Example 1: Differential Equation Application

The rate at which a snowball rolling down a hill gains weight is proportional to the difference between its weight at the bottom of the hill and its current weight. At time t=0 , when the snowball is weighed at the top of the hill, its weight is 10 grams. If S ( t ) is the weight of the snowball, in grams, at time t seconds after it is first weighed, then

dS dt = 1 10 ( 100 S ) .

Let y = S ( t ) be the solution to the differential equation above with initial condition S ( 0 ) = 10 .

Is the snowball gaining weight faster when it weighs 14 grams or when it weighs 17 grams?

Identifier: You are given to a differential equation to work with, and being asked a question about how a rate is behaving.

Step 1: Plug your given snowball weights, 14 grams and 17 grams , into the given differential equation.

dS dt = 1 10 ( 100 S )

Notice that the independent variable (what you need to plug into the differential equation) is the S-value , which is the weight of the snowball.

 

The given differential equation, dS dt , provides you information about the change in the S-variable (snowball weight in grams) per the change in the t-variable (seconds) .

dS dt = change in snowball weight change in time = gram second

 

dS dt = 1 10 ( 100 S )

dS dt | S = 14 = 1 10 (100 14) = 8 . 6 gram second

dS dt | S = 17 = 1 10 ( 100 17) = 8 . 3 gram second

Step 2: The question wants you to find which snowball was gaining weight faster. In other words, which one has the greatest amount of gram second .

 

To find that value, compare your differential equation results from Step 1.

dS dt | S = 14 = 8 . 6 gram second

dS dt | S = 17 = 8 . 3 gram second

8 . 6 gram second > 8 . 3 gram second

dS dt | S = 14 > dS dt | S = 17

 

 

Final Result:

The snowball is gaining weight at the rate of 8 . 6 gram second when it weighs 14 grams , and it is gaining weight at the rate of 8 . 3 gram second when it weighs 17 grams . The snowball weight is therefore growing faster (larger rate of change ) when it weighs 14 grams since it is adding weight at the rate of 8.6grams per second.

 

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