Problem 101 - Master Resource Book in JEE Main Physics

Question

If a graph is plotted taking the temperature in Fahrenheit along \(Y\)-axis and the corresponding temperature in celsius along the \(X\)-axis, it will be straight line (a) having a +ve intercept on \(Y\)-axis (b) having a tve intercept on \(X\)-axis (c) passing through the origin (d) having a -ve intercepts on both the axis

Step-by-Step Solution

Answer

It has a +ve y-intercept and a -ve x-intercept, answer is (a).

Step 1: Understand the Relationship Between Fahrenheit and Celsius

To solve this exercise, we begin by recalling the formula that relates temperatures in Fahrenheit (F) and Celsius (C). The equation is given by: \[ F = \frac{9}{5}C + 32 \]. Here, Fahrenheit is expressed as a linear function of Celsius.

Step 2: Identify the Slope and Intercept

In the equation \( F = \frac{9}{5}C + 32 \), compare it with the slope-intercept form of a line \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Here, the slope \( m = \frac{9}{5} \) and the y-intercept \( b = 32 \).

Step 3: Analyze the Y-axis Intercept

The y-intercept is the point where the line crosses the y-axis. This occurs when \( C = 0 \), giving \( F = 32 \). Therefore, the graph will intersect the y-axis at \( 32 \), which is a positive intercept.

Step 4: Examine the X-axis Intercept

To find where the graph intersects the x-axis, solve for \( C \) when \( F = 0 \). Set \( 0 = \frac{9}{5}C + 32 \) and solve for \( C \). This gives \( C = -\frac{160}{9} \), which is a negative value indicating a negative intercept on the x-axis.

Step 5: Draw Conclusions from Graph Properties

Given the intercepts found, the graph does not pass through the origin. It has a positive intercept on the y-axis and a negative intercept on the x-axis.

Key Concepts

Temperature Conversion

Temperature conversion between Celsius and Fahrenheit is an important concept in understanding linear equations in two variables. The equation that relates Celsius (C) and Fahrenheit (F) temperatures is \[ F = \frac{9}{5}C + 32 \].This formula shows the linear relationship between these two temperature scales.
To convert a temperature from Celsius to Fahrenheit, you multiply it by \( \frac{9}{5} \)and then add 32. Conversely, to convert Fahrenheit to Celsius, you would subtract 32 from the Fahrenheit temperature and then multiply by \( \frac{5}{9} \).

• Conversion from Celsius to Fahrenheit: \[ F = \frac{9}{5}C + 32 \]
• Conversion from Fahrenheit to Celsius: \[ C = \frac{5}{9}(F - 32) \]

These equations allow for an easy switch between the two scales, providing a clear example of linear equations in real-world scenarios. The understanding of temperature conversion is fundamental for analyzing how temperatures vary across different systems.

Graph Analysis

Analyzing a graph involves looking at various components like the slope, intercepts, and direction of the line. In our specific example of converting Celsius to Fahrenheit, the equation \( F = \frac{9}{5}C + 32 \) forms a straight line when plotted.
By setting values of Celsius and calculating the corresponding Fahrenheit temperatures, you can plot these points on a graph with Celsius on the x-axis and Fahrenheit on the y-axis.
One crucial part of graph analysis is understanding the intercepts. The line intersects the y-axis (Fahrenheit) when Celsius is zero, resulting in a y-intercept of 32. This means the line starts at 32 on the y-axis. On the x-axis (Celsius), when Fahrenheit is zero, the solution indicates an x-intercept at roughly -17.78, or \(-\frac{160}{9}\), demonstrating that the line will cross the x-axis at a negative Celsius value.
These intercepts and the consistent slope dictate the position and angle of the line and provide insights into the relationship between the variables.

Slope and Intercept

The slope and intercept are fundamental characteristics of the line formed when graphing linear equations. In the context of the temperature conversion equation \[ F = \frac{9}{5}C + 32 \], we can break it down as follows:

• Slope (\( m \)): The slope is \( \frac{9}{5} \), indicating that for every unit increase in Celsius, Fahrenheit increases by \( \frac{9}{5} \). This reflects the steepness and direction of the line on the graph.
• Y-intercept (\( b \)): This is the point where the line crosses the y-axis, which is 32. It tells us the Fahrenheit temperature when Celsius is zero.

The slope and y-intercept are part of the slope-intercept form of a line, \( y = mx + b \). The slope helps in predicting how two variables change in relation to each other, while the intercept answers the initial value of the dependent variable when the independent variable is zero.
Understanding these concepts helps in analyzing how a graph behaves and helps in making decisions based on linear relationships.