Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Dec 01, 2023

Construct a function to model the rain collected in a vial, given the initial volume of 12 cc, a volume of 27 cc at one point, and a final volume of 37 cc after 2 hours.

STEP 1

Assumptions
1. The initial amount of water in the vial is 12 cc.
2. The amount of water in the vial at some point during the storm is 27 cc.
3. The amount of water in the vial 2 hours after the previous measurement is 37 cc.
4. The rate of rain collection is constant between these two measurements.

STEP 2

First, we need to find the rate of rain collection. We can do this by subtracting the initial amount of water from the final amount of water, and then dividing by the time elapsed.
$Rate = \frac{Final\, amount - Initial\, amount}{Time\, elapsed}$

STEP 3

Now, plug in the given values for the initial amount, final amount, and time elapsed to calculate the rate.
$Rate = \frac{37\, cc - 27\, cc}{2\, hours}$

STEP 4

Calculate the rate of rain collection.
$Rate = \frac{10\, cc}{2\, hours} = 5\, cc/hour$

STEP 5

Now that we have the rate, we can construct a function that shows the amount of rain collected in the vial at any given time. This function will have the form:
$f(t) = Initial\, amount + Rate \times Time$

STEP 6

Plug in the values for the initial amount and the rate into the function.
$f(t) = 12\, cc + 5\, cc/hour \times t$
This function shows the amount of rain collected in the vial at any given time t hours after the first measurement was taken.

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