# Math Help - Question about forumula for weight loss

1. ## Question about forumula for weight loss

I was trying to come up with an equation which would calculate a person's weight over time as they add additional exercise. The problem I'm having is that the change in weight is dependent on the current weight. For example, two people at different weights will burn different amounts of calories for the same task. I can't figure out how to make an equation which can calculate the weight for a given day.

For someone of a given weight whose weight is constant, they take in a certain amount of calories and burn a certain amount of calories. The calories they burn are made up of their metabolic calories and activity calories. Metabolic calories are the calories used when your body is keeping all systems going. Activity calories are calories burned by moving around and doing stuff.

Let's say we have a situation with the following conditions:

Person weighs 250 pounds (W)
Calories intake per day = 3000
Metabolic burn rate per day = 8 calories per pound
Current activity burn rate per day = 4 calories per pound

Since the person is at equilibrium, we know that their intake calories and burn calories are the same:

3000 = 8w + 4w

At his current situation, he will stay 250 forever. But say he wants to lose weight by exercise alone. He will eat the same 3000 calories but he will increase his activity burn rate to 5 calories per pound per day. Each day his weight will change according to this equation:

change in weight = (3000 - 8w - 5w)/3600

The 3600 is there to convert calories to pounds (3600 calories per pound)

Here's where I get stuck. I can come up with this equation to describe his weight change over time:

w2 = w1 + (3000 - 8w1 - 5w1)/3600
or
w2 = w1 + (3000 - 13w1)/3600

Where:
w1 = His weight from the previous day
w2 = His weight today

I can put that formula into Excel and see how his weight changes over time. But what I'm struggling with is how to create a single equation which will calculate his weight on a given day. For example, at day 0 his weight is 250. At day 10 his weight is ? At day 100 his weight is ?

What would I need to do to make a single equation to calculate his weight on any given day?

2. ## Re: Question about forumula for weight loss

Welcome to MHF, sandyM!

What you have is called a differential equation:
change in weight per day = (3000 - 8w - 5w)/3600

The solution is:
$w(t) = c_1 e^{-13 t/3600}+3000/13$
as you can see here:
where $c_1$ is a constant such that at time t=0 the weight is the initial weight.

3. ## Re: Question about forumula for weight loss

Originally Posted by ILikeSerena
Welcome to MHF, sandyM!

What you have is called a differential equation:
change in weight per day = (3000 - 8w - 5w)/3600
The solution is:
$w(t) = c_1 e^{-13 t/3600}+3000/13$
as you can see here:where $c_1$ is a constant such that at time t=0 the weight is the initial weight.
That's awesome! I would have never figured that out. Thanks for the help and the link to the website.

4. ## Re: Question about forumula for weight loss

You're welcome!