A patient takes vitamin pills. Each day she must have at least 420 IU of vitamin \( \mathrm{A}, 12 \mathrm{mg} \) of vitamin \( \mathrm{B}_{1} \), and 90 mg of vitamin C . She can choose between pill 1 , which contains 90 IU of vitamin \( \mathrm{A}, 2 \mathrm{mg} \) of vitamin \( \mathrm{B}_{1} \), and 10 mg of vitamin C , and pill 2 , which contains 60 IU of vitamin \( \mathrm{A}, 2 \mathrm{mg} \) of vitamin \( \mathrm{B}_{1} \), and 25 mg of vitamin C . Pill 1 costs \( 25 \phi \), and pill 2 costs \( 75 \phi \). How many of each pill should she buy in order to minimize her cost? What is the minimum cost? She should buy \( \square \) of pill 1 and \( \square \) of pill 2 . The minimum cost is \( \$ \square \). (Simplify your answers. Type integers or decimals.)

Let \(x\) be the number of pill 1’s and \(y\) be the number of pill 2’s. The vitamin requirements give us the following constraints: 1. Vitamin \( \mathrm{A} \) (in IU): Pill 1 provides \(90\) IU and pill 2 provides \(60\) IU. \[ 90x + 60y \geq 420. \] Dividing through by \(30\) gives: \[ 3x + 2y \geq 14. \] 2. Vitamin \( \mathrm{B}_1 \) (in mg): Both pills provide \(2\) mg each. \[ 2x + 2y \geq 12 \quad \Longrightarrow \quad x + y \geq 6. \] 3. Vitamin \( \mathrm{C} \) (in mg): Pill 1 provides \(10\) mg and pill 2 provides \(25\) mg. \[ 10x + 25y \geq 90. \] Dividing through by \(5\) gives: \[ 2x + 5y \geq 18. \] The cost function to minimize is: \[ \text{Cost} = 25x + 75y. \] We now identify the intersection points of the constraint boundaries. **Intersection of \(x + y = 6\) and \(3x + 2y = 14\):** \[ x + y = 6 \quad \Longrightarrow \quad y = 6 - x. \] Substitute into \(3x + 2y = 14\): \[ 3x + 2(6-x) = 14 \quad \Longrightarrow \quad 3x + 12 - 2x = 14 \quad \Longrightarrow \quad x + 12 = 14, \] \[ x = 2, \quad y = 6 - 2 = 4. \] **Intersection of \(x + y = 6\) and \(2x + 5y = 18\):** Again, \(y = 6 - x\). Substitute into \(2x + 5(6-x) = 18\): \[ 2x + 30 - 5x = 18 \quad \Longrightarrow \quad -3x + 30 = 18 \quad \Longrightarrow \quad -3x = -12, \] \[ x = 4, \quad y = 6 - 4 = 2. \] Both points satisfy all constraints, as verified below: - For \((2, 4)\): - Vitamin \( \mathrm{A} \): \(90(2) + 60(4) = 180 + 240 = 420\) IU. - Vitamin \( \mathrm{B}_1 \): \(2(2) + 2(4) = 4 + 8 = 12\) mg. - Vitamin \( \mathrm{C} \): \(10(2) + 25(4) = 20 + 100 = 120\) mg. - For \((4, 2)\): - Vitamin \( \mathrm{A} \): \(90(4) + 60(2) = 360 + 120 = 480\) IU. - Vitamin \( \mathrm{B}_1 \): \(2(4) + 2(2) = 8 + 4 = 12\) mg. - Vitamin \( \mathrm{C} \): \(10(4) + 25(2) = 40 + 50 = 90\) mg. Now, calculate the cost at each candidate: - At \((2, 4)\): \[ \text{Cost} = 25(2) + 75(4) = 50 + 300 = 350. \] - At \((4, 2)\): \[ \text{Cost} = 25(4) + 75(2) = 100 + 150 = 250. \] The minimum cost is achieved at \((4, 2)\). Thus, she should buy \( \boxed{4} \) of pill 1 and \( \boxed{2} \) of pill 2. The minimum cost is \( \$\boxed{250} \).