To solve the problem, we need to determine the temperature on the additional day that allows the new mean temperature to be 30°F. Here is a detailed step-by-step solution:
1. Understand the original data:
Kurt has the following recorded temperatures:
[tex]\[
27, 28, 28, 28, 29, 29, 30, 31, 31
\][/tex]
Let's count the number of days for which temperatures were recorded:
[tex]\[
\text{Number of days} = 9
\][/tex]
2. Calculate the sum of the original temperatures:
[tex]\[
\text{Sum of original temperatures} = 27 + 28 + 28 + 28 + 29 + 29 + 30 + 31 + 31
\][/tex]
[tex]\[
= 27 + (3 \times 28) + (2 \times 29) + 30 + (2 \times 31)
\][/tex]
[tex]\[
= 27 + 84 + 58 + 30 + 62
\][/tex]
[tex]\[
= 261
\][/tex]
3. Determine the new mean temperature
The new mean temperature after including the temperature of the new day is 30°F.
4. Set up the equation to find the new temperature:
Let's assume the temperature on the additional day is \( x \).
The equation for the new mean temperature is:
[tex]\[
\text{New mean} = 30 = \frac{\text{Sum of original temperatures} + x}{\text{Number of days} + 1}
\][/tex]
Plugging the values into the equation:
[tex]\[
30 = \frac{261 + x}{10}
\][/tex]
5. Solve for \( x \):
Multiply both sides of the equation by 10 to isolate \( x \):
[tex]\[
30 \times 10 = 261 + x
\][/tex]
[tex]\[
300 = 261 + x
\][/tex]
Subtract 261 from both sides:
[tex]\[
x = 300 - 261
\][/tex]
[tex]\[
x = 39
\][/tex]
Thus, the temperature on the additional day is [tex]\( 39^\circ F \)[/tex].
