Problem 48 - College Algebra

Question

Use the four-step procedure for solving variation problems given on page 356 to solve. The electrical resistance of a wire varies directly as its length and inversely as the square of its diameter. A wire of 720 feet with \(\frac{1}{4}\)-inch diameter has a resistance of \(1 \frac{1}{2}\) ohms. Find the resistance for 960 feet of the same kind of wire if its diameter is doubled.

Step-by-Step Solution

Answer

The resistance of a 960 feet wire with its diameter doubled is 1 ohm.

Step 1: Express the Problem Formula

The direct and inverse variation is defined by the relation \(R = kL/D^2\). Here, R is the resistance, L is the length and D is the diameter of the wire. k is the constant of proportionality which stays the same for the same kind of wire.

Step 2: Calculate the Value of k

First plug in the values for the wire with a resistance of \(1 \frac{1}{2}\) ohms, L=720 feet and D=\(\frac{1}{4}\) inches into the equation to solve for k: \(k = R*D^2/L = 1.5*(\frac{1}{4})^2/720 = 0.00052083333\).

Step 3: Use k to Find the Resistance of the Other Wire

Now, use the proportionality constant k for the 960 feet wire, with its diameter doubled to \(2*\frac{1}{4}\). Plug these values into the equation to find the new resistance: \(R = k*L/D^2 = 0.00052083333*960/(2*\frac{1}{4})^2 = 1 ohms\).

Key Concepts

Direct Variation

In direct variation, two quantities increase or decrease together at the same rate. It is like a partnership between two variables. When one variable goes up, the other goes up too, and if one goes down, the other follows.

For example, in our exercise, the electrical resistance ( R ) of a wire varies directly with its length ( L ). This means that if you lengthen the wire, its resistance increases too.

• The relationship can be mathematically expressed as R = kL , where k is the constant of proportionality.
• If the length doubles, the resistance also doubles, as long as the diameter of the wire remains constant.

This concept helps in predicting how changes to the length will affect resistance, which can be very useful in fields like electrical engineering.

Inverse Variation

Inverse variation is when one quantity increases while the other decreases. It works the opposite way to direct variation. In this case, if one variable goes up, the other will definitely go down, and vice versa.

In the exercise, the resistance ( R ) varies inversely with the square of the wire's diameter ( D ).

• The mathematical relation is given by R = k/LD^2 .
• This means that when the diameter of the wire becomes larger, the resistance decreases, assuming the length stays the same.

This concept helps us to understand why thinner wires have higher resistance and how increasing diameter can reduce resistance, which is crucial for designing electronic circuits.

Proportionality Constant

The proportionality constant ( k ) is a key player in both direct and inverse variation problems. This constant gives a snapshot of the exact relationship between the variables for a specific situation.

In our problem, k stays the same for wires of the same type, which allows us to predict resistance for different lengths and diameters once we know k .

• To find k , you plug the known values into the formula: k = R*D^2/L .
• For the given wire with specific length and diameter, we calculated k , which was used for further calculations.

Understanding the proportionality constant simplifies your work since it applies to any wire of the same material. Once determined, k can be plugged back to find how changes in one variable affect another.

Electrical Resistance

Electrical resistance is an important concept in electronics and physics. It measures how much a material opposes the flow of electric current. The higher the resistance, the more the material resists current flow, causing energy loss in the form of heat.

In our original exercise, resistance relies on both the length and diameter of a wire.

• We determined that resistance varies directly with the length, meaning longer wires have higher resistance.
• It also varies inversely with the square of the diameter, so thicker wires have lower resistance.

This principle is fundamental for understanding how different wires and materials can impact an electrical circuit. It's essential for engineers when designing systems that require specific resistance levels to function correctly.