Let's break down the problem step by step. --- ## **Given Information** - The **rate** at which water flows through pipe P is given by: \[ v = 2.7t^2 \text{ (in L/min)} \] where \( t \) is in minutes. --- ## **Question** **What is the total water that passes through pipe P in the first 2. min?** --- ## **Step 1: Write the Formula** The total volume of water \( V \) that passes through in time interval \( [, 2.] \) is the **integral** of the flow rate: \[ V = \int_^{2.} v \, dt \] Given \( v = 2.7t^2 \): \[ V = \int_^{2.} 2.7 t^2 \, dt \] --- ## **Step 2: Evaluate the Integral** \[ \int 2.7 t^2 \, dt = 2.7 \int t^2 \, dt = 2.7 \left( \frac{t^3}{3} \right) = .9 t^3 \] So, \[ V = [.9 t^3]_{}^{2.} \] --- ## **Step 3: Calculate the Definite Integral** Plug in the limits: \[ V = .9 (2.)^3 - .9 ()^3 \] \[ V = .9 \times 8 - \] \[ V = 7.2 \] --- ## **Step 4: Final Answer** \[ \boxed{7.2 \ \text{L}} \] --- ### **Summary Table** | Step | Expression | Result | |---------|------------------------------------|-------------| | Integral| \( \int_^{2.} 2.7 t^2 dt \) | \( .9 t^3 \) | | Evaluate| \( .9 (2.^3) - .9 (^3) \) | 7.2 | --- ### **Final Answer** **7.2 L**