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    Applications of differentiation - C2 maths questions plz help

    - An aeroplane flying level at 250m above the ground suddenly swoops down to drop supplies, and then regains its former altitude. It is h m above the ground t s after beginning its dive where h = 8t^2 - 80t + 250. Find
    a) its least altitude during this operation
    b) the interval of time during which it was losing height.


    -An open tank is to be constructed with a square base and vertical sides as to contain 500m^3 of water.
    a) Given that the length of the side of the square base is x m, find expressions in terms of x, for the height of the tank, and for the external surface area of the tank.
    b)Find the value of x required so that the area of sheet mental used in constructing the tank is a minimum (Remember to show that the value of x found gives a minimum value.)
    c) Find the minimum area of metal


    - A rechtangular sheet of metal is 8 cm by 5cm and equal squares of side x are removed from each cotner and the edges are turned up to make an open box of volume V cm^3.
    Show that V= 40x - 26x^2 + 4x^3
    Hence find the maximum possible volume, and the corresponding value of x.

    -A sweet manufacturer estimates that if it sets the price of a box of speciality chocolates at £p it will sell n boxes per year, where n = 1000(84+12p-p^2) for 2.5 =/< p =/< 15.
    a) Find the price that will maximise the number of boxes sold.
    b) Write down the revenue recieved by selling n boxes at price £p.
    c) Hence show that the price that will maximise the manufacturer's revnue is £10.50, to the nearest 50 pence.

    Done all the other questions but really really struggling with just these questions. Please don't just give hints because I would really like full working out and an answer to each please. Will definitely give positive rep to those that help with any of the questions
    Thank you
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    Full solutions are discouraged in this subforum so you can either have hints (which should lead you to a solution) or nothing.

    I doubt that anyone would want to post full solutions to eight questions. We have our own homework to do.
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    1.
    a) What methods do you know for finding the minimum of a quadratic? You should have learnt 2 in C1 alone.
    b) The minimum of the quadratic is the final point where the plane is losing altitude. The first point is the point at which the plane began to lose altitude. Therefore the time needed is the time from the start to the minimum, and we can call the start t = 0.

    2.
    a) First find an expression, in terms of x and h, for the volume of the container. Equate this to 500.
    Now picture the container, and remember that it is open, so there is no top.
    b) We want the minimum area. Again, how do we find the minimum of a function of x?
    To answer the rest of the question, how do we find whether a stationary point is a maximum or a minimum?
    c) Just plug your value for x into your expression for h (height), and then plug both x and h into the expression you found for the surface area.


    And that's all I can be bothered to help you with. You should be able to do the rest, now that I've shown you how to do them.
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    (Original post by notnek)
    Full solutions are discouraged in this subforum so you can either have hints (which should lead you to a solution) or nothing.

    I doubt that anyone would want to post full solutions to eight questions. We have our own homework to do.
    Oh It's just that I've done the rest of the questions but have spent a lot of time thinking about how to answer this and still getting it wrong so I don't think hints will help me that much since I will probably have to keep asking the hint poster to keep posting more hints until I get it which is just tiresome for the person offering the hints..

    Also, I don't expect anyone to answer all the questions.. Even if each person could give me the worked solution to at least one question, I would very be really grateful.
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    (Original post by Princestia)
    Oh It's just that I've done the rest of the questions but have spent a lot of time thinking about how to answer this and still getting it wrong so I don't think hints will help me that much since I will probably have to keep asking the hint poster to keep posting more hints until I get it which is just tiresome for the person offering the hints..

    Also, I don't expect anyone to answer all the questions.. Even if each person could give me the worked solution to at least one question, I would very be really grateful.

