A Styrofoam cooler has a surface area of $2,700 \mathrm{~cm}^{2}$ and a wall thickness of $3.00 \mathrm{~cm}$. Styrofoam has a thermal conductivity of $0.010 \mathrm{~W} / \mathrm{m}{ }^{\circ} \mathrm{C}$. A $2.00 \mathrm{~kg}$ block of ice is placed inside the cooler that has a temperature inside of $2^{\circ} \mathrm{C}$. When the temperature outside is $30.0^{\circ} \mathrm{C}$, the ice lasts for 8.00 hours. If the temperature outside becomes $20.0^{\circ} \mathrm{C}$, then how long will the ice last? Ignore the effect of the air inside the cooler.

Question

Quizplus · Accepted Answer

The heat flow rate through the Styrofoam is given by $Q/t = kA(T_{\text{outside}} - T_{\text{inside}})/d$, where $k$ is the thermal conductivity, $A$ is the surface area, $T_{\text{outside}}$ and $T_{\text{inside}}$ are the temperatures outside and inside respectively, and $d$ is the thickness of the walls. When the outside temperature is $30.0^\circ C$, the temperature difference is $28.0^\circ C$, and when the outside temperature is $20.0^\circ C$, the temperature difference is $18.0^\circ C$. The time the ice lasts is inversely proportional to the temperature difference, so when the temperature difference decreases from $28.0^\circ C$ to $18.0^\circ C$, the time increases by a factor of $28/18$, or $1.555$. Multiplying the original 8.00 hours by this factor gives $8.00 \times 1.555 = 12.44$ hours, which is approximately $12.4$ hours.

A Styrofoam Cooler Has a Surface Area Of 2,700 cm22,700 \mathrm{~cm}^{2}2,700 cm2 And a Wall Thickness Of

A Styrofoam Cooler Has a Surface Area Of $2,700 \mathrm{~cm}^{2}$ And a Wall Thickness Of