top of the pole.

Step 3: Set up the trigonometric equations

From the first triangle (with x, y and z as the sides), we can write the following trigonometric equation using sine function: sin (9) = x / z From the second triangle (with h, 24 and y as the sides), we can write another trigonometric equation using tangent function: tan(53) = h / 24

Step 4: Solve for x and h

We can solve the equations from Step 3 for x and h: x = z * sin(9) h = 24 * tan(53)

Step 5: Use the Pythagorean theorem to find z

Now that we have expressions for x and h, we can use Pythagorean theorem on the first triangle to find the length of the tilted pole (z): x^2 + (y + 24)^2 = z^2 Substitute the expressions for x from Step 4: (z * sin(9))^2 + (y + 24)^2 = z^2

Step 6: Substitute h for y in the equation

We can now substitute the expression for h in the equation from Step 5 for y: (z * sin(9))^2 + (h + 24)^2 = z^2 Substitute the expression for h from Step 4: (z * sin(9))^2 + (24 * tan(53) + 24)^2 = z^2

Step 7: Solve for z

We can now solve the equation from Step 6 for z: z = sqrt((24 * tan(53) + 24)^2 / (1 - sin^2(9))) Calculate the values: z ≈ 27.61 feet The length of the pole is approximately 27.61 feet.

Key Concepts

Right Triangle

A right triangle is a type of triangle that includes a right angle, which is an angle of exactly 90 degrees. This means that the right triangle comprises two sides, one of which is always perpendicular to the other.
In the context of trigonometry, right triangles are crucial as they allow the use of trigonometric functions to solve for various unknowns.

• The 'base' is the side on the ground, such as the shadow in our earlier example.
• The 'height' is perpendicular to the base, such as the vertical part of the pole.
• The 'hypotenuse' is the longest side facing the right angle, often the direct line connecting the base of one end to the top of the perpendicular side.

Understanding these components helps in calculating related angles and lengths, making the problem-solving process more manageable.

Angle of Elevation

The angle of elevation refers to the angle formed between the horizontal plane and an upwards line of sight, typically when looking at an object above the observer.
In trigonometry, this angle helps to find various lengths when combined with right triangle properties, as seen in situations like shadow problems.

• This angle is crucial when measuring heights that are not directly accessible.
• It relates to other trigonometric functions (like sine or tangent) to aid in calculating unknown lengths.

For example, when the angle of elevation is given, along with one side of a right triangle, one can find another side using suitable trigonometric functions.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental relation in Euclidean geometry that works in right triangles. It states that the square of the hypotenuse (the longest side of the right triangle) is equal to the sum of the squares of the other two sides.

• The mathematical representation is: \[c^2 = a^2 + b^2\] where \(c\) is the hypotenuse, and \(a\) and \(b\) are the other two sides.
• This theorem allows you to find an unknown side if you know the lengths of the other two sides.

For instance, in our problem, we used the Pythagorean theorem to find the hypotenuse, which is the length of the tilted pole. By substituting the values obtained from trigonometric functions into the theorem's equation, we reach a solution efficiently.

Sine Function

The sine function is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the side opposite that angle to the hypotenuse. It is often used to calculate unknown lengths and angles in right triangles.

• In mathematical terms, the sine function is represented as \[\sin(\theta) = \frac{\text{Opposite Side}}{\text{Hypotenuse}}\]
• This function is especially useful when dealing with angles less than 90°, allowing you to determine side lengths if the angle and one other side are known.

In the initial exercise, the sine function was utilized to find one of the triangle's side lengths based on a given angle, demonstrating its practical use in problem-solving.

Tangent Function

The tangent function is a trigonometric function that correlates the angle within a right triangle with the ratio of the opposite side to the adjacent side. This function is particularly helpful when calculating different triangle measurements, such as heights and distances.

• In formula form, it is presented as \[\tan(\theta) = \frac{\text{Opposite Side}}{\text{Adjacent Side}}\]
• The tangent function is especially useful in dealing with problems involving steep or vertical angles.

In our solved problem, the tangent function was pivotal in determining a vertical height component based on the angle of elevation and a known horizontal distance (shadow length). This showcased how tangents assist in breaking down complex trigonometric questions into manageable equations.