Word Problem - Functions

[N0ximas]

Need help with the following word problem:

You are going to sell oranges by the box full. The first supplier will sell you a box of 20 oranges for \$16 a package, but there is an additional charge of \$300.00 per month to purchase oranges from the company.

1. Find and show you how you found the formula for C(N), which is what it costs your group for oranges, in which n is the # of boxes of oranges.

For this problem, I came up with the formula: C(N) = (\$16.00 N)+ \$300 Am I on the right track?

2. Determine P(p), which models your profit if you buy your oranges from this company. Do not forget to rewrite your cost function in terms of your selling price, p.3. What kind of function is P. 4. When you charge this price, how many boxes are sold?5. Find the maximum profit.I need the solutions and steps to derive the solutions to these five questions.
I sort of know how to get the answers, but I just want to make sure my conclusions are valid. All help is appreciated.

 

[MarkFL]

First, I wanted to let you know that the dollar sign is a tag for \(\displaystyle \LaTeX\), so you need to precede them with a backslash so that they display as simply dollar signs.

1.) Yes you are on the right track, however, I would simply state that the cost is in dollars, so that you can drop the dollar signs from the function definition:

\(\displaystyle C(n)=16n+300\)

The problem asks you to state how you came to this conclusion...can you state why this function works?

2.) Profit is revenue minus costs. Assume you sell all of the boxes you buy. What will the revenue be?

 

[N0ximas]

First, I wanted to let you know that the dollar sign is a tag for \(\displaystyle \LaTeX\), so you need to precede them with a backslash so that they display as simply dollar signs.

1.) Yes you are on the right track, however, I would simply state that the cost is in dollars, so that you can drop the dollar signs from the function definition:

\(\displaystyle C(n)=16n+300\)

The problem asks you to state how you came to this conclusion...can you state why this function works?

2.) Profit is revenue minus costs. Assume you sell all of the boxes you buy. What will the revenue be?

Thanks for the hint on problem 2. So would P(p) be:

P(n) = pn - 16n - 300 =-(p16)n - 300I know the function type is based on the power. If the power is 1, then the function is linear. Since the power is 1, the function is linear.For last question, I concluded that the max is positive infinity because there is no limit to how many oranges you can sell. But I also think it could be undetermined because we don't know what the maximum is. I'm not sure what to say, positive infinity or cannot be determined.

I could not find an answer to question 4. Price certainly does correlate with sales. However, determining a solid and exact price is impossible.

 

[MarkFL]

Have you left out a relationship between $n$ and $p$? The instructions say to write the cost function in terms of $p$ for part 2, but without some relationship between the two variables, I don't see how this can be done.

 

[CellCoree]

1. To find the formula for C(N), we need to consider the cost of purchasing the oranges and the additional charge per month. The cost of purchasing N boxes of oranges would be \$16.00 multiplied by N, which gives us \$16.00N. Then, we add the additional charge of \$300 per month, giving us a final formula of C(N) = \$16.00N + \$300. This formula represents the total cost of purchasing N boxes of oranges.

2. To determine P(p), we need to consider the profit we make from selling the oranges. The selling price, p, will be the revenue we make from selling the oranges. The cost function, C(N), can be rewritten in terms of p by dividing by the number of boxes, N, giving us C(N) = p. Therefore, the profit function can be written as P(p) = p - (\$16.00 + \$300). This formula represents the profit made from selling the oranges at a price of p.

3. The function P(p) is a linear function, as it has a constant rate of change and a straight line when graphed.

4. To determine the number of boxes sold, we need to find the value of N when the profit, P, is equal to 0. This is because when the profit is 0, it means that the cost and the revenue are equal, and we have broken even. We can set P(p) = 0 and solve for p to find the selling price at which we break even. Once we have the selling price, we can plug it back into the cost function, C(N), and solve for N to find the number of boxes sold.

5. To find the maximum profit, we need to find the value of p that will give us the highest value for P(p). This can be done by finding the vertex of the parabola formed when graphing the profit function. Alternatively, we can also find the value of p that will make the derivative of the profit function, P'(p), equal to 0. This will give us the point where the slope of the profit function is 0, which is also the point where the profit is maximized.