This is a long one, but it is a no solve.
Youʼre cooking four pancakes, one at a time, using a stove-top timer to judge the time Ät needed to cook each
side (i.e. half) of each circular cake (radius = r).
The timer emits a “ding,” a sound whose pressure variation is described by P = Pmaxcos(ùt ± êx). Your ear
is at a distance from the timer that is equal to exactly 7 wavelengths of the sound, and from that distance, the
loudness of the sound is âdinger.
The timerʼs sound power output is just .01% as much as any one of the stoveʼs burners. Youʼre using one of
those burners—but at just 80% of its maximum power —to heat the pancake griddle. That burnerʼs surface
temperature is T burner . The pancake batter (c pancake and âpancake ) is initially at temperature T i and thickness li. The griddle bottom (of thickness L and conductivity k) ) transmits only 25% of the burnerʼs total power to the pancake. Assume that each pancake is cooked evenly, achieving the same final temperature throughout as the inside bottom of the griddle.
As the pancakes are cooked, you stack them neatly and tightly in a dish in the oven (maintaining them at their
fi nal cooked temperature). When all are done, you pour enough syrup (density = ñsyrup), which is also kept at that final pancake temperature, around the pancakes so that the stack is fl oating freely (oh yeah).
What volume of well-chilled butter (density = ñwellchilledbutter) must you place on top so that the entire stack of cakes (but none of the butter) is submerged? (The pancakes do not, alas, soak up any syrup.)
To determine the volume of well-chilled butter required to submerge the stack of pancakes, we need to go through several steps. Let's break it down into a step-by-step process:
Step 1: Calculate the power output of the stove burner.
- Given that the timer's sound power output is just 0.01% of the burner's power output.
- Assuming the burner's power output at maximum is P_max, the timer's sound power is 0.0001 * P_max.
Step 2: Determine the temperature of the burner surface.
- Given that the burner is being operated at 80% of its maximum power.
- Assuming the maximum temperature of the burner surface as T_burner, then the actual surface temperature is 0.8 * T_burner.
Step 3: Calculate the power transmitted from the burner to the pancake griddle.
- Given that only 25% of the burner's total power is transmitted to the pancake.
- Assuming the transmitted power as P_transmitted, then P_transmitted = (0.25 * P_max).
Step 4: Calculate the heat received by the pancake griddle.
- Since power is proportional to heat, the heat received by the pancake griddle is P_transmitted.
Step 5: Calculate the temperature change of the pancake griddle.
- Using the formula Q = mLΔT, where Q is the heat received, m is the mass of the pancake griddle, L is the thickness of the griddle, and ΔT is the temperature change.
- Assuming the mass of the pancake griddle as M_griddle and its specific heat capacity as C_griddle, we can calculate ΔT.
Step 6: Calculate the final temperature of the pancake griddle.
- Assuming the initial temperature of the pancake batter as T_i and the thickness of the pancake batter as l_i.
- Using the formula T_final = T_i + ΔT * (l_i / L), we can calculate the final temperature of the pancake griddle.
Step 7: Calculate the volume of butter required.
- Assuming the density of the butter as ρ_butter and the density of the syrup as ρ_syrup.
- Since the stack of pancakes and syrup are floating freely, the buoyant force should equal the weight of the stack.
- The weight of the stack can be calculated by multiplying the total volume of the pancakes and syrup by their combined density.
- Assuming the combined volume of the pancakes and syrup as V_stack, we can calculate the volume of the syrup.
- Then, subtracting the volume of the syrup from V_stack will give us the volume of butter required.
Please note that some additional information, such as the radius of the pancake and the length of a wavelength, is missing in the question, which might affect the accuracy of the solution.
To determine the volume of well-chilled butter needed to submerge the entire stack of pancakes, we need to break down the problem into smaller steps and variables. Let's go through each step:
Step 1: Calculate the total volume of the stack of pancakes.
To determine the volume of the stack, we need to know the area of one pancake and multiply it by the number of pancakes in the stack. The area of a pancake is given by the formula:
A_pancake = π * r^2
where r is the radius of the pancake.
Step 2: Calculate the total volume of the stack and syrup combined.
Since the stack of pancakes is floating freely in the syrup, the volume of syrup needed is equal to the total volume of the stack. So, we need to calculate the volume of the stack of pancakes using the formula from Step 1.
V_pancakes = A_pancake * n
where n is the number of pancakes in the stack.
Step 3: Calculate the mass of the syrup.
To find the mass of the syrup, we need to know its density (ρ_syrup) and the volume of the syrup (V_pancakes calculated in Step 2).
m_syrup = ρ_syrup * V_pancakes
Step 4: Calculate the mass of the stack of pancakes.
We need to find the mass of the stack of pancakes. Assuming the pancakes do not soak up any syrup, their mass is equal to the mass of the syrup calculated in Step 3.
m_pancakes = m_syrup
Step 5: Calculate the mass of the butter needed.
To determine the mass of the butter needed, we need to make some assumptions. Let's assume that the entire stack of pancakes is fully submerged in the well-chilled butter. In this case, the mass of the butter would be equal to the mass of the stack of pancakes.
m_butter = m_pancakes
Step 6: Calculate the volume of the butter needed.
To find the volume of butter needed, we need to know the density of the well-chilled butter (ρ_wellchilledbutter) and the mass of the butter (m_butter calculated in Step 5).
V_butter = m_butter / ρ_wellchilledbutter
Now you can use these steps and equations to calculate the volume of well-chilled butter needed to submerge the entire stack of pancakes. Just plug in the known values for radius, number of pancakes, density of syrup, and density of well-chilled butter into the equations, and you'll get the answer.