The problem is as follows:
In an electronics factory, the owner calculates that the cost to
produce his new model of portable TV is $26$ dollars. After meeting
with the distributors, he agrees the sale price for his new product to
be $25$ dollars each and additionally $8\%$ more for each TV set sold
after $8000$ units. What is the least number of TV's he has to sell in
order to make a profit?.
The answers are:
- 16000
- 15001
- 16001
- 15999
- 17121
This problem has made me to go in circles on how to express it in a mathematical expression. I'm not sure if it does need to use of inequations.
What I tried to far is to think this way:
The first scenario is what if what he sells is $8000$ units, then this would become into:
$$\textrm{production cost:}\,26\frac{\$}{\textrm{unit}} \times 8000\,\textrm{units} = 208000\,\$$$
$$\textrm{sales:}\,25\frac{\$}{\textrm{unit}} \times 8000\,\textrm{units}=\,200000\,\$$$
Therefore there will be an offset of $8000\,\$$ as
$$208000\$-200000\$\,=\,8000\,\$$$
So I thought what If I consider the second part of the problem which it says that he will receive an additional of $8\%$ after $8000$ units.
Therefore his new sale price will be $27\,\$$ because:
$$25+\frac{8}{100}\left(25\right )=27\,\$$$
So from this I thought that this can be used in the previous two relations.
But how?.
I tried to establish this inequation:
$$26\left(8000+x\right)<25\left(8000\right)+27\left(8000+x\right)$$
But that's where I'm stuck at since it is not possible to obtain a reasonable result from this as one side will be negative and the other positive.
The logic I used was to add up $8000\,\$$ plus something which is the production cost must be less than what has been obtained from selling the first $8000$ units plus a quantity to be added to those $8000$.
However there seems to be an error in this approach. Can somebody help me to find the right way to solve this problem?
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