Mathematics

Mathematics

Problem

Water is pumped into a holding tank at a rate modeled by Q(t)=t1+t2(liters per minute),0≤t≤3.Q(t)=t\sqrt{1+t^{2}}\quad\text{(liters per minute)}, \qquad 0\le t\le 3. (a) Calculate the exact volume of water that enters the tank over the first 33 minutes. (b) The tank has a maximum capacity of 88 liters. Based on your result from part (a), determine whether the tank will overflow in this time period and justify your conclusion. (a) Evaluate ∫03Q(t) dt\displaystyle \int_{0}^{3} Q(t)\,dt exactly. (b) Compare your answer from part (a) with the tank capacity and state whether an overflow occurs, supporting your answer with quantitative reasoning.
Intuitive explanation

The key here is recognizing that a rate function tells you how fast something is happening at each instant, and to find the total amount that accumulates, you integrate the rate over your time interval. Once you have that total volume, comparing it to the tank's capacity is straightforward. The main challenge is evaluating the integral ∫03t1+t2 dt\int_0^3 t\sqrt{1+t^2}\,dt—look for a substitution that simplifies the expression under the square root.

Intuitive explanation

The key here is recognizing that a rate function tells you how fast something is happening at each instant, and to find the total amount that accumulates, you integrate the rate over your time interval. Once you have that total volume, comparing it to the tank's capacity is straightforward. The main challenge is evaluating the integral ∫03t1+t2 dt\int_0^3 t\sqrt{1+t^2}\,dt—look for a substitution that simplifies the expression under the square root.

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