You are working on a new temperature scale that will unify the earth. After some thought you decide to call it the Ava Abate-universal-scale. Let 𝐹
denote the temperature in degrees Fahrenheit, and let 𝑋
denote your new temperature scale. You want it to be such that if 𝐹=0
then 𝑋=21
and if 𝐹=100
then 𝑋=112
. You also want 𝑋
to be such that if you plot 𝑋
against 𝐹
you obtain a straight line. (This is described as "linear interpolation" in many textbooks.) You obtain the formula
𝑋=𝑚𝐹+𝑏

where 𝑚
=

and 𝑏
=

To find the constants \( m \) and \( b \) for the equation \( X = mF + b \), where \( F \) is the temperature in degrees Fahrenheit and \( X \) is the temperature in the Ava Abate-universal-scale, we can use the two given conditions:

1. When \( F = 0 \), \( X = 21 \).
2. When \( F = 100 \), \( X = 112 \).

These two points will allow us to determine the slope \( m \) and the intercept \( b \).

### Step 1: Find the slope \( m \)
The slope \( m \) for a straight line is given by the formula:
\[
m = \frac{X_2 - X_1}{F_2 - F_1}
\]
where:
- \( X_1 = 21 \) when \( F_1 = 0 \),
- \( X_2 = 112 \) when \( F_2 = 100 \).

Substitute the values:
\[
m = \frac{112 - 21}{100 - 0} = \frac{91}{100} = 0.91
\]

### Step 2: Find the intercept \( b \)
To find \( b \), substitute the value of \( m \) and one of the points (either \( (0, 21) \) or \( (100, 112) \)) into the equation \( X = mF + b \).

Using the point \( (0, 21) \):
\[
21 = 0.91(0) + b
\]
\[
b = 21
\]

### Final Formula
Thus, the formula for the Ava Abate-universal-scale is:
\[
X = 0.91F + 21
\]

### Summary:
- \( m = 0.91 \)
- \( b = 21 \)

Would you like further clarification on any part of this? Here are 5 related questions to explore further:

1. How can we use this formula to convert any temperature from Fahrenheit to the new Ava Abate-universal-scale?
2. What would the freezing point of water (32°F) be on the Ava Abate-universal-scale?
3. How would the boiling point of water (212°F) translate to the new scale?
4. Can we express Fahrenheit in terms of \( X \) on the Ava Abate-universal-scale?
5. How does this compare to other temperature scales like Celsius or Kelvin?

**Tip:** When creating any linear relationship, finding the slope is essential, as it defines how two variables change relative to one another.

You are working on a new temperature scale that will unify the earth. After some thought you decide to call it the Ava Abate-universal-scale. Let F denote the temperature in degrees Fahrenheit, and let X denote your new temperature scale. You want it to be such that if F=0 then X=21 and if F=100 then X=112. You also want X to be such that if you plot X against F you obtain a straight line. (This is described as 'linear interpolation' in many textbooks.) You obtain the formula X=mF+b where m= and b=

Learn how to create a linear temperature conversion scale using the Ava Abate-universal-scale. Explore how to use linear interpolation, calculate the slope (m), and determine the intercept (b) for temperatures in degrees Fahrenheit (F) and their corresponding values in the new scale (X). This problem is suitable for Grades 9-11.

Math Problem Statement

where 𝑚

and 𝑏

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