# Help with linear equation

#### San1998

##### New member
Hi,

I'm struggling with this question. I've finished all the subsections besides subsection (d.) Is that section asking for the intersection point of the cost and profit axis? Can anyone post a full solution for this?
Question
Auto-time, a manufacturer of 24-hour variable timers, has a monthly fixed cost of RM48,000 and a production cost of RM8 for each timer manufactured. The units sell for RM14 each.
(a) Sketch the graphs of the cost function and the revenue function, and hence find the break-even point graphically.
(b) Find the break-even point algebraically.
(c) Sketch the graph of the profit function.
(d) At what point does the graph of the profit function cross the x-axis? Interpret your result.

#### JeffM

##### Elite Member
Hi,

I'm struggling with this question. I've finished all the subsections besides subsection (d.) Is that section asking for the intersection point of the cost and profit axis? Can anyone post a full solution for this?
Question
Auto-time, a manufacturer of 24-hour variable timers, has a monthly fixed cost of RM48,000 and a production cost of RM8 for each timer manufactured. The units sell for RM14 each.
(a) Sketch the graphs of the cost function and the revenue function, and hence find the break-even point graphically.
(b) Find the break-even point algebraically.
(c) Sketch the graph of the profit function.
(d) At what point does the graph of the profit function cross the x-axis? Interpret your result.
Because part d is dependent on parts a, b, and c, we cannot possibly understand where you are having difficulty in part d without understanding what you did in those earlier parts. I knowthat it is difficult to show graphs, but you can answer questions about how you put those graphs together. So before we get to the specifics of what is confusing you about part d, please answer these questions about the parts you have done. Perhaps the reason that part d is hard is that you have made a mistake in part a or b or c.

In part a, your y-axis represents what? What does your x-axis represent? What expressions did you use to sketch the revenue and cost functions? So what did the graphs tell you about the breakeven point?

In part b, how did you algebraically define the breakeven point? What answer did you get?

In part c, what expression did you use to sketch the profit function?

#### San1998

##### New member
Because part d is dependent on parts a, b, and c, we cannot possibly understand where you are having difficulty in part d without understanding what you did in those earlier parts. I knowthat it is difficult to show graphs, but you can answer questions about how you put those graphs together. So before we get to the specifics of what is confusing you about part d, please answer these questions about the parts you have done. Perhaps the reason that part d is hard is that you have made a mistake in part a or b or c.

In part a, your y-axis represents what? What does your x-axis represent? What expressions did you use to sketch the revenue and cost functions? So what did the graphs tell you about the breakeven point?

In part b, how did you algebraically define the breakeven point? What answer did you get?

In part c, what expression did you use to sketch the profit function?
Hi Jeff!

My x-axis is number of timers sold and y-axis is price.
Cost = 8x + 48,000
Revenue = 14x
Profit = Revenue - Cost = 14x - (8x + 48,000) = 6x - 48,000
My break-even point was (8000,112000)

#### HallsofIvy

##### Elite Member
Auto-time, a manufacturer of 24-hour variable timers, has a monthly fixed cost of RM48,000 and a production cost of RM8 for each timer manufactured. The units sell for RM14 each.
So if they make and sell T timers, their cost is 48000+ 8T. Their income is 14T. Those are both "linear functions" so their graphs are straight lines. And straight lines are determined by two points.

When T= 0, the cost is 48000+ 8(0)= 48000. When T= 10000, the cost is 48000+ 8(10000)= 128000. Mark the points (0, 48000) and (10000, 128000) on your graph and draw the line through them. When T= 0, the income is 14(0)= 0. When T= 10000, the income is 14(10000)= 140000. Mark the points (0, 0) and (100, 140000) and draw the line between them.

The "break even point" is where there is neither profit nor loss- the cost of production is the same as the income. Where do those two lines intersect?

Algebraically, saying that cost is the same as income means that the two functions above are equal: 48000+ 8T= 14T. Solve that equation for T. You should get the same T as where the two lines intersect.

The profit is the income minus the cost: 14T- (48000+ 8T)= 14T- 8T- 48000= 6T- 48000. Again that is a straight line. When T= 0, the "profit" is 6(0)- 48000= -48000 (negative since this is actually a loss). When T= 10000 the profit is 6(10000)- 48000= 60000- 48000= 12000. Mark the points (0, -48000) and (10000, 12000) and draw the line between them. Notice that (0, -48000) is below the x-axis and (10000, 12000) is above it. A some place between T= 0 and T= 10000 that line must cross the x-axis. Of course, on the x-axis, the profit is 0- there is no profit and no loss. That is again the "break even point". To find that T set the profit equal to 0, 6T- 48000= 0, and solve for T. That will be the same as the two answers above.

#### JeffM

##### Elite Member
Hi Jeff!

My x-axis is number of timers sold and y-axis is price.
Cost = 8x + 48,000
Revenue = 14x
Profit = Revenue - Cost = 14x - (8x + 48,000) = 6x - 48,000
My break-even point was (8000,112000)
Cost function is correct.

