1.10 Exercises

20 points · 4 extra credit

Linear Models in Business, Science, and Engineering

1. 1. (2 pts) A juice blend is made by mixing $x$ liters of a $20\%$-juice drink with $y$ liters of a $50\%$-juice drink to obtain $9$ liters of a $40\%$-juice drink. Set up and solve the linear system for $x$ and $y$.

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2. 2. (3 pts) Two foods A and B are combined to meet nutrient targets. Per serving, Food A supplies $30$ units of protein and $10$ of calcium; Food B supplies $10$ of protein and $40$ of calcium. Find the numbers of servings $a$ and $b$ that provide exactly $90$ units of protein and $140$ units of calcium.

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3. 3. (3 pts) Each year $10\%$ of a city's population moves to its suburbs and $20\%$ of the suburban population moves to the city (no other change). Writing the population as $\mathbf{x} = \begin{bmatrix} \text{city} \\ \text{suburb} \end{bmatrix}$ (in thousands), the one-year update is $\mathbf{x}_{1} = M\mathbf{x}_0$ with

   $M = \begin{bmatrix} 0.9 & 0.2 \\ 0.1 & 0.8 \end{bmatrix}.$

   If $\mathbf{x}_0 = \begin{bmatrix} 600 \\ 400 \end{bmatrix}$, compute $\mathbf{x}_1$ and verify the total population is unchanged.

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4. 4. (3 pts) In a circuit, three currents satisfy Kirchhoff's laws:

   $\begin{aligned} I_1 - I_2 - I_3 &= 0 \\ 2I_1 + 3I_2 &= 12 \\ 3I_2 + 4I_3 &= 10. \end{aligned}$

   Find $I_1, I_2, I_3$.

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5. 5. (3 pts) An investor splits $10,000 among three funds A, B, C with annual returns $5\%$, $7\%$, $10\%$. She wants a total return of $680 and decides to place twice as much in A as in C. Letting $x, y, z$ be the amounts in A, B, C, set up and solve the system.

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6. 6. (3 pts) State whether each of the following is true or false, and justify each answer.

   (a) In a migration (stochastic) matrix like $M$ above, each column sums to $1$.

   (b) Every linear model has a unique solution.

   (c) In a diet model, each equation corresponds to one nutrient constraint.

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7. 7. (3 pts) Explain what each column of the migration matrix $M = \begin{bmatrix} 0.9 & 0.2 \\ 0.1 & 0.8 \end{bmatrix}$ represents, and explain why each column must sum to $1$.

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8. 8. (4 pts ••) Find the population distribution, expressed as fractions of the total, that is unchanged from one year to the next under $M$ above — i.e. the vector $\mathbf{x}$ with $M\mathbf{x} = \mathbf{x}$ and entries summing to $1$. Verify that $M\mathbf{x} = \mathbf{x}$ for your answer.

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