Juneau Average Temperature

If the average daily temperature in Juneau, Alaska, during the first 20 days of January was 30.9^\circ\text{F}, what must the daily average be during the remainder of the month if Juneau’s January average is to match the historical average of 25.7^\circ\text{F}?
Source: NCTM Mathematics Teacher 2008

SOLUTION
Let a_1,a_2,a_3,\cdots,a_{31} represent the respective daily temperatures of day 1,2,3,\cdots,31 of January in Juneau, Alaska.
By definition of average we want
\dfrac{a_1+a_2+a_3+\cdots+a_{20}+a_{21}+a_{22}+a_{23}+\cdots+a_{31}}{31}=25.7^\circ
Multiply by 31
a_1+a_2+a_3+\cdots+a_{20}+a_{21}+a_{22}+a_{23}+\cdots+a_{31}=796.7^\circ
Divide by 20
\dfrac{a_1+a_2+a_3+\cdots+a_{20}}{20}+\dfrac{a_{21}+a_{22}+a_{23}+\cdots+a_{31}}{20}=\dfrac{796.7^\circ}{20}
Substitute the value of the average daily temperature during the first twenty days
30.9^\circ+\dfrac{a_{21}+a_{22}+a_{23}+\cdots+a_{31}}{20}=39.835^\circ
Simplify
\dfrac{a_{21}+a_{22}+a_{23}+\cdots+a_{31}}{20}=8.935^\circ
Multiply by 20 and divide by 11
\dfrac{20}{11}\times \dfrac{a_{21}+a_{22}+a_{23}+\cdots+a_{31}}{20}=8.935^\circ\times \dfrac{20}{11}
\dfrac{a_{21}+a_{22}+a_{23}+\cdots+a_{31}}{11}=16.2^\circ
The average daily temperature for the remaining 11 days of January should be 16.2^\circ if the Juneau’s January average is to match the historical average of 25.7^\circ.

Answer: 16.2^\circ\text{F}

Advertisements

About mvtrinh

Retired high school math teacher.
This entry was posted in Problem solving and tagged , , . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s