Quadratics are not my strong point

J

James Smithson

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I have been stuck for a while on the following questions. They are all to do with quadratics.
It's about Bob who creates a economic model on the computer to predict ice cream sales. Models on the quadratic equation of the form

Y=ax^2 + bx + c
Y represents profit in thousands they are expected to make and
X represents the amount of ice cream in thousands of liters.

A b and c are determined by computer code

The quadratic produced by the computer is
Y= - 9.491x^2 + 7.125x - 0. 402

Ok that's all the information we get.
the questions are
1. Explain mathematically why the y intercept is 0.402.

(I can do that one)

2. Using the context of the question give a reason for a negative y intercept.

This confuses me

3. What's the minimum amount of smoothies Bob needs to produce to break even. Explain why

4. Bob wants to make £500 profit in next batch and decides to make 400 liters of ice cream will they exceed or fall short

. Thank you for any help

I really need it
 
James Smithson said:
I have been stuck for a while on the following questions. They are all to do with quadratics.
It's about Bob who creates a economic model on the computer to predict ice cream sales. Models on the quadratic equation of the form

Y=ax^2 + bx + c
Y represents profit in thousands they are expected to make and
X represents the amount of ice cream in thousands of liters.

A b and c are determined by computer code

The quadratic produced by the computer is
Y= - 9.491x^2 + 7.125x - 0. 402

Ok that's all the information we get.
the questions are
1. Explain mathematically why the y intercept is 0.402.

(I can do that one)

2. Using the context of the question give a reason for a negative y intercept.

This confuses me

3. What's the minimum amount of smoothies Bob needs to produce to break even. Explain why

4. Bob wants to make £500 profit in next batch and decides to make 400 liters of ice cream will they exceed or fall short

. Thank you for any help

I really need it
Click to expand...
First off, the y-intercept IS NOT 0.402.

They y-intercept is the value of the function if x equals zero.

So what is the correct answer?

Second, the second question is asking you to think about why it makes sense that the company is predicted to lose money if it sells no ice cream at all.

Can you think of a good reason for why that will be true?

The third question is not answerable at all if it is true that you have provided all the information. You have provided not an atom of information on smoothies so how can anyone answer a question about smoothies.

As for the fourth question, what is y if x is 400?
 
JeffM said:
First off, the y-intercept IS NOT 0.402.

They y-intercept is the value of the function if x equals zero.

So what is the correct answer?

Second, the second question is asking you to think about why it makes sense that the company is predicted to lose money if it sells no ice cream at all.

Can you think of a good reason for why that will be true?

The third question is not answerable at all if it is true that you have provided all the information. You have provided not an atom of information on smoothies so how can anyone answer a question about smoothies.

As for the fourth question, what is y if x is 400?
Click to expand...
I meant ice cream for question 3 but thanks for the help :)
 
JeffM said:
First off, the y-intercept IS NOT 0.402.

They y-intercept is the value of the function if x equals zero.

So what is the correct answer?

Second, the second question is asking you to think about why it makes sense that the company is predicted to lose money if it sells no ice cream at all.

Can you think of a good reason for why that will be true?

The third question is not answerable at all if it is true that you have provided all the information. You have provided not an atom of information on smoothies so how can anyone answer a question about smoothies.

As for the fourth question, what is y if x is 400?
Click to expand...
Q4. Since x represents icecream in thousands of litres, x=0.4 not 400
 
[MATH] [QUOTE="JeffM, post: 531013, member: 32081"] First off, the y-intercept [B][COLOR=rgb(41, 105, 176)]IS NOT[/COLOR][/B] 0.402. They y-intercept is the value of the function if x equals zero. So what is the correct answer? Second, the second question is asking you to think about why it makes sense that the company is predicted to lose money if it sells no ice cream at all. Can you think of a good reason for why that will be true? The third question is not answerable at all if it is true that you have provided all the information. You have provided not an atom of information on smoothies so how can anyone answer a question about smoothies. As for the fourth question, what is y if x is 400? [/QUOTE] I think its - 0.402 If they sell none then they won't make any profit so the curve will be below the x axis but I'm not sure why it crosses the y at a certain point maybe it's the business starts of with negative money as there is an initial outlay. Also I'm not sure what x^2 represents or what c represents I really do appreciate the help everyone. I had thought the questions were aimed at me to ask myself not for reply so please forgive my rudeness. English is a second language :)[/MATH]
 
Also will working out both x intercepts help me work out the minimum amount of ice cream to break even?
 
