Math Problem Statement

The relationship between the Celsius​ (°C) and Fahrenheit​ (°F) scales for measuring temperature is given by the equation Fequalsnine fifths Upper C plus 32 The relationship between the Celsius​ (°C) and Kelvin​ (K) scales is KequalsCplus273. Graph the equation Fequalsnine fifths Upper C plus 32 using degrees Fahrenheit on the​ y-axis and degrees Celsius on the​ x-axis. Use transformations to obtain the graph showing the relationship between Kelvin and Fahrenheit temperatures. Question content area bottom Part 1 Graph the equation Fequalsnine fifths Upper C plus 32 using degrees Fahrenheit on the​ y-axis and degrees Celsius on the​ x-axis. Choose the correct graph below. A. 0 100 0 320 degrees Upper Fdegrees Upper C

A coordinate system has a horizontal axis labeled degrees Celsius from 0 to 100 in increments of 10 and a vertical axis labeled degrees Fahrenheit from 0 to 320 in increments of 32. A line that rises from left to right passes through the points (0, 64) and (80, 208). B. 100 -192 0 192 degrees Upper Fdegrees Upper C

A coordinate system has a horizontal axis labeled degrees Celsius from 0 to 100 in increments of 10 and a vertical axis labeled degrees Fahrenheit from 0 to 320 in increments of 32. A line that rises from left to right passes through the points (40, 72) and (85, 153). C. 100 -192 0 192 degrees Upper Fdegrees Upper C

A coordinate system has a horizontal axis labeled degrees Celsius from 0 to 100 in increments of 10 and a vertical axis labeled degrees Fahrenheit from negative 192 to 192 in increments of 32. A line that rises from left to right passes through the points (0, negative 32) and (80, 112). D. 0 100 0 320 degrees Upper Fdegrees Upper C

A coordinate system has a horizontal axis labeled degrees Celsius from 0 to 100 in increments of 10 and a vertical axis labeled degrees Fahrenheit from 0 to 320 in increments of 32. A line that rises from left to right passes through the points (0, 32) and (90, 192). Part 2 Using transformations obtain the graph that shows the relationship between Kelvin and Fahrenheit temperatures. Choose the correct graph below. A. 200 440 0 384 degrees Upper FUpper K

A coordinate system has a horizontal axis labeled Kelvin from 200 to 440 in increments of 20 and a vertical axis labeled degrees Fahrenheit from 0 to 384 in increments of 32. A line that rises from left to right passes through the points (200, 87) and (240, 159). B. 200 440 0 384 degrees Upper FUpper K

A coordinate system has a horizontal axis labeled Kelvin from 200 to 440 in increments of 20 and a vertical axis labeled degrees Fahrenheit from 0 to 384 in increments of 32. A line that rises from left to right passes through the points (200, 0) and (240, 72). C. 200 440 0 384 degrees Upper FUpper K 0

A coordinate system has a horizontal axis labeled Kelvin from 200 to 440 in increments of 20 and a vertical axis labeled degrees Fahrenheit from 0 to 384 in increments of 32. A line that rises from left to right passes through the points (305, 25.6) and (380, 160.6).

Solution

Let's break this down into the two parts of your problem:

Part 1: Graph the equation F=95C+32F = \frac{9}{5}C + 32

This equation represents the relationship between Celsius (C) and Fahrenheit (F). To create a graph of this equation, you can calculate several values of Fahrenheit for corresponding Celsius values. Here's how we can approach this:

  • When C = 0:
    F=95(0)+32=32°FF = \frac{9}{5}(0) + 32 = 32 \, \text{°F}

  • When C = 100:
    F=95(100)+32=180+32=212°FF = \frac{9}{5}(100) + 32 = 180 + 32 = 212 \, \text{°F}

So, we know that the points (0, 32) and (100, 212) are key points on the graph. The line should start at (0, 32) and rise to (100, 212).

Now, let's look at the options:

  • Option A:
    The graph passes through the points (0, 64) and (80, 208), which doesn't match the expected points. This is not correct.

  • Option B:
    The graph passes through (40, 72) and (85, 153). These points don't match either. This is not correct.

  • Option C:
    The graph passes through (0, -32) and (80, 112), which doesn't fit the relationship for Celsius and Fahrenheit. This is not correct.

  • Option D:
    The graph passes through (0, 32) and (90, 192). This matches the expected relationship. This is the correct graph.

Part 2: Transformation to show the relationship between Kelvin and Fahrenheit

The relationship between Celsius and Kelvin is given by: K=C+273K = C + 273 Now, to transform the equation for Celsius into one for Kelvin, we substitute C=K273C = K - 273 into the Fahrenheit equation:

F=95(K273)+32F = \frac{9}{5}(K - 273) + 32 Simplifying this: F=95K9×273/5+32F = \frac{9}{5}K - 9 \times 273/5 + 32 F=95K491.4+32F = \frac{9}{5}K - 491.4 + 32 F=95K459.4F = \frac{9}{5}K - 459.4

So, this transformed equation gives us the relationship between Kelvin (K) and Fahrenheit (F).

Let's check the options for Part 2:

  • Option A:
    The points (200, 87) and (240, 159) don't match the equation F=95K459.4F = \frac{9}{5}K - 459.4. This is not correct.

  • Option B:
    The points (200, 0) and (240, 72) are too far off from the expected values. This is not correct.

  • Option C:
    The points (305, 25.6) and (380, 160.6) are closer to the expected values. This is the correct graph.

Summary:

  • Part 1: The correct graph is option D.
  • Part 2: The correct graph is option C.

Would you like further clarification or additional details on these transformations? Here are some related questions that might interest you:

  1. How can you derive the formula for Fahrenheit from Celsius?
  2. What is the historical background of the Celsius and Fahrenheit scales?
  3. How does temperature conversion relate to real-life applications (e.g., cooking, weather)?
  4. How would you graph other temperature scales, like Rankine or Réaumur?
  5. Can you explain the difference between absolute zero in Kelvin and Celsius?

Tip: Always pay attention to the slope and intercept in linear equations. In temperature conversions, the slope tells you how much one scale increases relative to the other, and the intercept tells you the baseline shift (e.g., Celsius to Fahrenheit starts at a different point).

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Graphing
Temperature Conversion

Formulas

F = (9/5)C + 32
K = C + 273
F = (9/5)(K - 273) + 32

Theorems

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Suitable Grade Level

Grades 9-12