Problem 16 - The Calculus with Analytic Geometry

Question

Suppose a weight is to be held \(10 \mathrm{ft}\) below a horizontal line \(A B\) by a wire in the shape of a \(Y\). If the points \(A\) and \(B\) are \(8 \mathrm{ft}\) apart, what is the shortest total length of wire that can be used?

Step-by-Step Solution

Answer

21.54 ft

Step 1: Understand the Geometry

Visualize the setup: A horizontal line AB is 8 ft apart, and the weight is 10 ft vertically below this line, forming a Y with wires extending to points A and B.

Step 2: Identify Point C

Let point C be the location of the weight directly below the midpoint of AB. Since AB is 8 ft, the midpoint is at 4 ft from either A or B. Thus, C is 10 ft below this midpoint.

Step 3: Apply the Pythagorean Theorem

For each wire from C to A and from C to B, form right triangles. The horizontal leg is 4 ft (half of AB's length), and the vertical leg is 10 ft (distance below AB).

Step 4: Calculate the Length of Each Wire

Use the Pythagorean theorem to find the hypotenuse of each triangle:\[ L = \sqrt{(4 \text{ ft})^2 + (10 \text{ ft})^2} \] Simplify inside the square root:\[ L = \sqrt{16 + 100} = \sqrt{116} \approx 10.77 \text{ ft} \] Thus, each wire has a length of approximately 10.77 ft.

Step 5: Sum the Total Length

Since there are two identical wires, double the length of one wire:\[ \text{Total Length} = 2 \times 10.77 \text{ ft} = 21.54 \text{ ft} \]

Key Concepts

Geometry

Geometry is the branch of mathematics that studies shapes, sizes, and properties of space. It involves understanding different figures, their dimensions, and how they relate to each other. In this problem, we are dealing with a simple geometric setup involving a horizontal line AB, a point C below it, and the wires forming a Y-shape extending to A and B.

It's essential to visualize the arrangement clearly:

• The line AB is horizontal and measures 8 ft.
• The point C is located 10 ft directly below the midpoint of AB.
• We use the properties of triangles to solve the problem.

This setup allows us to apply fundamental geometric principles to find the shortest length of the wires.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be expressed as:

\[c^2 = a^2 + b^2\]

In our problem, each wire forms the hypotenuse of a right triangle. We know the other two sides:

• The horizontal leg is 4 ft (half the distance of AB).
• The vertical leg is 10 ft (the distance from AB to C).

Applying Pythagoras' theorem:
\[L = \sqrt{(4\text{ ft})^2 + (10\text{ ft})^2} = \sqrt{16 + 100} = \sqrt{116} \approx 10.77\text{ ft}\]

Thus, each wire measures approximately 10.77 ft.

Right Triangle

A right triangle is a triangle in which one of the angles is a right angle (90 degrees). The side opposite the right angle is called the hypotenuse, and the other two sides are called the legs. Right triangles are essential in geometry because of properties like the Pythagorean theorem.

In this problem, the wires and the segment form right triangles:

• The legs: 4 ft horizontally (half of AB) and 10 ft vertically (from midpoint of AB to C).
• The hypotenuse: the wires extending from C to A and C to B.

Understanding the properties of right triangles helps us calculate the wire length using the Pythagorean theorem.

Given the dimensions, we calculated the hypotenuse (wire length) to be approximately 10.77 ft. Since there are two such wires, the total wire length is:
\[2 \times 10.77 \text{ ft} = 21.54 \text{ ft}\]