Hello! I came across the following two problems that appear easy but I was not able to construct the needed equations.
(1) The profit (in thousand of dollars) on x thousand units of a specialty item is p = 0.6x - 14.5. The cost c of manufactoring x thousand items is given by c = 0.8x + 14.5.
(A) Find an equation that gives the revenue r from selling x thousand items.
The correct equation is r = 1.4x but where did 1.4 come from?
Is 1.4 the slope? If so, how is the slope produced using the given information above?
Someone suggested that profit = revenue - cost and that I should add the two given slopes 0.6 + 0.8 to get 1.4 as the new slope. However, profit = revenue MINUS cost not PLUS cost. Why do I need to add 0.6 and 0.8?
(B) How many items must be sold for the company to break-even?
I understood breaking-even to means when revenue equals cost.
I decided to let p = 0 in the profit equation given above and solve for x.
I got x = 24 but I am so far from the right answer. What did I do wrong?
Hello! Let's go through your questions step by step:
(A) To find the equation that gives the revenue r from selling x thousand items, we need to understand the relationship between revenue and the number of items sold. Revenue is calculated by multiplying the number of items sold by the selling price per item. In this case, the selling price per item is not directly given, but we can calculate it using the profit equation.
The profit equation p = 0.6x - 14.5 tells us that for every x thousand units sold, the profit made is 0.6x - 14.5 thousand dollars. We can rearrange this equation to isolate the selling price per item:
p = 0.6x - 14.5
0.6x = p + 14.5
x = (p + 14.5) / 0.6
Now, we know that the selling price per item is given by (p + 14.5) / x. To calculate the revenue, we multiply the selling price per item by the number of items sold, x:
r = x * [(p + 14.5) / x]
Simplifying the above expression, the x's cancel out:
r = p + 14.5
So, the equation that gives the revenue r from selling x thousand items is r = p + 14.5.
Now, let's talk about why 1.4 is the slope in this case. The slope represents the rate of change of one variable with respect to another. In this scenario, the slope represents the change in revenue per change in the number of items sold. From the equation r = p + 14.5, we can see that for every unit increase in the number of items sold, the revenue increases by 1.4 (since the coefficient of x in the profit equation is 0.6 and the constant term is -14.5). Therefore, 1.4 is the slope in this context.
(B) To find the number of items that must be sold for the company to break even, we need to set the revenue equal to the cost.
The revenue equation, as we found earlier, is r = p + 14.5. The cost equation is given as c = 0.8x + 14.5.
Setting r equal to c and substituting the revenue equation, we have:
p + 14.5 = 0.8x + 14.5
Simplifying the equation, we can cancel out the 14.5 on both sides to get:
p = 0.8x
Now, if we set p = 0 (indicating no profit), we can solve for x:
0 = 0.8x
This equation gives us x = 0. However, this would mean selling zero items, which doesn't make sense in the context of breaking even. So, it seems there was an error made. Let's revisit the equations and calculations to identify the mistake:
The original profit equation is p = 0.6x - 14.5, where p represents the profit and x represents the number of items sold. Setting p = 0 to find the break-even point, we have:
0 = 0.6x - 14.5
Now, let's solve for x:
0.6x = 14.5
x = 14.5 / 0.6
Evaluating this expression, we find x ≈ 24.17.
Therefore, the correct number of items that must be sold for the company to break even is approximately 24, not 0.
I hope this explanation helps clarify the concepts and resolves any confusion. Let me know if you have any further questions!
To answer your questions:
(A) To find the equation for revenue, we need to understand that revenue is the total amount of money generated from selling a certain quantity of items. Mathematically, revenue (r) is calculated by multiplying the number of items sold (x) by the price per item (p). In this case, the price per item is not explicitly given in the problem, but we can infer it from the given information.
Given that profit (p) is equal to revenue (r) minus cost (c), we can rewrite the profit equation as p = r - c. Substituting the given equations for profit (p) and cost (c), we have:
0.6x - 14.5 = r - (0.8x + 14.5)
Simplifying this equation, we get:
0.6x - 14.5 = r - 0.8x - 14.5
Combining like terms, we have:
1.4x = r
So, the correct equation for revenue is r = 1.4x. The value 1.4 is the coefficient of x, which represents the price per item (in thousand dollars) at which the items are being sold.
(B) To find the break-even point, we need to determine the quantity of items sold (x) at which the revenue is equal to the cost. In other words, we set the revenue equal to the cost and solve for x. Using the equations for revenue (r) and cost (c), we have:
1.4x = 0.8x + 14.5
Subtracting 0.8x from both sides of the equation, we get:
0.6x = 14.5
Dividing both sides of the equation by 0.6, we have:
x = 14.5 / 0.6
Evaluating this expression, we find that x is approximately 24.17.
So, the correct answer for the number of items needed to break even is x = 24.17 (rounded to two decimal places).
It seems like you made a calculation error when solving for x in part (B), which led to the incorrect answer of x = 24. Double-check your calculations.