Solving Physics Problem: Hot & Cold Liquids w/Formulas

[Zygotic Embryo]

Hello, I am just starting Physics. ( Sophmore in high school)

I need some help, understanding and solving this problem.


When you mix hot and cold liquids you can find the temperature of the mixture by using the formula T= ah+bc divided by a+b, where T is the temperature of the mixture, h is the temperature of the hot liquid, c is the temperature of the cold liquid, a and b respresent the amounts of hot and cold liquids. Suppose you mix hot tea and cold milk in a ratio a:b of 9:1 and find that the temperature of the mixture is 117degrees You then change the trea:milk ratio to 2:1 and the temperature drops 96degrees. Find the initial temperatures of the tea and Milk.

Again, I can solve it Logically. But using Formula's ( showing your work ) I am not very found of..

Can someone show how to go about this. Thanks!

 

[Mk]

Can't you solve it using simple algebra?

For your first ratio, you would just plug in your values, and it would look something like this:

117=(9h/10)+(c/10)
for the second:
96=(2h/3)+(c/3)

Simplify them and work them out separately, until you have both of the variables (h and c) on one side for both equations.

You can then have "h" equal something ±c, then in the second equation inject what h equals into h's place. Solve the equation for c, then solve the first equation as you would a one-variable algebraic equation.

 

[quicknote]

Hello, welcome to the world of physics! It's great that you are starting to learn about this fascinating subject. Let's dive into this problem and see how we can solve it using formulas.

First, let's start with the given information: the ratio of hot tea to cold milk is 9:1, and the temperature of the mixture is 117 degrees. We can represent this using the formula T= ah+bc/a+b. Plugging in the values, we get:

117 = (9h + c)/10

Next, we are asked to change the ratio to 2:1 and the temperature drops to 96 degrees. We can set up a second equation using the same formula:

96 = (2h + c)/3

Now, we have two equations and two unknowns (h and c). We can solve this system of equations using algebra. First, let's multiply the first equation by 3 to get rid of the fractions:

351 = 27h + 3c

Next, let's multiply the second equation by 10 to get rid of the fractions:

960 = 20h + 10c

Now, we have two equations with the same variables, so we can subtract them to eliminate one variable:

351 - 960 = 27h - 20h + 3c - 10c

-609 = 7h - 7c

Finally, we can solve for one variable in terms of the other:

h = (7c + 609)/7

Now, we can substitute this into one of the original equations to solve for c. Let's use the second equation:

96 = (2(7c+609)/7 + c)/3

Simplifying this, we get:

96 = (14c + 609 + 3c)/21

96 = (17c + 609)/21

Now, we can solve for c:

1616 = 17c + 609

1007 = 17c

c = 59.24 degrees

Now, we can plug this value back into our equation for h to solve for h:

h = (7(59.24) + 609)/7

h = 95.06 degrees

So, the initial temperatures of the tea and milk were approximately 95.06 degrees and 59.24 degrees, respectively.

I understand that using formulas may seem daunting, but they are a