(Solved): The rate at which a body cools also depends on its exposed surface area \( S \). If \( S \) is a co ...

The rate at which a body cools also depends on its exposed surface area \( S \). If \( S \) is a constant, then a modification of ( 2 ), given in Section 3.1, is \[ \frac{d T}{d t}=k S\left(T-T_{m}\right) \text {, } \] where \( k<0 \) and \( T_{m} \) is a constant. Suppose that two cups \( A \) and \( B \) are filled with coffee at the same time. Initially, the temperature of the coffee is \( 165^{\circ} \mathrm{F} \). The exposed surface area of the coffee in cup \( B \) is twice the surface area of the coffee in cup A. After \( 30 \mathrm{~min} \) the temperature of the coffee in cup \( A \) is \( 115^{\circ} \mathrm{F} \). If \( T_{\mathrm{m}}=85^{\circ} \mathrm{F} \), then what is the temperature of the coffee in cup B after \( 30 \mathrm{~min} \) ? (Round your answer to two decimal places.)