How Do You Calculate the Power of a Heater in a Vending Machine?

[RoryP]

Just revising for my practical physics exam tomorrow and got stuck on this! not sure if its correct, just annpyed me that i was unsure!

Homework Statement

A vending machine for serving hot drinks has a tank of capacity 7.5 litres, and the manufactuurers say that 45 cups of hot water (at 99^o C) can be produced at anyone time.
a) what is the volume of one cup
b) if 170 cups can be produced in one hour, what is the power of the heater? assume that the water starts at a temperature of 20 ⁰ C and that there are no energy losses.

Homework Equations

Q=mc delta t
P=E/t
s.h.c of water = 4200 j/kg/k

The Attempt at a Solution

a) 7.5/45 = 0.16 litres
b) 1 litre of water = 1 kg
7.5 litres = 7.5 kg
Q=mc delta t= 7.5*4200*79= 2488500 J ( that's for 45 cups)
so for 170 cups Q= (2488500/45)*170= 9401000 J
P=E/t= 9401000/3600 = 2611.4 W
not sure if this is the correct method, any help will be greatly appreciated!

 

[Amok]

I don't see anything wrong with what you did.

 

[nlightner]

Your calculations and method seem correct. Specific heat capacity is the amount of energy needed to raise the temperature of a substance by 1 degree Celsius. In this case, the specific heat capacity of water is 4200 J/kg/K, which means that it takes 4200 joules of energy to raise the temperature of 1 kg of water by 1 degree Celsius. Using this equation, you were able to calculate the amount of energy needed to heat 7.5 kg of water from 20 degrees Celsius to 99 degrees Celsius.

For part (a), you correctly calculated the volume of one cup of water to be 0.16 litres, which is equivalent to 0.16 kg. This means that each cup of hot water produced by the vending machine contains 0.16 kg of water.

For part (b), you correctly used the equation P=E/t to calculate the power of the heater. This equation states that power (P) is equal to the energy (E) divided by the time (t). In this case, the energy needed to heat 170 cups of water is 9401000 J, and the time it takes to produce 170 cups is 3600 seconds (1 hour). Therefore, the power of the heater is 2611.4 watts.

Overall, your understanding and application of specific heat capacity is correct. Keep up the good work and good luck on your exam!