James Smithson said:
[MATH] I think its - 0.402 If they sell none then they won't make any profit so the curve will be below the x axis but I'm not sure why it crosses the y at a certain point maybe it's the business starts of with negative money as there is an initial outlay. Also I'm not sure what x^2 represents or what c represents I really do appreciate the help everyone. I had thought the questions were aimed at me to ask myself not for reply so please forgive my rudeness. English is a second language :)[/MATH]
Click to expand...
Your English is fine.

In algebra, you can think of f(x) as an instruction on what to do arithmetically once you learn what number the letter x represents.

[MATH]f(x) = ax^2 + bx + c \text { and } a \ne 0[/MATH] represents a whole family of functions called quadratic functions. The members of the family have certain common characteristics.

An obvious common characteristic is the value of f(0). What is 0 squared? 0. What is anything times 0? 0. What is z + 0? Just z. So

[MATH]f(0) = a * 0^2 + b * 0 + c = a * 0 + 0 + c = 0 + c = c.[/MATH]
Another pair of common characteristics (not provable through algebra alone) are these:

[MATH]\text {If } a < 0, \text { then } f(x) \text { has its maximum value when } x = - \dfrac{b}{2a}.[/MATH]
[MATH]\text {If } a > 0, \text { then } f(x) \text { has its minimum value when } x = - \dfrac{b}{2a}.[/MATH]
In other words, it is possible to reach certain conclusions that apply to all quadratics even though we cannot work with them arithmetically until we know a, b, c, and x.

In your problem, a = -9.491, b = 7.12, and c = - 0.402.

“What is the y-intercept” simply means “What is f(0)” Just a different form of words. And if f(x) is a quadratic function in the form of ax^2+bx+c, then f(0) = c.

So your answer of -0.402 is correct. It reflects what y-intercept means when applied to a quadratic.

The second question is a whole lot more subtle. We say: if x represents thousands of liters of ice cream sold and y represents thousands of pounds in profits, then, IN THIS CASE, it is true that

[MATH]y = -9.491x^2 + 7.12x - 0.402.[/MATH]
OK. But the formula says that profits are negative (the firm is losing money) if it sells nothing. You are very, very close to the right answer.

If they want to sell ice cream, they must rent a refrigerator to keep it cold. They must buy cream and vanilla and chocolate as ingredients. In short, they have to spend money (incur expenses) to have any hope of making a profit. And if they can sell no ice cream at all, every penny they spent is lost.

So the second question is asking you to think about why the mathematical relationship makes sense when applied to the real-world situation.

Now your question about what x2 means is a good question. Clearly ice cream cannot be squared. In terms of the formula, however, we are thinking about numbers, and numbers can be squared. So your question is a much more subtle version of the second question you were asked. Why is it reasonable that profits should turn into losses as sales increase?

The reason is that costs can skyrocket as the number of liters sold increase and the price realized is likely to fall to get sales to reach huge levels.
 
James Smithson said:
[MATH] I think its - 0.402 If they sell none then they won't make any profit so the curve will be below the x axis but I'm not sure why it crosses the y at a certain point maybe it's the business starts of with negative money as there is an initial outlay. Also I'm not sure what x^2 represents or what c represents I really do appreciate the help everyone. I had thought the questions were aimed at me to ask myself not for reply so please forgive my rudeness. English is a second language :)[/MATH]
Click to expand...
The business has expenses even if they do not sell any ice cream. Then need to pay the rent for their location. They need to pay for electricity. They need to pay for water. They need to pay their workers. They need to pay the loan for any equipment they purchased on credit. All of these are called overhead or fixed costs--they must be paid whether business is good or not.
 
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