To solve this problem, we need to apply Newton's Law of Cooling, which states that the rate of heat loss of a body is proportional to the difference in temperature between the body and its surroundings, provided this difference is small. In this case, we have two solid copper spheres with different sizes and temperatures, and we want to find the value of x in the cooling rate of the larger sphere.
Understanding the Problem
We have two spheres:
- The first sphere has a temperature of 127°C and cools at a rate of 2.8°C/min.
- The second sphere has twice the radius, a temperature of 327°C, and its cooling rate is given as 9.72/x.
The room temperature is maintained at 27°C for both spheres. We need to find the value of x.
Applying Newton's Law of Cooling
According to Newton's Law of Cooling, the rate of temperature change can be expressed as:
Rate of cooling = k(T - T_room)
Where:
- k is a constant that depends on the properties of the material and the environment.
- T is the temperature of the object.
- T_room is the ambient temperature (27°C in this case).
Calculating the Cooling Rate for the First Sphere
For the first sphere:
- Temperature (T) = 127°C
- Room Temperature (T_room) = 27°C
- Cooling Rate = 2.8°C/min
Using the formula:
2.8 = k(127 - 27)
2.8 = k(100)
From this, we can solve for k:
k = 2.8 / 100 = 0.028
Calculating the Cooling Rate for the Second Sphere
Now, for the second sphere:
- Temperature (T) = 327°C
- Room Temperature (T_room) = 27°C
- Cooling Rate = 9.72/x
Using the same formula:
9.72/x = k(327 - 27)
9.72/x = k(300)
Substituting the value of k we found earlier:
9.72/x = 0.028(300)
9.72/x = 8.4
Solving for x
Now, we can rearrange the equation to find x:
9.72 = 8.4x
x = 9.72 / 8.4
x ≈ 1.15
Final Result
The value of x is approximately 1.15. This means that the cooling rate of the second sphere, when expressed as 9.72/x, indicates that it cools at a rate of about 8.4°C/min when its temperature is 327°C.
