Question

{Use of Tech} A mixing tank A 500-liter (L) tank is filled with pure water. At time t=0, a salt solution begins flowing into the tank at a rate of 5 L/min. At the same time, the (fully mixed) solution flows out of the tank at a rate of 5.5 L/min. The mass of salt in grams in the tank at any time t≥0 is given by M(t) = 250(1000−t)(1−10−³⁰(1000−t)¹⁰) and the volume of solution in the tank is given by V(t) = 500-0.5t.
b. Graph the volume function and verify that the tank is empty when t=1000 min.

Accepted Answer

Step 1: Understand the volume function V(t) = 500 - 0.5t, which represents the volume of the solution in the tank at any time t. The initial volume is 500 L, and it decreases by 0.5 L per minute due to the outflow rate being higher than the inflow rate. Step 2: To graph the volume function V(t), identify key points. Start with the initial condition at t = 0, where V(0) = 500 L. Then, find the volume at other significant times, such as t = 1000 min. Step 3: Calculate V(1000) to verify when the tank is empty. Substitute t = 1000 into the volume function: V(1000) = 500 - 0.5(1000). Simplify this expression to find the volume at t = 1000. Step 4: Plot the graph of V(t) using the points calculated. The graph should be a straight line with a negative slope, starting at (0, 500) and ending at (1000, 0). Step 5: Verify that the tank is empty at t = 1000 by confirming that V(1000) = 0. This confirms that the volume function correctly models the situation where the tank is empty at 1000 minutes.

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Key Concepts

Volume Function

Graphing Functions

Limits and Continuity