Let's tackle the problem step-by-step.
a.) Total Cost Function [tex]\( C(x) \)[/tex]:
To determine the total cost function, we need to consider both the initial setup cost and the variable cost per candle. The initial setup cost is a fixed amount of \[tex]$343, and the variable cost is \$[/tex]0.17 per candle.
The total cost function [tex]\( C(x) \)[/tex], where [tex]\( x \)[/tex] represents the number of candles made, can be formulated as:
[tex]\[ C(x) = 343 + 0.17x \][/tex]
b.) Total Revenue Function [tex]\( R(x) \)[/tex]:
If each candle is sold for \$0.78, the revenue function represents the total income from selling [tex]\( x \)[/tex] candles. This can be formulated as:
[tex]\[ R(x) = 0.78x \][/tex]
c.) Finding the Breakeven Point:
To find the smallest number of candles needed to break even, the total revenue must be equal to or greater than the total cost. We set up the inequality:
[tex]\[ R(x) \geq C(x) \][/tex]
Substituting the expressions for [tex]\( R(x) \)[/tex] and [tex]\( C(x) \)[/tex]:
[tex]\[ 0.78x \geq 343 + 0.17x \][/tex]
Next, solve for [tex]\( x \)[/tex]:
[tex]\[ 0.78x - 0.17x \geq 343 \][/tex]
[tex]\[ 0.61x \geq 343 \][/tex]
[tex]\[ x \geq \frac{343}{0.61} \][/tex]
Evaluating [tex]\(\frac{343}{0.61}\)[/tex] gives approximately 562.295.
Since we can't sell a fraction of a candle, we'll need to round up to the next whole number:
[tex]\[ x \geq 563 \][/tex]
Hence, the smallest number of candles needed to be sold in order for the total revenue to be at least the total cost is 563 candles.
In summary:
a.) Total cost function [tex]\( C(x) = 343 + 0.17x \)[/tex]
b.) Total revenue function [tex]\( R(x) = 0.78x \)[/tex]
c.) The smallest number of candles needed to break even is 563 candles.
