Horizon BP in Antelope Park, Alaska, has noticed that utility bills are substantially higher when the average monthly temperature is colder. The only thing in the shop that uses natural gas is the furnace. Because of prevailing low temperatures, the furnace is used every month of the year (though less in the summer months and very little in August). Everything else in the shop runs on electricity, and electricity use is fairly constant throughout the year.

For a year, Horizon has been recording the average daily temperature and the cost of its monthly utility bills for natural gas and electricity.

 	Average Temperature	Utility Cost
January	31°F	$ 780
February	41	649
March	43	563
April	44	430
May	46	295
June	50	253
July	53	240
August	60	230
September	50	325
October	40	550
November	30	770
December	10	940
Required:
Use the high-low method to estimate utility cost for the upcoming months of January and February. The forecast for January is a near record average temperature of 5°F; temperatures in February are expected to average 40°F.



The **high-low method** is a simple technique to estimate the variable and fixed portions of costs, particularly useful when costs fluctuate with an underlying factor (in this case, temperature). This method uses the highest and lowest data points in the dataset to create an estimate of the cost relationship.

### Step-by-Step Solution:

1. **Identify the highest and lowest data points:**
   - The lowest average temperature is **10°F** in December with a utility cost of **$940**.
   - The highest average temperature is **60°F** in August with a utility cost of **$230**.

2. **Calculate the variable cost per degree (slope):**

   The formula to find the variable cost per unit (temperature in this case) is:

   \[
   \text{Variable cost per degree} = \frac{\text{Cost at lowest temp} - \text{Cost at highest temp}}{\text{Lowest temp} - \text{Highest temp}}
   \]

   Substituting the given values:

   \[
   \text{Variable cost per degree} = \frac{940 - 230}{10 - 60} = \frac{710}{-50} = -14.2 \, \text{(dollars per degree)}
   \]

   This indicates that for every 1°F increase in temperature, the utility cost decreases by **$14.20**.

3. **Calculate the fixed cost:**

   Now, using either the highest or lowest data point, we can determine the fixed cost. Using the low point (December, 10°F, $940):

   \[
   \text{Fixed cost} = \text{Cost at 10°F} - (\text{Variable cost per degree} \times 10°F)
   \]

   \[
   \text{Fixed cost} = 940 - (-14.2 \times 10) = 940 + 142 = 1082
   \]

   So, the fixed cost (the cost that does not vary with temperature) is **$1082**.

4. **Estimate utility cost for January (5°F) and February (40°F):**

   Using the cost equation:

   \[
   \text{Utility Cost} = \text{Fixed cost} + (\text{Variable cost per degree} \times \text{Temperature})
   \]

   - For **January** (forecasted temperature: 5°F):

     \[
     \text{Utility Cost} = 1082 + (-14.2 \times 5) = 1082 - 71 = 1011
     \]

     So, the estimated utility cost for January is **$1011**.

   - For **February** (forecasted temperature: 40°F):

     \[
     \text{Utility Cost} = 1082 + (-14.2 \times 40) = 1082 - 568 = 514
     \]

     So, the estimated utility cost for February is **$514**.

### Summary:
- Estimated utility cost for January (5°F): **$1011**.
- Estimated utility cost for February (40°F): **$514**.

Would you like any additional details on this calculation?

---

Here are 5 related questions for further exploration:
1. How does the high-low method compare to regression analysis in terms of accuracy?
2. What factors besides temperature might impact utility costs in this scenario?
3. How would adding data for an unusually warm month affect the high-low method estimates?
4. Can the high-low method be used to predict electricity usage alone? Why or why not?
5. What other methods could Horizon BP use to forecast utility costs?

**Tip:** The high-low method is best used when the cost relationship is linear and there are no outliers affecting the extremes.

Horizon BP in Antelope Park, Alaska, has noticed that utility bills are substantially higher when the average monthly temperature is colder. The only thing in the shop that uses natural gas is the furnace. For a year, Horizon has been recording the average daily temperature and the cost of its monthly utility bills for natural gas and electricity. Use the high-low method to estimate utility cost for the upcoming months of January and February. The forecast for January is a near record average temperature of 5°F; temperatures in February are expected to average 40°F.

Estimate utility costs for Horizon BP using the high-low method based on temperature data in Alaska. This problem involves calculating variable and fixed costs from temperatures ranging from 10°F to 60°F, and forecasting utility costs for January (5°F) and February (40°F). Suitable for Grades 10-12 students.

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