You've decided to buy a printer and have narrowed it down to two choices. The laser printer costs
$150 but the average cost of each page is just 1.5 cents. The other option is an inkjet printer, which
costs $30, but each page has an average cost of 6 cents. Determine the conditions (e.g. the number
of pages) where each printer is the better buy.

Answer:

The conditions are;

Where the number of paper printed = x (x ∈ Z)

For 0 ≤ x ≤ 2,666, It costs less to own and therefore it is better to buy the ink jet printer

For 2,666 < x ≤ ∞, it costs less to own and therefore it is better to buy the lase jet printer

Step-by-step explanation:

The given parameters are'

The cost of the laser printer = $150

The average cost of each page printed with the laser printer = 1.5 cents

The cost of the inkjet printer = $30

The average cost of each page printed with the inkjet printer = 6 cents

Let 'y', represent the total cost of ownership each printer and let 'x', represent the number of paper printed, we have;

For the laser printer, we have;

[tex]y_{laser \ jet}[/tex] = 150 + 0.015·x

For the ink jet printer, we have;

[tex]y_{ink\ jet}[/tex] = 30 + 0.06·x

Therefore, the initial cost of ownership of the ink jet printer is lesser than the initial cost of ownership of the laser jet printer

However, where the cost, 'y', of the laser jet printer and the ink jet printer are equal, we have;

[tex]y_{laser \ jet}[/tex]  = [tex]y_{ink\ jet}[/tex]

∴ 150 + 0.015·x = 30 + 0.06·x

120 = 0.06·x - 0.015·x = 0.045·x

x = 120/0.045 = 2,666.[tex]\overline 6[/tex]

Therefore, given that the cost of ownership and the number of sheets printed is a straight line relationship, after printing 2,666.[tex]\overline 6[/tex] sheets or when 2,667 sheets are printed, the cost of ownership of the laser jet printer will become lesser than the cost of ownership of the ink jet printer

Therefore, the conditions where each printer is better to buy are;

For printing the number of papers in the range 0 ≤ x ≤ 2,666, the ink jet printer is better to buy

When the number of paper printed, 'x', is in the range, 2,666 < x ≤ ∞, the laser jet printer is better to buy.

We want to see when is better to buy which printer.

Printer 1 is better for 2667 pages or more.

Printer 2 is better for less than 2667 pages.

Printer 1 costs $150 pls 1.5 cents per page, so the cost for x pages is:

C₁(x) = $150 + $0.015*x

Printer 2 costs only $30, but each page costs 6 cents, so we have the cost:

C₂(x) = $30 + $0.06*x

First, we need to find the value of x such that the two costs are equal:

C₁(x) = C₂(x)

$150 + $0.015*x = $30 + $0.06*x

$150 - $30 = $0.06*x -  $0.015*x

$120 = $0.045*x

$120/$0.045 = x = 2,666.67

Rounding to the next whole number we have x = 2,667

Then, if you plan to use less than x = 2,667, you should buy the second printer because it has a smaller y-intercept.

If you plan to use 2,667 or more, you should buy the first printer, because it has a smaller slope.

If you want to learn more, you can read

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