Question

In Exercises \(29-32,\) express the desired quantity as a definite integral and evaluate the integral using Theorem \(2 .\) Find the amount of water lost from a bucket leaking 0.4 liters per hour between \(8 : 30\) A.M. and \(11 : 00\) A.M.

Short answer

The bucket lost 1 liter of water between 8:30AM to 11:00AM.

Step-by-step solution

Understanding the problem

The given problem tells about a bucket leaking at a constant rate of 0.4 liters per hour from 8:30AM to 11:00AM. The task is to find the total amount of water leaked during this time range. So, imagine that each hour is a small slice of time during which water is leaked and when we sum up all these slices, we get the total leaked amount. This is what definite integration does.

Express the quantity as a definite integral

The water leakage is a constant rate, so we can represent this as an integral over the time range. In mathematics, it's typical to use 't' for time, and 'dt' to denote a small slice of time. The constant leak rate is 0.4 liters per hour, so we express the total water leaked as the integral from 0 to 2.5 (8:30AM to 11:00AM equal to 2.5 hours) of 0.4 dt. This can be written mathematically as \(\int_{0}^{2.5} 0.4\, dt\).

Evaluate the definite integral

Now, you can evaluate the integral using basic integral rules. The integral of a constant (in this case 0.4) multiplied by time (t) is the constant times t, evaluated between the limits of integration. Thus, we have \(0.4 * [t]_{0}^{2.5}\), which simplifies to \(0.4(2.5 - 0)\).

Calculate the total quantity of water leaked

The final step is to calculate the result of the expression \(0.4 * (2.5 - 0)\), which gives a total of 1 liter.