You and your friend are trying to heat up your identical fully insulated dinners using desk lamps.

You put your food 25 cm from the lamp and your friend puts their food 35 cm away from their lamp. If your food takes 5 minutes to heat up, how long does it take your friend's food to heat up?

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Given the system, we can assume that the dinner of your friend is farther from yours. Since there are no values for amount of heat that was used for heating, we can relatively term it as the intensity that your food can receive. The intensity, I, is inversely proportional to the square of the distance (R). The expression can be written as:

`(I_1)/(I_2) = (R_2^2)/(R_1^2)`

`I_1` = Intensity at 35 cm

`I_2` = Intensity at 25 cm

With this, we can see the ratio of intensity of the light. By substituting the values of R, we can have:

`(I_1)/(I_2) = (R_2^2)/(R_1^2)`

`(I_1) =(I_2)*((R_2^2)/(R_1^2))`

`(I_1) =(I_2)*((25^2)/(35^2))`

`(I_1) =(I_2)*0.51`

This means that the intensity at 35 cm is 0.51 or 51% of the intensity at 25 cm. This means that the time needed for the dinner at 35 cm will be longer. Therefore:

`T_1 = Time at 35 cm = (T_2)/0.51 = 5/0.51` = **9.8 minutes**

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