Answer:
a) C = 20x + 200
b) R = 50x
The Break-even point is at (6.67, 333.33)
See the graph below
Explanation:
Given:
The cost to hire equipment and facilities = $200
The cost to paint each board = $20
The charge per decoration = $50
To find:
the cost equation and revenue equation
break-even point using graph and equation
a) For the cost equation:
let the number of boards = x
The equation given for the cost equation is C = mx + c
where m = cost to paint each board = $20
c = cost to hire equipment and facilities = 200
The equation becomes:
[tex]\begin{gathered} C\text{ = 20\lparen x\rparen + 200} \\ C=\text{ 20x + 200} \end{gathered}[/tex]
b) For the revenue equation:
let the number of boards decorated = x
The equation given for the revenue equation is R = mx + c
m = charge to decorate each board = $50
c = additional payment = 0
The equation becomes:
[tex]\begin{gathered} R=\text{ 50\lparen x\rparen + 0} \\ R\text{ = 50x} \end{gathered}[/tex]
c) Plotting the 2 points for cost equation: C = 20x + 200
when x = 0
C = 20(0) + 200 = 200
C = 200
when x = 10
C = 20(10) + 200 = 200 + 200
C = 400
Plotting the 2 points for the revenue equation: R = 50x
when x = 0
R = 50(0)
R = 0
when x = 10
R = 50(10)
R = 500
d) Plotting the lines:
On the y-axis, each box represents 100 units
On the x-axis, each box represents 2 units
The 2 points for each equation are on the graph
e) Using the graph to get the break-even point;
The point of intersection of both equations will be the break-even point
Break-even point on the graph (x, y): (6.67, 333.33)
They need to decorate 6.67 boards to break even
f) At break-even, cost = revenue
To determine the number of boards they need to break even, we will equate the equation for the cost and the revenue
[tex]\begin{gathered} C\text{ = R} \\ 20x\text{ + 200 = 50x} \end{gathered}[/tex][tex]\begin{gathered} subtract\text{ 20x from both sides:} \\ 20x\text{ - 20x + 200 = 50x - 20x} \\ 200\text{ = 30x} \\ \\ divide\text{ both sides by 30:} \\ \frac{200}{30}=\text{ }\frac{30x}{30} \\ x\text{ = 6}\frac{2}{3} \end{gathered}[/tex]
They need to decorate 6.67 boards to break even
when x = 6 2/3 = 6.67
R = 50(6 2/3) = 333.33
C = 20(6 2/3) + 200 = 333.33
Hence, the break-even point is (6.67, 333.33)
