# Microwaving two products based on individual instructions

Suppose I have a cake, which has instructions to heat it for 60 seconds in the microwave. I also have some custard, which has instructions to heat it for 30 seconds in the microwave. For convenience, I put the cake in a bowl, pour the custard over the cake, and then heat them together in one go.

How long should the bowl be placed in the microwave for, such that the custard and cake reach the temperature they would have reached based on their individual instructions? (Let's assume that after heating individually based on their individual instructions, the cake and custard would have reached the same temperature.)

Should it be:

1) 90 seconds?

2) Longer than 90 seconds?

3) Shorter than 90 seconds?

Assume the power of the microwave oven to be $P$ and that the instructions for cake and custard lead to the same temperature ($T$) of both when they are heated separately, then:

$t_1=\frac{m_1c_1(T-T_0)}{\epsilon P}$

and :

$t_2=\frac{m_2c_2(T-T_0)}{\epsilon P}$

where in the indices $1$ and $2$ refer to cake and custard, $m$ the mass, $c$ the specific heat capacity, $T_0$ the initial temperature and $\epsilon$ some efficiency factor.

Obviously we can simply add these together:

$t_1+t_2=\frac{m_1c_1(T-T_0)}{\epsilon P}+\frac{m_2c_2(T-T_0)}{\epsilon P}$

$t_1+t_2=\frac{(m_1c_1+m_2c_2)(T-T_0)}{\epsilon P}$

So bar any unexpected interactive effects between cake and custard, simply adding the prescribed heating times together is a safe bet. The efficiency factor $\epsilon$ may however differ slightly between cake and custard as microwave absorptivity tends to depend a little on water content of the food stuff.