    Have you tried questions 1 and 2 using my hints?
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    As you have already been given hints for questions 1 and 2, I'll help you out with 3. You are given that a rectangular sheet of dimensions 8x3 has squares of side x cut out of each corner and the protruding edges are turned up to make a box. If you can visualise this box, it will have a depth of x, and will measure (8-2x)cm long and (5-2x)cm wide (if you are having trouble seeing this, draw the 8x5 rectangle and then cut out arbitrary but equal squares from each corner... hopefully you will now see where these dimensions come from if you fold the resultant shape into a box). Given these dimensions, how would you work out the volume of the box V? By now you should have shown the statement given. Now simply differentiate this to find the stationary point of the cubic which is a maximum (hint: you will find two stationary points, one max and one min), the coordinates of which give you the max volume and corresponding value of x.
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    3rd one would be where i started if unsure about topic
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    (Original post by AnonyMatt)
    1.
    a) What methods do you know for finding the minimum of a quadratic? You should have learnt 2 in C1 alone.
    b) The minimum of the quadratic is the final point where the plane is losing altitude. The first point is the point at which the plane began to lose altitude. Therefore the time needed is the time from the start to the minimum, and we can call the start t = 0.

    2.
    a) First find an expression, in terms of x and h, for the volume of the container. Equate this to 500.
    Now picture the container, and remember that it is open, so there is no top.
    b) We want the minimum area. Again, how do we find the minimum of a function of x?
    To answer the rest of the question, how do we find whether a stationary point is a maximum or a minimum?
    c) Just plug your value for x into your expression for h (height), and then plug both x and h into the expression you found for the surface area.


    And that's all I can be bothered to help you with. You should be able to do the rest, now that I've shown you how to do them.
    Hi,
    Thanks so much for the help Sorry, I didn't see your post then because I posted my reply to the previous poster around 30 seconds after you so I didn't see your post.

    Regarding question 1:
    I've tried to use completing the square to find the minimum but the answer is not correct . Can you please please tell me of the other ways (/easier) ways to find the minimimum? E.g. maybe using differentiation in some way?

    part b of question 1; I sorta understand now .. I think .

    Question2:
    I understand question 2a now, thanks a lot
    Still don't understand part B due to problems finding the minimum (just like in question 1) so hence haven't done part c yet.

    Thanks so much for the help so far AnonyMatt but I would appreciate more hints/answers to the parts I've pointed out Thanks a lot Will give positive rep .


    Also, thanks for the person who posted hints for question 3 - I'll have a look at that post in a few minutes x
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    (Original post by Princestia)
    Hi,
    Thanks so much for the help Sorry, I didn't see your post then because I posted my reply to the previous poster around 30 seconds after you so I didn't see your post.

    Regarding question 1:
    I've tried to use completing the square to find the minimum but the answer is not correct . Can you please please tell me of the other ways (/easier) ways to find the minimimum? E.g. maybe using differentiation in some way?

    part b of question 1; I sorta understand now .. I think .
    Completing the square should work...
    Since you gave the correct method, I'll give you the answer!
    8(t-5)^2 + 250 - 25 = 8(t-5)^2 + 225
    If you are unsure of how I got this, quote me again and I'll explain.

    For part b), draw a sketch so you can see clearly what is happening at all times. As I said before, the coefficient of t^2 is positive, so it is 'U' shaped, and it's stationary point will always be a minimum. We have now found that minimum using part a). Can you see what it is?
    Now, we're looking for the time that the plane is losing height, which if you look at your sketch, is time (x-axis) from which height (y-axis) is 250 and the minimum value. So write down the value for t which gives h = 250, and then the value for t which gives h = minimum.
    The difference between the two will be the time the plane is losing height.


    Question2:
    I understand question 2a now, thanks a lot
    Still don't understand part B due to problems finding the minimum (just like in question 1) so hence haven't done part c yet.
    So what did you get for 2a)?
    I think you should have got h = 500x^-2 for the height.
    The surface area on the outside is simply the area of the square base, plus 4 times the area of the sheets around the outside. You should have: x^2 + 4xh = x^2 + 2000x^-1, I think.
    Now, this is aimed at differentiation. Can you differentiate this function?
    Minimums and maximums are stationary points. What happens to dy/dx at stationary points? So what is the value of x for this minimum/maximum? How do we show that it is a minimum, or a maximum?
    For c), we just plug our x value into the expression for surface area we gave in part a).

    If you need any more help, quote!
 
 
 
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