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 is the weight of the snowball, in grams, at time t seconds after it is first weighed, then . Let be the solution to the differential equation above with initial condition . Is the snowball gaining weight faster when it weighs 14 grams or when it weighs 17 grams? |
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Identifier: You are given to a differential equation to work with, and being asked a question about how a rate is behaving. |
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Step 1: Plug your given snowball weights, 14 grams and 17 grams , into the given differential equation.
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, , provides you information about the change in the S-variable (snowball weight in grams) per the change in the t-variable (seconds) .
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Step 2: The question wants you to find which snowball was gaining weight faster. In other words, which one has the greatest amount of .
To find that value, compare your differential equation results from Step 1. |
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Final Result: The snowball is gaining weight at the rate of when it weighs 14 grams , and it is gaining weight at the rate of 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. |