Revenue function is correct.

Profit function is correct.

Breakeven point is correct.

All section d is doing is to let you see that the x-value where the cost and revenue functions are equal is the same x-value where the profit function is zero.

#### San1998

##### New member
Is the graph supposed to look like this?

#### JeffM

##### Elite Member
No. Define the y-axis as number of monetary units (RMs in this case) and the x-axis as number of physical units (timers in this case).

(Economists like to put multiple functions into the same graph. The way to do that is to define the axes in common units.)

Look at your cost function. Measured in RM, it is:

$$\displaystyle c = 48000 + 8x$$

It has POSITIVE slope, and, for positive x, is everywhere positive.

Look at your revenue function. Measured in RM, it is:

$$\displaystyle r = 14x$$

It has POSITIVE slope, and, for positive x, is everywhere positive.

Look at your profit function. Measured in RM, it is:

$$\displaystyle p = 14x - (48000 + 8x) = 6x - 48000$$

It has POSITIVE slope, but it is not positive for all positive x.

So none of your graphs is even close to correct.

I see what you did for your x and y column, but

$$\displaystyle 14 * 100 = 1400 \ne 140000.$$

As far as I can see, your graphs do not represent your x and y values. That is an excel issue.

#### San1998

##### New member
When T= 0, the cost is 48000+ 8(0)= 48000. When T= 10000, the cost is 48000+ 8(10000)= 128000. Mark the points (0, 48000) and (10000, 128000) on your graph and draw the line through them. When T= 0, the income is 14(0)= 0. When T= 10000, the income is 14(10000)= 140000. Mark the points (0, 0) and (100, 140000) and draw the line between them.
Sorry don't you mean (10000, 140000)?

#### San1998

##### New member
No. Define the y-axis as number of monetary units (RMs in this case) and the x-axis as number of physical units (timers in this case).

(Economists like to put multiple functions into the same graph. The way to do that is to define the axes in common units.)

Look at your cost function. Measured in RM, it is:

$$\displaystyle c = 48000 + 8x$$

It has POSITIVE slope, and, for positive x, is everywhere positive.

Look at your revenue function. Measured in RM, it is:

$$\displaystyle r = 14x$$

It has POSITIVE slope, and, for positive x, is everywhere positive.

Look at your profit function. Measured in RM, it is:

$$\displaystyle p = 14x - (48000 + 8x) = 6x - 48000$$

It has POSITIVE slope, but it is not positive for all positive x.

So none of your graphs is even close to correct.

I see what you did for your x and y column, but

$$\displaystyle 14 * 100 = 1400 \ne 140000.$$

As far as I can see, your graphs do not represent your x and y values. That is an excel issue.
So you're saying that besides my last point, my points are correct, but not my graph? Sorry but I can't follow.

#### JeffM

##### Elite Member
So you're saying that besides my last point, my points are correct, but not my graph? Sorry but I can't follow.
Your points, not all of which are correct, are not always even on your graphs. Each graph has a negative slope, but your formulas all have positive slopes. In other words, your points are partially wrong, and your graphs are completely wrong.

Because you have not labeled your points and have no legend on your graph, it is difficult to do more than ask you to check how you calculated your y values and whether each point is on the appropriate line. It appears that you have actually marked the points on the graph, and it is obvious by inspection that there are points not on any line.

I suggest you do the following.

Put your formulas that you calulated algebraically into the "y" points so you do not multiply 14 and 100 and get 140000.

I believe you can put legends on a graph. If that is correct, do so in order that people can correlate points in your table with the appropriate graph.

Finally, you need to review how excel translates from table to graph. However you are doing it, it is not working properly. I am not good enough in using the presentation features of excel to give reliable advice on that topic.

#### San1998

##### New member
D
No. Define the y-axis as number of monetary units (RMs in this case) and the x-axis as number of physical units (timers in this case).

(Economists like to put multiple functions into the same graph. The way to do that is to define the axes in common units.)

Look at your cost function. Measured in RM, it is:

$$\displaystyle c = 48000 + 8x$$

It has POSITIVE slope, and, for positive x, is everywhere positive.

Look at your revenue function. Measured in RM, it is:

$$\displaystyle r = 14x$$

It has POSITIVE slope, and, for positive x, is everywhere positive.

Look at your profit function. Measured in RM, it is:

$$\displaystyle p = 14x - (48000 + 8x) = 6x - 48000$$

It has POSITIVE slope, but it is not positive for all positive x.

So none of your graphs is even close to correct.

I see what you did for your x and y column, but

$$\displaystyle 14 * 100 = 1400 \ne 140000.$$

As far as I can see, your graphs do not represent your x and y values. That is an excel issue.
Dearest Jeff,
Thank you for all your help! I figured out my mistake. I have a basic knowledge of linear equations and I made the mistake of drawing the points of cost, revenue and profit as two lines for each two sets of coordinates. I didn't know that a straight line indicates that I should graph each two sets as one straight line!