Questions: Joel assumes that the average predicted high in Destin will be 10° warmer than the average predicted high in Kingsport for Sunday through Thursday. In the table below, predicted daily high temperatures are indicated with H, while predicted daily low temperatures are indicated with L. - Sunday - Kingsport, TN: H:68, L: 40 - Destin, FL: H:84, L: 65 - Monday - Kingsport, TN: H: 72, L: 41 - Destin, FL: H:82, L:60 - Tuesday - Kingsport, TN: H: 73, L: 45 - Destin, FL: H: 78, L: 64 - Wednesday - Kingsport, TN: H: 75, L: 48 - Destin, FL: H:80, L:66 If the predicted high in Kingsport is 72° on Thursday, what would Thursday's predicted high for Destin need to be for Joel's assumption to be correct?

Joel assumes that the average predicted high in Destin will be 10° warmer than the average predicted high in Kingsport for Sunday through Thursday. In the table below, predicted daily high temperatures are indicated with H, while predicted daily low temperatures are indicated with L. - Sunday - Kingsport, TN: H:68, L: 40 - Destin, FL: H:84, L: 65 - Monday - Kingsport, TN: H: 72, L: 41 - Destin, FL: H:82, L:60 - Tuesday - Kingsport, TN: H: 73, L: 45 - Destin, FL: H: 78, L: 64 - Wednesday - Kingsport, TN: H: 75, L: 48 - Destin, FL: H:80, L:66 If the predicted high in Kingsport is 72° on Thursday, what would Thursday's predicted high for Destin need to be for Joel's assumption to be correct?
Transcript text: Joel assumes that the average predicted high in Destin will be $10^{\circ}$ warmer than the average predicted high in Kingsport for Sunday through Thursday. In the table below, predicted daily high temperatures are indicated with an H , while predicted daily low temperatures are indicated with an L. \begin{tabular}{|c|c|c|c|c|} \hline & Sunday & Monday & Tuesday & Wednesday \\ \hline \multirow[t]{2}{*}{\begin{tabular}{l} Kingsport, \\ TN \end{tabular}} & H:68 & H: 72 & H: 73 & H: 75 \\ \hline & L: 40 & L: 41 & L: 45 & L: 48 \\ \hline \multirow[t]{2}{*}{Destin, FL} & H:84 & H:82 & H: 78 & H:80 \\ \hline & L: 65 & L:60 & L: 64 & L:66 \\ \hline \end{tabular} If the predicted high in Kingsport is $72^{\circ}$ on Thursday, what would Thursday's predicted high for Destin need to be for Joel's assumption to be correct?
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Solution

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Solution Steps

To solve this problem, we need to calculate the average predicted high temperatures for both Kingsport and Destin from Sunday to Thursday. We will then determine the required high temperature for Destin on Thursday such that its average high is 10 degrees warmer than Kingsport's average high.

  1. Calculate the average high temperature for Kingsport from Sunday to Thursday.
  2. Calculate the average high temperature for Destin from Sunday to Wednesday.
  3. Determine the required high temperature for Destin on Thursday to make its average 10 degrees higher than Kingsport's average.
Step 1: Calculate the Average High Temperature for Kingsport

The predicted high temperatures for Kingsport from Sunday to Thursday are \( 68, 72, 73, 75, 72 \). The average high temperature can be calculated as follows:

\[ \text{Average}_{\text{Kingsport}} = \frac{68 + 72 + 73 + 75 + 72}{5} = \frac{360}{5} = 72.0 \]

Step 2: Calculate the Sum of High Temperatures for Destin

The predicted high temperatures for Destin from Sunday to Wednesday are \( 84, 82, 78, 80 \). The sum of these temperatures is:

\[ \text{Sum}_{\text{Destin}} = 84 + 82 + 78 + 80 = 324 \]

Step 3: Determine the Required High Temperature for Destin on Thursday

To find the required high temperature for Destin on Thursday, we need to ensure that the average high temperature for Destin is \( 10 \) degrees warmer than that of Kingsport. Thus, we set up the equation:

\[ \text{Average}_{\text{Destin}} = \frac{\text{Sum}_{\text{Destin}} + \text{Required}_{\text{Thursday}}}{5} \]

We want:

\[ \text{Average}_{\text{Destin}} = \text{Average}_{\text{Kingsport}} + 10 \]

Substituting the known values:

\[ \frac{324 + \text{Required}_{\text{Thursday}}}{5} = 72.0 + 10 \]

This simplifies to:

\[ \frac{324 + \text{Required}_{\text{Thursday}}}{5} = 82.0 \]

Multiplying both sides by \( 5 \):

\[ 324 + \text{Required}_{\text{Thursday}} = 410 \]

Solving for \( \text{Required}_{\text{Thursday}} \):

\[ \text{Required}_{\text{Thursday}} = 410 - 324 = 86.0 \]

Final Answer

The required high temperature for Destin on Thursday is \\(\boxed{86.0}\\).

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