Linear Algebra Quiz: Modeling With Linear Systems

A water treatment facility has three treatment stages connected in series with recycling loops. Fresh water enters at 1000 L/min, and treated water exits at 800 L/min. Stage 2 recycles 150 L/min back to stage 1, and stage 3 recycles 100 L/min back to stage 2. If $f_1, f_2, f_3$ represent the total flow rates through stages 1, 2, and 3 respectively, what is $f_1$?

A traffic network has four intersections connected by one-way streets. Traffic flows are measured in vehicles per hour. At intersection 2, the conservation equation is $x_1 - x_3 + x_4 = 50$, where $x_1, x_3, x_4$ represent flows on streets 1, 3, and 4 respectively, and 50 represents net traffic generation at that intersection. If street 1 carries 200 vehicles/hour toward intersection 2, and street 4 carries 120 vehicles/hour away from intersection 2, what is the flow on street 3?

A supply chain network distributes products from 2 warehouses to 3 retail stores. The cost matrix $C$ has entry $c_{ij}$ representing the cost per unit to ship from warehouse $i$ to store $j$. If warehouse 1 ships quantities $x_{11}, x_{12}, x_{13}$ to stores 1, 2, 3 respectively, and warehouse 2 ships quantities $x_{21}, x_{22}, x_{23}$, which constraint ensures that store 2 receives exactly its demand $d_2$?

A metabolic pathway involves three enzymes (E1, E2, E3) that process substrates at rates $v_1, v_2, v_3$ respectively. At steady state, the concentration of each intermediate remains constant. The pathway can be represented as: $S \xrightarrow{v_1} A \xrightarrow{v_2} B \xrightarrow{v_3} P$, where enzyme E2 also has a side reaction that removes intermediate A at rate $0.2v_2$. What is the correct steady-state balance equation for intermediate A?

A logistics network ships products between 4 cities using a hub-and-spoke model with city 1 as the hub. Direct shipments between non-hub cities are not allowed. The network is represented by a flow matrix where positive entries indicate outbound flows and negative entries indicate inbound flows. At city 2 (a non-hub city), the flow balance equation is $-x_{12} + x_{21} = d_2$, where $d_2 = -50$ represents net outbound demand. If $x_{21} = 200$ units/day, what does this imply about the logistics operation at city 2?

A manufacturing company produces three types of electronic components (A, B, C) using four different machines (M1, M2, M3, M4). Each machine can produce multiple component types but with different efficiency rates. The production requirements are 500 units of A, 300 units of B, and 200 units of C per day. The company wants to minimize total machine hours while meeting all production demands.

If the efficiency matrix $E$ has entry $e_{ij}$ representing units of component $i$ produced per hour on machine $j$, and $\mathbf{h} = [h_1, h_2, h_3, h_4]^T$ represents hours each machine operates, which matrix equation correctly models this optimization problem's constraints?

An economic input-output model for a three-sector economy (agriculture, manufacturing, services) uses the equation $(I - A)\mathbf{x} = \mathbf{d}$, where $A$ is the technology matrix, $\mathbf{x}$ is total output, and $\mathbf{d}$ is final demand. If $a_{21} = 0.3$ and the manufacturing sector increases its output by 1000 units, how much additional input does manufacturing require from agriculture?