Question

A company's manager estimated that the cost C, in dollar for producing n items is $C=7n+350$. The Company sells each item for 12 dollar. The company makes profit when total income from selling a quantity of item is greater than the total cost of producing that quantity of item. Which of the following inequality gives all possible values of n for which the manager estimates that the company will make profit.
\[\begin{align}
  & \text{A}.\text{ n }<\text{ 7}0 \\
 & \text{B}.\text{ n }<\text{ 84} \\
 & \text{C}.\text{ n }>\text{ 7}0 \\
 & \text{D}.\text{ n }>\text{ 84} \\
\end{align}\]

Answer 1

Hint: We are given the cost of production of n items as $C=7n+350$. The selling price of 1 item is 12 dollars. So, firstly we will find the total income for selling n items and mark it as S. As the selling price for 1 item is 12 dollars, so, for n items, total income will be S=12n. Now, we know profit occurs when S > C. So, we will compare $S=12n\text{ and }C=7n+350$ to get the value n for which the company will always have a profit.

Complete step-by-step solution
We are given that, the cost of producing n items is $C=7n+350$
We are also given that, the selling price for each item is 12 dollars.
The company has prepared n items. So, the selling price of each item is the same and equal to 12.
So, the selling price for n items will be given as \[n\times \text{selling price of item}\].
The selling price of 1 item is 12 dollars. So, the selling price of n items will be $12\times n$.
The selling price of n items is 12n.
Let's denote the selling price of n item by S. So, we have:
$S=12n$
We are given that profit will happen when the income from selling “n” items is greater than the total cost of production of n items. So, we are given that, the company will have a profit if S is greater than C. So,
Company will have profit is $S\text{ }>\text{ C}$
Now, we have $S=12n\text{ and }C=7n+350$.
We are asked to find those possible values of n where the company will have profit.
We know that the company will have profit if $S\text{ }>\text{ C}$.
Now, we put the value of S and C, so we get:
\[12n\text{ }>\text{ }7n+350\]
Now, simplifying further, we get:
\[12n-7n\text{ }>\text{ }350\]
Solving for n, we get:
\[5n\text{ }>\text{ 350}\]
Divide both sides with 5, we get:
\[\dfrac{5n}{5}\text{ }>\text{ }\dfrac{\text{350}}{5}\]
Therefore, $n\text{ }>\text{ 70}$
So, we get $S\text{ }>\text{ C}$ for all n greater than 70.
Means company will have profit for all $n\text{ }>\text{ 70}$
Hence, option C is the correct answer.

Note: We can cross-check that for n=70 or less than 70, company won't be in profit.
Now, we put n = 70, we first find the value of the cost price for producing n = 70 items.
\[\begin{align}
  & C=7n+350 \\
 & C=7\times 70+350 \\
\end{align}\]
So, we get: \[C=490+350\]
So total cost price is C = 840
Now we will find the total income earned by selling n = 70 items.
So, we put the value of n = 70 in S = 12n. So, we get:
\[S=12\times 70=840\]
So, we get the total income by selling 70 items as 840.
Now, by definition, profit is there if income from selling is more than the cost of the production.
Here, we get S = C = 840
So, S is not greater than C. So, no profit of n = 70 or less than 70.
Hence, the answer is correct.