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The problem provides us with the following information:

- \( f(T) \) is the cost to heat the house when the outside temperature is \( T \) degrees Fahrenheit.
- \( f(23) = 11.88 \), meaning that at \( T = 23^\circ F \), the cost is 11.88 dollars per day.
- \( f'(23) = -0.23 \), meaning that the rate of change of the cost with respect to temperature at \( T = 23^\circ F \) is \(-0.23\), meaning the cost decreases by 0.23 dollars per day for each degree drop in temperature around \( 23^\circ F \).

We are asked to approximate the cost when the temperature is \( 20^\circ F \).

### Solution:

We can approximate the cost using the linear approximation formula, which is:

\[
f(20) \approx f(23) + f'(23) \cdot (20 - 23)
\]

Substitute the given values:

\[
f(20) \approx 11.88 + (-0.23) \cdot (20 - 23)
\]
\[
f(20) \approx 11.88 + (-0.23) \cdot (-3)
\]
\[
f(20) \approx 11.88 + 0.69
\]
\[
f(20) \approx 12.57
\]

Thus, the approximate cost to heat the house when the temperature is \( 20^\circ F \) is **12.57 dollars per day**.

Would you like any further details or have any questions?

Here are some related questions you might find useful:
1. What does \( f'(T) \) represent in this context?
2. How would the cost change if the temperature increased instead of decreased?
3. What is the significance of using linear approximation in this problem?
4. How can we improve the accuracy of the cost approximation?
5. What would happen to the cost if \( f'(T) \) were a positive value?

**Tip:** Linear approximation is most accurate when the temperature change is small. For larger changes, a more detailed model might be needed.

Suppose that f(T) is the cost to heat my house, in dollars per day, when the outside temperature is T°F. If f(23) = 11.88 and f'(23) = -0.23, approximately what is the cost to heat my house when the temperature is 20°F?

Learn how to approximate the cost to heat a house when the temperature drops, using the derivative and linear approximation method. We solve the problem where f(23) = 11.88 and f'(23) = -0.23 to estimate the heating cost at 20°F, suitable for high school calculus students.

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