Questions: A robotic machine fills containers with an acid solution at the rate of 60+4 t milliliters ( mL ) per second, where t is in seconds and 0 ≤ t ≤ 60. How many mL are put into a container in 60 seconds? Round to the nearest whole number.

A robotic machine fills containers with an acid solution at the rate of 60+4 t milliliters ( mL ) per second, where t is in seconds and 0 ≤ t ≤ 60. How many mL are put into a container in 60 seconds? Round to the nearest whole number.

Transcript text: A robotic machine fills containers with an acid solution at the rate of $60+4 t$ milliliters ( mL ) per second, where $t$ is in seconds and $0 \leq t \leq 60$. How many mL are put into a container in 60 seconds? Round to the nearest whole number.

Solution

Solution Steps

To find the total amount of acid solution put into a container in 60 seconds, we need to integrate the rate function $60 + 4t$ with respect to time $t$ from 0 to 60 seconds. This will give us the total volume of the solution in milliliters.

Step 1: Define the Rate Function

The rate at which the robotic machine fills the containers is given by the function: \[ R(t) = 60 + 4t \quad \text{(mL/second)} \] where $t$ is the time in seconds and $0 \leq t \leq 60$.

Step 2: Set Up the Integral

To find the total volume of the acid solution filled in 60 seconds, we need to compute the definite integral of the rate function from $t = 0$ to $t = 60$: \[ V = \int_{0}^{60} R(t) \, dt = \int_{0}^{60} (60 + 4t) \, dt \]

Step 3: Calculate the Integral

Calculating the integral: \[ V = \left[ 60t + 2t^2 \right]_{0}^{60} \] Evaluating this at the bounds: \[ V = \left( 60 \cdot 60 + 2 \cdot 60^2 \right) - \left( 60 \cdot 0 + 2 \cdot 0^2 \right) = 3600 + 7200 = 10800 \, \text{mL} \]