Problem 76 - Algebra and Trigonometry with Analytic Geometry

Question

The figure illustrates the apparatus for a tightrope walker. Two poles are set 50 feet apart, but the point of attachment \(P\) for the rope is yet to be determined. (a) Express the length \(L\) of the rope as a function of the distance \(x\) from \(P\) to the ground. (b) If the total walk is to be 75 feet, determine the distance from \(P\) to the ground.

Step-by-Step Solution

Answer

The height from the point of attachment to the ground should be approximately 27.94 feet.

Step 1: Define the Problem

We need to determine how the length of a tightrope between two poles relates to the height from the ground where the rope is fixed, and then calculate at which height the rope should be fixed for it to be 75 feet long.

Step 2: Establish Rope Geometry

Assume the two poles of equal height are placed on a flat surface 50 feet apart. Let the point \( P \) represent the height at which the rope is attached from the ground. Then the two lengths formed are each half of the span (25 feet horizontally each) plus their individual height components.

Step 3: Consider Each Triangle

The length of the rope forms the hypotenuse of a right triangle, with one leg being 25 feet (half the distance between the poles) and the other leg \( x \), the height \( P \) from the ground. Thus, we use the Pythagorean theorem, finding that the length of the rope \( L \) can be expressed as \( L = 2\sqrt{x^2 + 25^2} \).

Step 4: Express Rope Length as Function of Height

The function describing the length \( L \) of the rope as a function of the height \( x \) is: \[ L(x) = 2\sqrt{x^2 + 25^2} \]

Step 5: Solve for Specific Rope Length

To find the height \( x \) for which the rope is 75 feet long, set \( L = 75 \) and solve the equation: \[ 75 = 2\sqrt{x^2 + 25^2} \]

Step 6: Isolate the Square Root

Divide both sides by 2: \[ \frac{75}{2} = \sqrt{x^2 + 625} \] which simplifies to \( 37.5 = \sqrt{x^2 + 625} \).

Step 7: Solve for x

Square both sides to eliminate the square root: \[ (37.5)^2 = x^2 + 625 \] Simplify: \[ 1406.25 = x^2 + 625 \] Subtract 625 from both sides: \[ x^2 = 781.25 \] Take the square root of both sides: \[ x = \sqrt{781.25} \approx 27.94 \]

Step 8: Conclusion

The distance from the point of attachment \( P \) to the ground should be approximately 27.94 feet for the rope to be 75 feet long.

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that relates the lengths of the sides of a right triangle. This theorem states that in a right triangle, the sum of the squares of the lengths of the two shorter sides (known as the legs) is equal to the square of the length of the longest side (known as the hypotenuse). Mathematically, this is expressed as:\[ a^2 + b^2 = c^2 \]where:

• \( a \) and \( b \) are the lengths of the legs
• \( c \) is the length of the hypotenuse

In the problem involving the tightrope, the rope itself represents the hypotenuse of a right triangle. By applying the Pythagorean Theorem, we can determine the relationship between the rope's length and the attachment height from the ground. This allows us to solve practical problems involving distances and heights in a straightforward manner.

Right Triangle

A right triangle is a type of triangle that includes one 90-degree angle. This property makes them very useful in many areas of mathematics, including trigonometry and geometry. In a right triangle, the side opposite the right angle is known as the hypotenuse, while the other two sides are referred to as the legs.
In our exercise, each half of the tightrope tied between two poles forms a right triangle. Here, the horizontal leg is the distance between the pole and the midpoint of the span, which is 25 feet. The vertical leg is the height from the ground to the point of attachment of the rope, denoted as \( x \). The rope sections themselves are the hypotenuses.
By understanding the properties of right triangles, we can set up equations using the Pythagorean Theorem to find the required lengths and positions.

Distance Calculation

Distance calculation, particularly in a geometric sense, involves using known formulas and theorems to find unknown lengths and dimensions. When dealing with problems like the tightrope walker scenario, it's about determining the length of the rope based on given parameters like the separation between poles and attachment height.
In this case, we calculated the rope length using a distance calculation formula derived from the Pythagorean Theorem:\[ L(x) = 2\sqrt{x^2 + 25^2} \]This function describes how the length \( L \) of the rope changes as the height \( x \) varies, with \( 2\sqrt{x^2 + 25^2} \) being twice the length of the hypotenuse formed by the pole distance and attachment height.

• The challenge is to manipulate the equation to find specific values, such as determining \( x \) when the rope is 75 feet.
• By representing the physical dimensions mathematically, we can easily solve for unknowns.

Geometry of Triangles

The geometry of triangles provides a broad framework where the properties of various types of triangles, including right triangles, are studied. This field examines aspects like angles, side lengths, and relationships between these elements.
Triangles are one of the simplest but most common shapes in geometry. In the context of the exercise, understanding the formation of triangles when a rope is suspended between two points is crucial.

• By recognizing each sub-section of the tightrope as a triangle, we employ geometric principles to solve for unknown lengths.
• This understanding allows us to dissect complex real-world setups into simpler geometric forms and solve them using established mathematical principles.

Using geometry, we apply known theorems to calculate dimensions and understand the structural setup of objects like ropes, bridges, and various architectural designs.