UpStudy Free Solution:

To find out how much the temperature increases in degrees Fahrenheit when it increases by 2.10 kelvins, we need to determine the change in the function \(F( x) \) when \(x\) increases by 2.10.

Given the function:

\[F( x) = \frac { 9} { 5} ( x - 273.15) + 32\]

Let’s denote the initial temperature in kelvins as \(x_ 1\) and the increased temperature as \(x_ 2 = x_ 1 + 2.10\).

The change in temperature in degrees Fahrenheit, \(\Delta F\), is given by:

\[\Delta F = F( x_ 2) - F( x_ 1) \]

First, calculate \(F( x_ 2) \):

\[F( x_ 2) = \frac { 9} { 5} ( x_ 1 + 2.10 - 273.15) + 32\]

\[F( x_ 2) = \frac { 9} { 5} ( x_ 1 - 273.15 + 2.10) + 32\]

\[F( x_ 2) = \frac { 9} { 5} ( x_ 1 - 273.15) + \frac { 9} { 5} \cdot 2.10 + 32\]

\[F( x_ 2) = F( x_ 1) + \frac { 9} { 5} \cdot 2.10\]

Now, calculate the term \(\frac { 9} { 5} \cdot 2.10\):

\[\frac { 9} { 5} \cdot 2.10 = 1.8 \cdot 2.10 = 3.78\]

Therefore, the increase in temperature in degrees Fahrenheit is:

\[\Delta F = 3.78\]

So the temperature increased by 3.78 degrees Fahrenheit. The correct answer is:

A) 3.78

Supplemental Knowledge

Temperature Conversion Formula

The given function converts temperature from Kelvin (\(x\)) to Fahrenheit (\(F( x) \)):

\[F( x) = \frac { 9} { 5} ( x - 273.15) + 32\]

To find how much the temperature in degrees Fahrenheit increases when the temperature in Kelvin increases by a certain amount, we can analyze the linear relationship of the function.

Linear Relationship Analysis

The function \(F( x) = \frac { 9} { 5} ( x - 273.15) + 32\) is linear, meaning any change in \(x\) (Kelvin) will result in a proportional change in \(F( x) \) (Fahrenheit). The coefficient of \(x\), which is \(\frac { 9} { 5} \), represents how much \(F( x) \) changes for each unit change in \(x\).

If the temperature increases by 2.10 Kelvins:

\[\Delta x = 2.10\]

The corresponding increase in Fahrenheit can be calculated using the coefficient of \(x\):

\[\Delta F = \frac { 9} { 5} \times 2.10 = \frac { 9} { 5} \times 2.10 = 1.8 \times 2.10 = 3.78\]

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