Help with a homework assignment: profit equation

[gte]

Hello,

My wife went back to school and is taking Algebra. I haven't been to school in years and am trying to help her, I'm not looking for the answers, but to be reminded of the steps to do the math so that I can in turn show her how.

The original problem is at the bottom.

If I remember correctly, P(x) means the profit of x and it is not a multiplication equation.

Formula:
P (x) = - 3.2 x² + 268.2 x + 257 or read it as the Profit of x = - 3.2 x² + 268.2 x + 257

x = items produced per day

If that is correct, I have the following

2.)
x = 40

P(40) = -3.2(40²) + 268.2(40) + 257

P(40) = -5120 + 10728 + 257

Profit of (40 items) = 5865

3.)
x = 50

P(50) = -3.2(50²) + 268.2(50) + 257

P(50) = -8000 + 13410 + 257

Profit of (50 items) = 5667

4.)
Should this be as follows? 5423 = -3.2(x²) + 268.2(x) + 257 . I tried this way, and then I tried to convert the numbers to standard equation, making them all whole numbers, but I can't figure out a way to do that for both values that have a decimal place.

5.)
I'm not sure how to calculate this, besides going through it via attrition

A manufacturing company found that the equation P(x) = - 3.2x2 +268.2x + 257 could be used to approximate their profit per day for x items produced.

For each of the following, show all calculation methods and explain your results in complete sentences.

1. This is an equation for “profit”. What, besides selling price, determines the “profit” from the sale of an item? Explain using at least 3 specific ideas.
2. Find the profit if 40 items are made.
3. Find the profit if 50 items are made. How does this compare to the profit from making 40 items? Does this result seem reasonable? Explain.
4. How many objects would need to be made to get a profit of $5,423? Be sure to show the solution process step-by-step.
5. How many objects should be sold to maximize the profit? What is the maximum profit?
6. Show a graph of the profit equation. Be sure to label the horizontal and vertical axes.
• Label the y-intercept with an ordered pair of numbers. Explain these values in the context of the problem.
• Label both x-intercepts with ordered pairs of numbers, approximating values to the nearest tenth. Explain these values in the context of the problem.
• Label the vertex with an ordered pair of numbers. Explain these values in the context of the problem.
7. Explain why a profit equation could be useful to a manufacturing company in planning their business.

[buddy]

If I remember correctly, P(x) means the profit of x and it is not a multiplication equation.

Yeh, it's function notation. You can review that here.

1. This is an equation for “profit”. What, besides selling price, determines the “profit” from the sale of an item? Explain using at least 3 specific ideas.

Formula: P (x) = - 3.2 x² + 268.2 x + 257 or read it as the Profit of x = - 3.2 x² + 268.2 x + 257
x = items produced per day

Yeh, it's P=profit for x=units but it's probably asking for something like expenses, etc.

2. Find the profit if 40 items are made

2.) x = 40
P(40) = -3.2(40²) + 268.2(40) + 257
P(40) = -5120 + 10728 + 257
Profit of (40 items) = 5865

i didn't check the math but yeh, you plug in 40 for x and find the answer for P.

3.Find the profit if 50 items are made. How does this compare to the profit from making 40 items? Does this result seem reasonable? Explain.

3.) x = 50
P(50) = -3.2(50²) + 268.2(50) + 257
P(50) = -8000 + 13410 + 257
Profit of (50 items) = 5667

what's your answer for the reasonable part?

