Question

A biologist is developing two new strains of bacteria. Each sample of type i bacteria produces 4 new viable bacteria, and each sample of type il produces 3 new viable bacteria. Altogether, at least 240 new viable bacteria must be produced. At least 30, but not more than 60 of the original samples must be type i. Not more than 70 of the samples are to be type il. A sample of type i costs P500 and a sample of type il costs P700. If both types are to be used, how many samples of each should be used to minimize the cost?

Answer 1

To minimize the cost, the biologist should use 30 samples of type i bacteria and 70 samples of type ii bacteria.

Step-by-step explanation:

The goal is to minimize the cost while meeting the requirements of producing at least 240 new viable bacteria and using at least 30 but not more than 60 of type i samples and not more than 70 of type ii samples.

Let's denote the number of type i samples as x and the number of type ii samples as y. The total number of new viable bacteria produced is given by the equation
`\( 4x + 3y \)`, and this should be at least 240.

`So, \( 4x + 3y \geq 240 \).`

The constraints are:

`1. \( 30 \leq x \leq 60 \)` (number of type i samples)

2.
`\( y \leq 70 \)` (number of type ii samples)

To minimize the cost, we consider the cost function. A type i sample costs P500, and a type ii sample costs P700. The total cost
`(\( C \)) is given by \( C = 500x + 700y \).`

Now, we can set up the objective function to minimize c subject to the constraint
`\( 4x + 3y \geq 240 \) and the given constraints for \( x \) and \( y \).`

After solving this linear programming problem, the optimal solution is
`\( x = 30 \) and \( y = 70 \),` resulting in the minimum cost while satisfying all the conditions. Therefore, using 30 samples of type i bacteria and 70 samples of type ii bacteria minimizes the cost.

Answer 2

To minimize the cost, the biologist should use 30 samples of type i bacteria and 70 samples of type ii bacteria.The problem is visualized by plotting inequalities to find the minimum cost at the vertices of the feasible region.

Step-by-step explanation:

To minimize the cost of producing at least 240 new viable bacteria using two different strains, we need to set up and solve a linear programming problem. The constraints given in the question are: each sample of type i bacteria produces 4 new viable bacteria, and each sample of type ii bacteria produces 3 new viable bacteria, with a necessity of at least 240 new viable bacteria.

At least 30 but not more than 60 of the original samples must be type i, and not more than 70 samples of type ii are allowed. The cost of a sample of type i is P500 and type ii is P700.

Let's denote the number of type i samples as x and the number of type ii samples as y. We can set up the following inequalities based on the problem's constraints:

• 4x + 3y ≥ 240 (to produce at least 240 viable bacteria)
• x ≥ 30 (at least 30 samples of type i)
• x ≤ 60 (no more than 60 samples of type i)
• y ≤ 70 (no more than 70 samples of type ii)

The cost function to be minimized is:

Cost = 500x + 700y

To minimize the cost, we would plot the inequalities on a graph to find the feasible region, then calculate the cost at each vertex of the feasible region to determine the minimum cost.