Question

nicole needed room-temperature butter for a cookie recipe. she took two sticks of butter out of the refrigerator and put them on the counter. each stick had the same amount of butter, but one stick was long and thin, and one stick was short and wide. this table shows how the temperature of each stick of butter changed over 30 minutes.

stick of butter	initial temperature (°c)	final temperature (°c)
long stick of butter	4	13

the next time nicole needs butter for a cookie recipe, she wants it to warm up to room temperature quickly. what should she do to maximize the rate of thermal energy transfer into the stick of butter?

• use a short and wide stick of butter
• use a long and thin stick of butter
• either; the rate of thermal energy transfer is the same whether or not the butter is long or short

Explanation:

First, calculate the temperature change rate for each stick. For the short stick: $\frac{10 - 3}{30}=\frac{7}{30}\approx0.23^\circ\text{C/min}$. For the long stick: $\frac{13 - 4}{30}=\frac{9}{30} = 0.3^\circ\text{C/min}$. Wait, no—wait, surface area affects heat transfer. A long and thin stick has more surface area relative to volume (since same amount of butter, so same volume). Wait, no, the initial calculation was wrong. Wait, the problem is about maximizing thermal energy transfer rate. Thermal energy transfer (heat) depends on surface area (more surface area, more heat transfer). A long and thin stick has greater surface area than a short and wide one (same volume, as same amount of butter). Wait, but wait the table: short stick initial 3, final 10 (change 7 in 30 min). Long stick initial 4, final 13 (change 9 in 30 min). Wait, maybe I miscalculated. Wait, no—wait, the question is about what to do to maximize the rate. Wait, no, maybe the error is in my first thought. Wait, no—wait, the long and thin stick: let's recalculate the rate. Short stick: (10 - 3)/30 = 7/30 ≈ 0.233 °C per minute. Long stick: (13 - 4)/30 = 9/30 = 0.3 °C per minute. So the long and thin stick has a higher rate. Wait, but the options: "use a long and thin stick of butter" is an option. Wait, but maybe my initial surface area thought was wrong. Wait, the problem is about thermal energy transfer rate. The rate here is (final - initial)/time. So for short: (10-3)/30 = 7/30 ≈0.23. Long: (13-4)/30=9/30=0.3. So the long stick (long and thin) has a higher rate. So to maximize the rate, she should use a long and thin stick. Wait, but the options: the second option is "use a long and thin stick of butter". Wait, but maybe I made a mistake. Wait, let's check again. The short stick: initial 3, final 10: change 7 in 30 min. Long stick: initial 4, final 13: change 9 in 30 min. So 9/30 is higher than 7/30. So the long stick (long and thin) warms up faster. So the answer should be "use a long and thin stick of butter".

Answer:

use a long and thin stick of butter