4. How many objects would need to be made to get a profit of $5,423? Be sure to show the solution process step-by-step.

4.) Should this be as follows? 5423 = -3.2(x²) + 268.2(x) + 257 . I tried this way, and then I tried to convert the numbers to standard equation, making them all whole numbers, but I can't figure out a way to do that for both values that have a decimal place.

i don't know what you mean by "convert the numbers to standard equation". try plugging this into the quadratic formula. then pick "reasonable" rounded answers (like "i can't make 2.65 units, but I can 2 or 3 units" kind of thing)

[buddy]

5. How many objects should be sold to maximize the profit? What is the maximum profit?

5.) I'm not sure how to calculate this, besides going through it via attrition

find the vertex and interpret to get the answer

6. Show a graph of the profit equation. Be sure to label the horizontal and vertical axes.
• Label the y-intercept with an ordered pair of numbers. Explain these values in the context of the problem.
• Label both x-intercepts with ordered pairs of numbers, approximating values to the nearest tenth. Explain these values in the context of the problem.
• Label the vertex with an ordered pair of numbers. Explain these values in the context of the problem.

graphing is here

7. Explain why a profit equation could be useful to a manufacturing company in planning their business.

look in the book for examples like this to see what they probably mean by "explain"

[gte]

Hi Buddy,

Thanks for the replies

My answer for the reasonable part is that it is not reasonable, because the more products you make, the more your profit decreases. With volume your profit should increase, not decrease.

40 items equal $5865, 50 items equals $5667 .

I will work on your other suggestions now.

[gte]

i don't know what you mean by "convert the numbers to standard equation". try plugging this into the quadratic formula. then pick "reasonable" rounded answers (like "i can't make 2.65 units, but I can 2 or 3 units" kind of thing)

This is what I have for 4 although I don't see anything in her book that deals with decimals in the quadratic equation so I believe I've done something wrong here.

[gte]

I've been trying for the past 10 minutes to upload my equations and the forum keeps giving me an sql error, so I'm going to try a screen capture of the values I worked out, sorry for any inconvenience.

[buddy]

I've been trying for the past 10 minutes to upload my equations and the forum keeps giving me an sql error

yeh, it sometimes goes sideways like that. i don't understand it myself.

5423 = -3.2x^2 + 268.2x + 257
P = -3.2x^2 + 268.2x - 5166

no, it's 0 = -3.2x^2 + 268.2x - 5166. you put the 5423 in for the P so it's not in there any more.

after the first square-root thing, you have a line i don't understand??? and i don't get what you're doing in the second picture???? just do the quadratic formula to 0 = -3.2x^2 + 268.2x - 5166 to get x.

[gte]

So is the below correct? How do you work with the decimal portion? (The forum doesn't allow me to use the square root symbol or the squared symbol without a sql database error, so I had to use a V and a ^2 instead)


0 = -3.2(x)^2 + 268.2(x) -5166



a = 3.2
b = 268.2
c = -5166

____________________________________________
x = -268.2 ± V(268.2)^2 - 4(3.2)(-5166)
------------------------------------------------
2(3.2)

or

________________________________________
x = -268.2 ± V71931.24 - (-66124.8)
------------------------------------------------
6.4
or

_______________________________
x = -268.2 ± V138056.04
------------------------------
6.4

[gte]

I just had an epiphany. Can I simplify the problem more than this?

0 = -3.2(x)^2 + 268.2(x) -5166

0 = (-3.2(x)^2 + 268.2(x) -5166) 10 or 0 = 32x^2 + 2682 - 51660



a = 32
b = 2682
c = -51660

____________________________________________
x = -2682 ± V(2682)^2 - 4(32)(-51660)
------------------------------------------------
2(32)

or

________________________________________
x = -2682 ± V7193124 - (-6612480)
------------------------------------------------
64

or

_______________________________
x = -2682 ± V13805604
------------------------------
64


or

_______________________________
x = -2682 ± 6V383489
------------------------------
64


or

_______________________________
x = -1341 ± 3V383489
------------------------------
32



With the answer being

_______________________________
x = -1341 + 3V383489
------------------------------
32

AND

_______________________________
x = -1341 - 3V383489
------------------------------
32

[buddy]

it doesn't matter if you times by 10 or whatever to get rid of decimals. the answer will be the same because the 10s all cancel off. decimals are fine for the answer. most quadratics will end up with decimals!!