Sometimes I lie too long in my bath and the water cools down. When this happens I add more hot water and mix it in. Sometimes I wonder if the bath will overflow before the water becomes warm enough.
A mathematical model of this situation can be made by expressing the temperature of the water as a function of the amount of hot water added. Suppose that the bath contained 100L of water at 36oC, and x L if hot water at the temperature of 60oC was mixed with it.
To set up the mathematical model, you need to know something about heat. The total heat in an amount of warm water depends on the quantity of water and its temperature. A unit of heat is the amount of heat required to warm 1L of water by 1oC. The amount of heat in the bath at first was 100 x 36 units. Adding x liters of hot water at 60oC would increase the amount of heat by x times 60 units.
When the hot water was mixed in, there would be 100+ x litres. If the temperature of the mixture was ToC, the quantity of heat in it would be T x (100+x). From this we can write down an equation and use it to express T as a function of x:
The graph of the function is shown below.
a) The above graph is a transformation of the basic curve y = 1 . Describe how this basic curve has been transformed to produce the above graph.
b) The equation T = 60 - 2400 | 1 | is a model for the temperature of the bathwater, what is the domain?
| x + 100|
c) As x increases, what value does the temperature of the water approach?
d) Determine how much hot water should be added to bring the temperature of the water to 40oC. ( Do this algebraically, so that your answer is accurate according to your model.)
Question 2 ( this relates to Q5)
Another time, I turned the hot tap on too much when I began to run my bath, and ended up with a bath containing 80 liters of water at 50oC. I turned off the hot tap, and mixed in cold water at 20oC.
a) find the formula expressing the temperature (ToC) of the water in the bath as a function of the volume of cold water added.
b) Use transformations to sketch a graph of this function.
c) How much water is needed to bring the water temperature down to 40oC?
d) State any assumptions or limitations that may affect your model.
I know its long, but if you could solve this, you're my hero.