High School

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Assuming that the amount of water leaking from the faucet every hour increases uniformly by \(\frac{1}{12}\) of the amount that leaked in the previous hour, write a function representing the amount of water leaked each hour.

* The faucet leaked 76 ml of water in the first hour.

Answer :

The amount of water leaked from a faucet that increases uniformly by 1/12 each hour can be represented by the function f(t) = 76 * (13/12)^(t-1), which defines a geometric sequence with the first term being 76 ml and the common ratio being 13/12.

The student is asking for a function that represents the amount of water leaked each hour from a faucet, with the leak rate increasing uniformly by 1/12 of the amount leaked in the previous hour. Since the faucet leaked 76 ml in the first hour, we can define a function f(t), where f(t) is the amount of water leaked in hour t. The initial amount leaked f(1) is 76 ml. For subsequent hours, f(t) = f(t-1) + 1/12 * f(t-1), meaning that f(t) = (1 + 1/12) * f(t-1) = (13/12) * f(t-1). Therefore, f(t) can be expressed as a geometric sequence with the first term a = 76 and the common ratio r = 13/12.

The formula for the nth term of a geometric series is a * r^(n-1), so the leakage function can be written as f(t) = 76 * (13/12)^(t-1).

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