Problem 23 - Technical Mathematics with Calculus

Question

A resistance, now \(7250 \Omega\), is to be increased by \(15.0 \% .\) How much resistance should be added?

Step-by-Step Solution

Answer

To increase the resistance by 15%, we add 1087.5 ohms to the original 7250 ohms.

Step 1: Understand the Percentage Increase

To increase the resistance by 15%, we need to first find out what 15% of the original resistance is. Calculate 15% of 7250 ohms by multiplying 7250 by 0.15.

Step 2: Calculate the Additional Resistance

After finding 15% of the original resistance, that value represents the amount of resistance that needs to be added to the original to achieve the 15% increase.

Key Concepts

Understanding Percentage Increase

Calculating a percentage increase is often needed when you want to find out how much a certain value has changed in terms of a percentage. In the context of electrical circuits, you might need to increase the resistance, current, or voltage by a specific percentage for various reasons, such as adjusting the power output, or matching the performance characteristics of a particular component.

To compute a percentage increase, first determine the percentage in decimal form. For a 15% increase, this would be 0.15. Then, multiply the original value by this decimal to find the actual increase amount. If we have an original resistance of 7250 Ohms and want to increase it by 15%, we calculate the increase by multiplying 7250 by 0.15, yielding the additional resistance required.

• Decimal form of 15% is 0.15
• Multiply the original value: 7250 Ohms * 0.15
• The product is the added resistance needed for the increase

This approach is a straightforward and essential skill for anyone working with electronic components and circuits.

Applying Ohm's Law

Ohm's Law is one of the most fundamental principles in the field of electronics and electrical engineering, establishing a relationship between voltage, current, and resistance in a circuit. Denoted by the equation \( V = IR \), it explains that the voltage (V) across a resistor is equal to the product of the current (I) flowing through it and its resistance (R).

When you are working with circuits, understanding Ohm's Law helps you determine how changes in resistance affect the current and voltage, and vice versa. For instance, if you increase the resistance in a circuit while keeping the voltage constant, the current will decrease. This can have various practical applications like dimming lights or reducing the speed of an electric motor.

• Ohm's Law Equation: \( V = IR \)
• Voltage remains constant when resistance increases, current decreases
• The law is crucial for designing and analyzing circuits

Mastering the use of Ohm's Law is essential for anyone who wants to work effectively with electrical systems.

Electrical Resistance in Circuits

Electrical resistance is a measure of how much a material opposes the flow of electric current. It is an inherent property of all conductive materials, and it plays a critical role in how electrical circuits operate. Resistance is usually denoted by the symbol 'R' and measured in ohms (\( \Omega \)).

In practical terms, when you add resistance to a circuit (for instance, adding a resistor), you limit the amount of current that can flow through the circuit — much like pinching a water hose reduces the flow of water. The effect of the added resistance can be calculated using Ohm's Law as reference. In our original exercise, finding the amount of resistance to add to achieve a 15% increase allows for precise control over the current, which can be important for protecting components or achieving the desired electrical performance.

• Resistance is the opposition to current flow
• Measured in ohms (\( \Omega \))
• Adding resistance reduces current if voltage is constant
• Important for controlling and protecting circuits

Understanding resistance and how to calculate changes in resistance is essential for anyone designing, building, or troubleshooting electrical circuits.