Sure, let's solve each part of the question step by step:
Given the expense function for Wonder Widget:
[tex]\[ E = 4.14q + 55,789 \][/tex]
### a. What is the fixed cost in the expense function?
The fixed cost is the part of the expense function that does not depend on the quantity [tex]\( q \)[/tex]. From the given function, the fixed cost is \[tex]$55,789.
### Answer:
The fixed cost is \$[/tex]55,789.
### b. What is the cost of producing 500 Wonder Widgets?
To find the cost of producing 500 Wonder Widgets, we substitute [tex]\( q = 500 \)[/tex] into the expense function:
[tex]\[ E = 4.14 \times 500 + 55,789 \][/tex]
[tex]\[ E = 2,070 + 55,789 \][/tex]
[tex]\[ E = 57,859 \][/tex]
### Answer:
The cost of producing 500 Wonder Widgets is \[tex]$57,859.
### c. What is the average cost per widget of producing 500 Wonder Widgets? Round to the nearest cent.
To find the average cost per widget, we divide the total cost by the number of widgets:
\[ \text{Average Cost} = \frac{E}{q} = \frac{57,859}{500} \]
\[ \text{Average Cost} = 115.718 \]
Rounded to the nearest cent, the average cost per widget is \$[/tex]115.72.
### Answer:
The average cost per widget of producing 500 Wonder Widgets is \[tex]$115.72.
### d. What is the total cost of producing 600 Wonder Widgets?
To find the cost of producing 600 Wonder Widgets, we substitute \( q = 600 \) into the expense function:
\[ E = 4.14 \times 600 + 55,789 \]
\[ E = 2,484 + 55,789 \]
\[ E = 58,273 \]
### Answer:
The total cost of producing 600 Wonder Widgets is \$[/tex]58,273.
### e. What is the average cost per widget of producing 600 Wonder Widgets? Round to the nearest cent.
To find the average cost per widget, we divide the total cost by the number of widgets:
[tex]\[ \text{Average Cost} = \frac{E}{q} = \frac{58,273}{600} \][/tex]
[tex]\[ \text{Average Cost} = 97.1217 \][/tex]
Rounded to the nearest cent, the average cost per widget is \[tex]$97.12.
### Answer:
The average cost per widget of producing 600 Wonder Widgets is \$[/tex]97.12.
### f. As the number of widgets increased from 500 to 600, did the average expense per widget increase or decrease?
To determine whether the average expense per widget increased or decreased, we compare the average costs:
- Average cost for producing 500 widgets: \[tex]$115.72
- Average cost for producing 600 widgets: \$[/tex]97.12
Since \[tex]$97.12 is less than \$[/tex]115.72, the average expense per widget decreased.
### Answer:
As the number of widgets increased from 500 to 600, the average expense per widget decreased.
