1. A Pyrex plate is filled to the brim with liquid pumpkin pie filling, then baked at 190 degrees Celsius. The pie plate is h=3.60 cm deep, but the sides of the dish taper from the top to bottom so that the bottom is a circle of diameter d2=25.8cm, while the top opening is a Circe diameter d1=27.4cm.
The filling is prepared at 20 degrees Celsius, but when it is baked it expands and .236 L of it flows over the edge of the pie pan and ends up in the bottom of the oven. Compute the thermal expansion for the volume of the pie plate itself, and then use the overflow amount for the filling to get an estimate for the thermal expansion coefficient (alpha pumpkin) of pumpkin pie filling.
Note: volume of the pie plate can be found using v=(1/3)( a1+a2+(square root of a1xa2))(h)
To calculate the thermal expansion for the volume of the pie plate, we need to calculate the initial volume and the final volume after baking.
1. Calculating the initial volume (V_initial):
- The top opening of the pie plate is a circle with diameter d1 = 27.4 cm, so the radius is r1 = d1/2 = 27.4/2 = 13.7 cm.
- The bottom of the pie plate is a circle with diameter d2 = 25.8 cm, so the radius is r2 = d2/2 = 25.8/2 = 12.9 cm.
- Using the formula for volume of a frustum of a cone:
V_initial = (1/3)(πr1^2 + πr2^2 + √(πr1^2 * πr2^2))(h)
= (1/3)(π*(13.7)^2 + π*(12.9)^2 + √(π*(13.7)^2 * π*(12.9)^2))(3.60 cm)
≈ 1017.64 cm^3
2. Calculating the final volume (V_final):
- The initial volume (V_initial) plus the overflow amount is the final volume (V_final), so:
V_final = V_initial + overflow amount
= 1017.64 cm^3 + 0.236 L
3. Converting the overflow amount to cm^3:
- 1 L = 1000 cm^3, so 0.236 L = 0.236 * 1000 cm^3 = 236 cm^3.
V_final = 1017.64 cm^3 + 236 cm^3
= 1253.64 cm^3
Now, we can calculate the thermal expansion for the volume of the pie plate:
Thermal expansion = (V_final - V_initial) / V_initial
= (1253.64 cm^3 - 1017.64 cm^3) / 1017.64 cm^3
≈ 0.2327
To estimate the thermal expansion coefficient (alpha pumpkin) of pumpkin pie filling, we can use the overflow amount and the change in temperature.
1. Convert the overflow amount to liters:
- 1 cm^3 = 0.001 L, so 236 cm^3 = 236 * 0.001 L = 0.236 L.
2. Calculate the change in volume and change in temperature:
- Change in volume = overflow amount = 0.236 L
- Change in temperature = final temperature - initial temperature
3. Rearrange the thermal expansion equation:
Thermal expansion = (change in volume) / (initial volume * change in temperature)
4. Plug in the values and solve for the thermal expansion coefficient (alpha pumpkin):
0.2327 = (0.236 L) / (V_initial * change in temperature)
alpha pumpkin = (0.236 L) / (V_initial * change in temperature * 0.2327)
Note: The value of change in temperature is missing from the given information, so you need to input that value to calculate the thermal expansion coefficient (alpha pumpkin).