# Math used in this equation rearrangement?

• MarchON
In summary, the conversation is discussing a problem involving the use of water to raise the temperature of a car from -25°C to 0°C. The equation being used is mwcwΔTw + mwLf,w = mcccΔTc, which has been rearranged to solve for the mass of water, mw. The inclusion of latent heat of fusion is due to the water transitioning from liquid to solid as it freezes on the car. However, there may be an error in the algebra of the equation and further clarification is needed.

#### MarchON

I'm trying to determine how much water it takes to raise a car's temperature from -25°C to 0°C. The water is at 10°C.

What I apparently need to have set up is:

-ΔUint,water = ΔUint,car

-(mwcwΔTw) - mwLf,w = mcccΔTc

The resultant rearranged equation looking for mass of water gives this:

mw = -(mcccΔTc)/cwΔTw - mwLf,w

I don't understand how this was done. Also, why are you subtracting Latent heat of fusion x Mass from mwcwΔTw?

Looks like there's a mistake in the algebra, then. What do you think that eqn should be?

Q for you: why does latent heat of fusion enter into the picture at all?

MarchON said:
The resultant rearranged equation looking for mass of water gives this:

mw = -(mcccΔTc)/cwΔTw - mwLf,w

Says who?

(Hint: are the units consistent?)

Latent heat of fusion is in the picture because the water is going from a liquid to a solid. It freezes when it hits the car, then at a certain point it doesn't because the car warms up to 0 degrees. And I don't know what's up with the equation, but that's what my professor's solution says. Based on my math, I got something that makes no sense:

0= mcarccarΔTcar/CwΔTw + Lf

The left side should be mw. You need a pair of brackets on the right side, and then it should look right.

I don't understand how. Is there any way (and I know this is no easy task) to break down the algebra step by step for me?
Also, I made a mistake with the resultant equation in my first post. It's actually mw = -(mcccΔTc)/cwΔTw - Lf,w (no - mwLf,w)

I realize that ends up being the same thing that you said and there is no error (but he kept the negatives in, whereas we canceled them out), but I still don't understand how.

You started with this: -mwcwΔTw - mwLf,w = mcccΔTc
Taking out a common factor -mw we have
-mw(cwΔTw +Lf,w)= mcccΔTc

Now divide both sides by (cwΔTw +Lf,w)
and we are left with
-mw = mcccΔTc / (cwΔTw +Lf,w)

Multiplying both sides by -1 so that we end up with mw by itself,
mw = -mcccΔTc / (cwΔTw +Lf,w)

The brackets I said you needed are those in the denominator; the ones you added in the numerator make no difference.

## 1. How is math used in equation rearrangement?

Equation rearrangement involves manipulating mathematical expressions in order to solve for a specific variable. This requires the use of various mathematical operations such as addition, subtraction, multiplication, and division.

## 2. Why is it important to rearrange equations?

Rearranging equations is important because it allows us to solve for a specific variable and gain a better understanding of the relationship between different variables in an equation. It also helps in simplifying complex equations and finding more efficient solutions.

## 3. What are some common techniques used in equation rearrangement?

Some common techniques used in equation rearrangement include the distributive property, combining like terms, isolating variables on one side of the equation, and using inverse operations to solve for a specific variable.

## 4. How does algebra play a role in equation rearrangement?

Algebra is the branch of mathematics that deals with manipulating and solving equations. It provides the rules and techniques for rearranging equations and finding solutions to complex problems.

## 5. Can equation rearrangement be used in real-world applications?

Yes, equation rearrangement is used extensively in real-world applications such as engineering, physics, economics, and many other fields. It allows us to model and solve real-world problems by rearranging equations to find the desired variable.