# The sales of air-conditioners of a manufacturer is 50000 in 2014. For each air-conditioner sold, the manufacturer gains $500. It is known that the sales increases steadily at a rate of 10% per... The sales of air-conditioners of a manufacturer is 50000 in 2014. For each air-conditioner sold, the manufacturer gains$500. It is known that the sales increases steadily at a rate of 10% per year, while the gain of each air-conditioner decreases steadily at a rate of 4% per year.

(a) What is the increase in the profit from 2014 to 2015?

(b) Will the profit in 2017 exceed that in 2014 by 20%? Explain your answer.

tiburtius | High School Teacher | (Level 2) Educator

Posted on

Let AC denote number of sales of air-conditioners and G/AC gain per air-conditioner sold.

2014

AC:        50 000

G/AC:    $500 Total: 50 000*500 =$25 000 000

To get the numbers for 2015 and 2017 we will use the formula for compound interest

C_n=C_0(1+r/100)^n

where C_0 is the starting value (50 000 for the AC) r is the rate (10 for the AC and -4 in case of gain because it is decreasing) and n  is the number of years past (since 2014  in this case).

(a)

2015

AC:       50000\cdot(1+10/100)^1=50000\cdot1.1=55000

G/AC:     500(1-4/100)^1=500\cdot0.96=$480 Total: 55000\cdot500=$26400000

We can get the increase in profit by dividing the total of year 2015 by total of year 2014.

26400000/25000000=1.056

This means that the profit in year 2015 is 105.6% that of the profit in year 2014. In other words profit has increased by 5.6%.

(b)

2017

AC:       50000(1+10/100)^3=50000\cdot1.1^3=66550

G/AC:    500(1-4/100)^3=500\cdot0.96^3=$442.37 Total: 66550\cdot442.37=$29439723.50

Again, to gain the increase in profit we divide the totals from 2017 and 2014.

29439723.5/25000000=1.1776

Therefore, the increase in profit is 17.76%, meaning that the profit from 2017 will not exceed the profit from 2014 by 20%.

Sources: