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. 0 The total amount of water that will drain from the pool is gallons. (Simplify your answer.)

To find the total amount of water that drains from the pool indefinitely, we need to set up the integral of the rate function \( R(t) \) from \( t = 0 \) to \( t = \infty \). This can be expressed as: \[ \int_0^{\infty} R(t) \, dt = \int_0^{\infty} 30 e^{-0.1 t} \, dt \] Now, plugging in the rate function, we can compute: \[ \int_0^{\infty} 30 e^{-0.1 t} \, dt \] This integral will yield the total gallons of water that drain from the pool over an indefinite period. Evaluating this integral, we can further simplify: \[ = 30 \left[ -\frac{1}{0.1} e^{-0.1t} \right]_0^{\infty} = 30 \left( 0 - (-10) \right) = 300 \text{ gallons} \] So, the total amount of water that will drain from the pool is \( 300 \) gallons.