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.

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tiburtius | High School Teacher | (Level 2) Educator

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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:

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