A company that makes thing-a-ma-bobs has a start up cost of $\$ 15116$. It costs the company $\$ 1.65$ to make each thing-a-ma-bob. The company charges $\$ 3.15$ for each thing-a-ma-bob. Let $x$ denote the number of thing-a-ma-bobs produced.
Write the cost function for this company.
$C(x)=\square$ (Your answer needs to be an expression containg the variable $x$ )
Write the revenue function for this company.
$R(x)=\square$ (Your answer needs to be an expression containing the variable $x$ )
What is the minumum number of thing-a-ma-bobs that the company must produce and sell to make a profit? $\square$ (Your answer needs to be rounded up to the nearest number.)
Answer
Since the company cannot produce a fraction of a thing-a-ma-bob, we round up to the nearest whole number. So, the minimum number of thing-a-ma-bobs that the company must produce and sell to make a profit is \(\boxed{10078}\).
Step 1 ：The cost function for the company is the start up cost plus the cost to make each thing-a-ma-bob times the number of thing-a-ma-bobs produced. So, the cost function is \(C(x)=1.65x + 15116\).
Step 2 ：The revenue function for the company is the price the company charges for each thing-a-ma-bob times the number of thing-a-ma-bobs sold. So, the revenue function is \(R(x)=3.15x\).
Step 3 ：To find the minimum number of thing-a-ma-bobs the company must produce and sell to make a profit, we need to find the point where the revenue function is greater than the cost function. This can be done by setting the cost function equal to the revenue function and solving for x, then rounding up to the nearest whole number.
Step 4 ：Solving the equation \(1.65x + 15116 = 3.15x\), we get \(x = 10077.3333333333\).
Step 5 ：Since the company cannot produce a fraction of a thing-a-ma-bob, we round up to the nearest whole number. So, the minimum number of thing-a-ma-bobs that the company must produce and sell to make a profit is \(\boxed{10078}\).
