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Suppose \( f(t) \) is the temperature (in degrees F) of a potato \( t \) minutes after it has been placed into a heated oven. You are told that \( f(10) = 257 \) and \( f'(10) = 22.5 \).

Which of the following statements is true?

A. 11 minutes after it has been placed in the oven, the potato has a temperature of approximately 279.5° F.

B. The potato must be increasing in temperature at a constant rate of 22.5° per minute while it is in the oven.

C. 10 minutes after it has been placed in the oven, the temperature of the potato is 22.5° F.

D. 10 minutes after being placed in the oven, the temperature of the potato is changing at a rate of 257° F/minute.

E. None of the above statements is true.

Answer :

Final answer:

The correct statement is a) 11 minutes after it has been placed in the oven, the potato has a temperature of approximately 279.5° F.

Explanation:

The correct statement is a) 11 minutes after it has been placed in the oven, the potato has a temperature of approximately 279.5° F.

The given information tells us that the function f(t) represents the temperature of the potato after t minutes in the oven. Therefore, at t = 10 minutes, the temperature is f(10) = 257° F. f'(10) = 22.5 tells us the rate of change of temperature at t = 10 minutes. This means that the temperature is increasing at a rate of 22.5° F per minute at that moment.

To find the temperature at t = 11 minutes, we can use the derivative to approximate the change. Since the rate of change is constant, we can use the equation: f(11) ≈ f(10) + f'(10) * (11 - 10). Plugging in the values, we get: f(11) ≈ 257 + 22.5 * (11 - 10) which simplifies to f(11) ≈ 257 + 22.5 and gives us a temperature of approximately 279.5° F.

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