Question
Water is drained from a swimming pool at a rate given by \( R(t)=30 e^{-0.1 t} \mathrm{gal} / \mathrm{hr} \). If the drain is left open indefinitely, how much water drains from the pool? Set up the integral needed to compute the total amount of water that will drain from the pool. (Typ) dt (Type exact answers.)
Ask by Hodgson Olson. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The total amount of water that drains from the pool is given by the integral \( W = \int_{0}^{\infty} 30e^{-0.1t} \, dt \).
Solution
Define the total amount of water \( W \) that drains as the integral of the rate \( R(t) \) from \( t = 0 \) to \( t = \infty \):
\[
W = \int_{0}^{\infty} 30e^{-0.1t} \, dt
\]
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Simplify this solution Extra Insights
To find the total amount of water that will drain from the pool, you'll want to set up the integral of the rate function \( R(t) = 30 e^{-0.1 t} \) over time as \( t \) approaches infinity. The integral expression for the total amount of water drained, \( A \), can be written as: \[ A = \int_0^{\infty} R(t) \, dt = \int_0^{\infty} 30 e^{-0.1 t} \, dt \] Now, calculating this integral will give you the total amount of water drained from the pool.
