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Maths functions/algebra help

HI, m stuck in this last question of my assignment, any help wud be greatly appreciated. thank you :smile:

A sweet manufacturer found that the sales figures for a certain item depended on its selling price. The company’s market research department advised that the maximum number of items that could be sold weekly was 20000 and that the number decreased by 100 for every 1p increase in its price. The total production cost consists of a set-up cost of £200 plus 50p for every item manufactured.

If the price increase one week is p pence, find expressions in terms of p for:

the number N of items sold weekly;

(b) the production cost £C;

(c) the revenue £R from the sales.

(d) Show that the profit £P is given by

P = 250p – 10200 – p2

(e) What values of p will generate no profit? Give your answers to the nearest penny.

(f) From your knowledge of the behaviour of quadratic functions and using your answers to part (e), what value of p will generate the maximum profit and what is this maximum profit? (No calculus is needed.)

Reply 1

the bear

20

Original post by baller23

HI, m stuck in this last question of my assignment, any help wud be greatly appreciated. thank you :smile:

A sweet manufacturer found that the sales figures for a certain item depended on its selling price. The company’s market research department advised that the maximum number of items that could be sold weekly was 20000 and that the number decreased by 100 for every 1p increase in its price. The total production cost consists of a set-up cost of £200 plus 50p for every item manufactured.

If the price increase one week is p pence, find expressions in terms of p for:

the number N of items sold weekly;

(b) the production cost £C;

(c) the revenue £R from the sales.

(d) Show that the profit £P is given by

P = 250p – 10200 – p2

(e) What values of p will generate no profit? Give your answers to the nearest penny.

(f) From your knowledge of the behaviour of quadratic functions and using your answers to part (e), what value of p will generate the maximum profit and what is this maximum profit? (No calculus is needed.)

did you mean P = 250p – 10200 – p² ?

Reply 2

baller23

OP

2

Original post by the bear

did you mean P = 250p – 10200 – p² ?

yepp its p squared.

Reply 3

the bear

20

for part e) you need to make quadratic function = 0

for part f) you could differentiate it & find where dP/dp = 0

Reply 4

ghostwalker

17

For (f), the mean of the two values you got in (e) gives the desired result.

Reply 5

baller23

OP

2

thanks for your help. all done and dusted :smile:

Reply 6

Transporter07

I too have this question and im stuck, any pointers where to start with this one? Thanks for any help in advance

Reply 7

ghostwalker

17

Original post by Transporter07

I too have this question and im stuck, any pointers where to start with this one? Thanks for any help in advance

What have you managed so far? Which bit are you stuck on?

Reply 8

Transporter07

Im stuck at the first part finding N of items sold weekly.

Reply 9

ghostwalker

17

Original post by Transporter07

Im stuck at the first part finding N of items sold weekly.

I don't know if the original question as posted is worded exactly, however I think we must assume the maximum number of items sold weekly is the number of items sold weekly.

So, this is 20000 initially.

Then for each 1p increase in price, that goes down by 100.

So, what happens if the price is increased by 1p? How many are sold?

And if it were increased by 2p?

And finally if it were increased by "p" pence.

Reply 10

Transporter07

So if the price is increased by 1p, 19,900 are sold and 19,800 are sold if there is a 2p increase.

It it was increased by p would it result in p - 100?

Reply 11

ghostwalker

17

Original post by Transporter07

So if the price is increased by 1p, 19,900 are sold and 19,800 are sold if there is a 2p increase.

Yep.

It it was increased by p would it result in p - 100?

Don't know how you've gotten that.

You're starting with 20000, and subtracting 100 for every penny.

So your formula becomes....

N= 20,000 -100(p)

Original post by Transporter07

N= 20,000 -100(p)

Yep.

Reply 14

Takireo

I've got the same issue. But from question C onwards.

Reply 15

ghostwalker

17

Original post by Takireo

I've got the same issue. But from question C onwards.

If I sold 6 items at 45p each can you tell me what the revenue is?

Now apply the same principle here.

Reply 16

Takireo

Original post by ghostwalker

If I sold 6 items at 45p each can you tell me what the revenue is?

Now apply the same principle here.

Assuming s=items sold I figured:

R=ps ?

But that seems too easy...

And when it comes to (d), I'm thinking that by "Show that the profit £P is given by" we have to show how our previous expressions relate to this equation.

Am I at all warm? :tongue:

Thanks for all the help.

Reply 17

ghostwalker

17

Original post by Takireo

Assuming s=items sold I figured:

R=ps ?

But that seems too easy...

Well you need to express s in terms of p.

However, I've just realised that p is the increase in price, not the actual price. No where in the question does it tell you the actual price, so this part is undoable! I presume the questioner has ignored this and assumed the price is actually p. I'd go with what and see what comes out.

And when it comes to (d), I'm thinking that by "Show that the profit £P is given by" we have to show how our previous expressions relate to this equation.

Am I at all warm?

Yep. Profit = Income - Costs, and apply that here.

Note that they're asking for the profit in pounds.

Reply 18

Rhys808

hey!

I've got to do this question and I'm a bit stuck on how to work it out.

2. A mass on the end of a spring which is hanging vertically is raised up and let go. It then oscillates between 2m and 1.5m above the floor and completes 32 cycles in one minute. The height, h metres, of the mass above the floor after t seconds can be modelled by the function

h=acos((pi*b*t)/180) + c

where a, b and c are constants.

(a) Determine exactly the period, T, of the oscillation in seconds per cycle and hence find the value of b. (4 marks)

(b) By considering the extremes of the oscillation, work out the values of a and c. (4 marks)

(c) Calculate exactly the value of h when t = 25 seconds. (2 marks)

(d) Find the first time when h = 1.75 metres. (5 marks)

For part a I got T=1.875 and b=192. I'm not sure if they're right, but my problem is on part b. I can't figure out how to work it out.

Any help on this would be great.

Reply 19

Chlorophile

20

Original post by Rhys808

hey!

I've got to do this question and I'm a bit stuck on how to work it out.

2. A mass on the end of a spring which is hanging vertically is raised up and let go. It then oscillates between 2m and 1.5m above the floor and completes 32 cycles in one minute. The height, h metres, of the mass above the floor after t seconds can be modelled by the function

h=acos((pi*b*t)/180) + c

where a, b and c are constants.

(a) Determine exactly the period, T, of the oscillation in seconds per cycle and hence find the value of b. (4 marks)

(b) By considering the extremes of the oscillation, work out the values of a and c. (4 marks)

(c) Calculate exactly the value of h when t = 25 seconds. (2 marks)

(d) Find the first time when h = 1.75 metres. (5 marks)

For part a I got T=1.875 and b=192. I'm not sure if they're right, but my problem is on part b. I can't figure out how to work it out.

Any help on this would be great.

Your answer to part a is correct. For part b, think about the cosine function. The period of the cosine function is 2pi. The period of our function is 1.875.

Maths functions/algebra help

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