A potato is placed in an oven heated to a temperature of 200°C. The temperature of the potato, in °C, is modeled by the function [tex]p(t) = 200 - 190(0.97^t)[/tex], where [tex]t[/tex] is the time, in minutes, that the potato has been in the oven.

1. Write down the temperature of the potato at the moment it is placed in the oven.

2. Find the temperature of the potato half an hour after it has been placed in the oven.

3. After the potato has been in the oven for [tex]k[/tex] minutes, its temperature is 40°C. Find the value of [tex]k[/tex].

The temperature of the potato when first placed in the oven is 10°C. After 30 minutes, its temperature rises to approximately 35.2°C. The potato reaches a temperature of 40°C at around 53 minutes.

Explanation:

The temperature of the potato at the moment it is placed in the oven is determined by substituting t=0 into the equation, yielding p(0) = 200 - 190(0.97^0), which simplifies to 200 - 190 = 10°C. Half an hour after it has been placed in the oven, or t=30 minutes, we substitute into the equation to find p(30) = 200 - 190(0.97^30), which rounds to 35.2°C. Finally, when the potato is 40°C, we set the equation equal to 40 and solve for t to get k = log(160/190) / log(0.97), which is approximately 53 minutes.

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