Chapter P.FOM, Problem 1P

Renting versus buying a photocopier A certain office can purchase a photocopier for $5800 with a maintenance fee of $25 a month. On the other hand, they can rent the photocopier for $95 a month (including maintenance). If they purchase a photocopier, each copy would cost 3¢; if they rent, the copy is 6¢ per copy. The office estimates that they make 8000 copies a month.

(a) Find a formula for the cost $C$ of purchasing and using the copier for $n$ months.

(b) Find a formula for the cost $C$ of renting and using the copier for $n$ months.

(c) Make a table of the cost of each method for 1 year to 6 years of use, in 1-year increments.

(d) After how many moths of use would the cost be the same for each method?

To determine

(a)

To Find:

A formula for the cost of purchasing a photocopier for n months.

Answer to Problem 1P

Solution:

$C=5800+265n$.

Explanation of Solution

The cost of the photocopier is $5800.

The maintenance cost per month is $25.

When the photocopier is purchased, the cost per copy is 3¢.

The office makes 8000 copies per month.

Then, the cost of 8000 copies is $8000\cdot 3=\frac{24000}{100}=240$.

Because, $1=100¢.

We need to find a formula for purchasing and using the photocopier for $n$ months.

The total cost for purchasing and using a photocopier can be obtained by adding the cost of photocopier, maintenance cost and the cost of the copies.

Therefore, $C=5800+\left(25+240\right)n=5800+265n$.

Hence, the formula for purchasing and using a photocopier is $C=5800+265n$.

To determine

(b)

To Find:

A formula for the cost of renting a photocopier.

Answer to Problem 1P

Solution:

$C=575n$.

Explanation of Solution

The cost for rent and maintenance of the photocopier per month is $95.

When the office rent a photocopier, the cost per copy is 6¢.

The office makes 8000 copies per month.

Then, the cost of 8000 copies is $8000\cdot 6=\frac{48000}{100}=480$.

We need to find a formula for renting and using the photocopier for $n$ months.

Therefore, $C=\left(95+480\right)n=575n$.

Hence, the formula for purchasing and using a photocopier is $C=575n$.

To determine

(c)

To Find:

A table for the cost of each methods.

Answer to Problem 1P

Solution:

n	Purchase (in dollars)	Rent (in dollars)
12	8980	6900
24	12160	13800
36	15340	20700
48	18520	27600
60	21700	34500
72	24880	41400

Explanation of Solution

The formula for purchasing and using a photocopier is $C=5800+265n$.

Substitute $n=12$.

Then, $C=5800+265\cdot 12=8980$.

Substitute $n=24$.

Then, $C=5800+265\cdot 24=12160$.

Substitute $n=36$.

Then, $C=5800+265\cdot 36=15340$.

Substitute $n=48$.

Then, $C=5800+265\cdot 48=18520$.

Substitute $n=60$.

Then, $C=5800+265\cdot 60=21700$.

Substitute $n=72$.

Then, $C=5800+265\cdot 72=24880$.

The formula for renting and using a photocopier is $C=575n$.

Substitute $n=12$.

Then, $C=575\cdot 12=6900$.

Substitute $n=24$.

Then, $C=575\cdot 24=13800$.

Substitute $n=36$.

Then, $C=575\cdot 36=20700$.

Substitute $n=48$.

Then, $C=575\cdot 48=27600$.

Substitute $n=60$.

Then, $C=575\cdot 60=34500$.

Substitute $n=72$.

Then, $C=575\cdot 72=41400$.

To determine

(d)

To Find:

A formula for the cost of renting a photocopier.

Answer to Problem 1P

Solution:

After 19 months the cost is same for each method.

Explanation of Solution

The formula for purchasing and using a photocopier is $C=5800+265n$.

The formula for renting and using a photocopier is $C=575n$.

It is given that the cost becomes same for both the methods.

Therefore, $5800+265n=575n$.

Now, $575n-265n=5800$.

Then, $310n=5800$.

Hence, $n=\frac{5800}{310}=18.71\approx 19$.

Hence, after 19 months the cost is same for each method.