Bradley, working in his 23 C kitchen, is cooking himself a crepe in an iron skillet that has a circular bottom with a diameter of .30 m. How hot must the skillet be in order for Bradley to make a .07103 m^2 crepe that just fills the bottom of the pan?
(coefficient of expansion of iron=
12x10^-6 C^-1)
225.8 C
To find the required temperature of the skillet, we can use the formula:
Final area = Initial area x (1 + coefficient of expansion x change in temperature)
Let's break down the problem step-by-step:
Step 1: Calculate the initial area of the skillet.
The initial area of the skillet can be calculated using the formula for the area of a circle:
Initial area = π x (radius)^2
Given that the diameter of the skillet is 0.30 m, the radius will be half of that:
Radius = 0.30 m / 2 = 0.15 m
Using this value, we can calculate the initial area:
Initial area = π x (0.15 m)^2
Step 2: Calculate the change in area.
The change in area will be the final area of the crepe minus the initial area of the skillet:
Change in area = Final area - Initial area
Given that the final area of the crepe is 0.07103 m^2 and we calculated the initial area in step 1, we can now find the change in area.
Change in area = 0.07103 m^2 - Initial area
Step 3: Set up and solve the equation using the coefficient of expansion.
The equation for the change in area can be written as:
Change in area = Initial area x (coefficient of expansion x change in temperature)
Substituting the values we have, the equation becomes:
0.07103 m^2 - Initial area = Initial area x (coefficient of expansion x change in temperature)
Step 4: Solve for the change in temperature.
Rearranging the equation, we can solve for the change in temperature:
Change in temperature = (0.07103 m^2 - Initial area) / (Initial area x coefficient of expansion)
Step 5: Calculate the required temperature of the skillet.
To find the required temperature of the skillet, we need to add the change in temperature to the initial temperature of the skillet:
Required temperature = Initial temperature + Change in temperature
Assuming the initial temperature of Bradley's 23°C kitchen, we can calculate the required temperature of the skillet.
To determine how hot the skillet must be, we need to consider the change in diameter due to the coefficient of expansion of iron.
Here are the steps to calculate the required temperature:
1. Calculate the radius of the skillet's bottom:
The diameter of the skillet is given as 0.30 m. Divide it by 2 to get the radius.
Radius = Diameter / 2 = 0.30 m / 2 = 0.15 m.
2. Calculate the initial area of the skillet's bottom:
The initial area of the skillet's bottom is given by the formula: A = πr^2, where r is the initial radius.
Initial Area = π * (0.15 m)^2.
3. Calculate the final area of the crepe, which is the same as the skillet's bottom:
Final Area = 0.07103 m^2.
4. Calculate the linear expansion of the skillet:
Linear Expansion = Coefficient of Expansion * Original Length * Temperature Change.
The original length is the initial diameter, which is equal to 2 * initial radius.
Linear Expansion = 12 x 10^-6 C^-1 * (2 * 0.15 m) * Temperature Change.
5. Use the linear expansion to find the change in area:
Change in Area = 2 * Linear Expansion * Initial Area.
Note that we multiply by 2 since the expansion occurs equally in both directions.
Change in Area = 2 * Linear Expansion * Initial Area.
6. Determine the final diameter of the skillet:
Final Diameter = Initial Diameter + Change in Diameter.
Change in Diameter = 2 * Change in radius.
Final Diameter = Initial Diameter + 2 * Change in radius.
7. Solve for the required temperature:
Set the final area (crepe area) equal to the final area of the skillet and solve for the temperature:
Final Area (Crepe Area) = π * (Final Diameter / 2)^2 = 0.07103 m^2.
From this equation, we can isolate the temperature as the unknown variable.
These steps should allow you to calculate the temperature needed for Bradley to make a crepe that just fills the bottom of the pan.