Problem page a chef is going to use a mixture of two brands of italian dressing. the first brand contains 7% vinegar, and the second brand contains 15% vinegar. the chef wants to make 240 milliliters of a dressing that is 13% vinegar. how much of each brand should she use?

Let be the amount (in mL) of the first brand (7% vinegar) and the amount of the second brand (15% vinegar). She wants to end up with a mixture with volume 240 mL, so that

and she wants it contain 13% vinegar. Each mL of the first brand contributes 0.07 mL (i.e. 7% of 1 mL) vinegar, while each mL of the second brand contributes 0.15 mL (i.e. 15% of 1 mL). The final mixture needs to contribute 0.13 mL (i.e. 13% of 1 mL) for each mL of dressing, so that

Now solve:

The chef needs to use 60 mL of the first brand and 180 mL of the second brand.

First brand: 60 milliliters

Second brand: 180 milliliters

Step-by-step explanation:

Let's call m the amount of the first dressing mark that contains 7% vinegar

Let's call n the amount of the second dressing mark that contains 15% vinegar

The resulting mixture should have 13% vinegar and 240 milliliters.

Then we know that the total amount of mixture will be:

Then the total amount of vinegar in the mixture will be:

Then we have two equations and two unknowns so we solve the system of equations. Multiply the first equation by -0.15 and add it to the second equation:

              +

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We substitute the value of m into one of the two equations and solve for n.

(a) {a₁, a₁r, a₁r², … a₁rⁿ⁻¹, …}

(b) $59 398

step-by-step explanation:

(a) the annual sequence of profits

aₙ = a₁rⁿ⁻¹

the model for the sequence of profits is {a₁, a₁r, a₁r², … a₁rⁿ⁻¹, …} .

(b) profit after 15 yr

  a₁ = $ 30 000

  r = 1.05

a₁₅ = 30 000 × 1.05¹⁴

a₁₅ = 30 000 × 1.979 93

a₁₅ = $59 398

the profit in year 15 will be $59 398.