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Math Help - Maths SAC on shape and measurement.. really urgent!

  1. #1
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    Maths SAC on shape and measurement.. really urgent!

    You find your younger sister under her bed, upset. It is halloween and her witch costume is missing a hat. between tears she asks you for a hat 60cm tall with a 10cm brim. You measure the circumference of her head (50am)
    Looking at the black cardboard you decide the hat must be made in two parts:
    1) An open-ended cone for the hat part and
    2) An annulus for the brim.

    To make a cone you need to draw a circle, take out a sector and then join the sides of the remainging sector together(no overlap is catered for)

    1) Given the above constraints, calculate to the nearest degree the angle of the sector that needs to be removed to leave you with the portion that you will use for the hat.
    2) Is the hat made out of the major sector or minor sector?
    3) Calculate the surface area of the cone
    4) Calculate the area of cardboard needed for the brim
    5) Overall, what us the TSA of the witch's hat?
    6) Your sister's best friend would also like a hat- like the witch's hat without the brim. From the leftover sector, are you able to construct a cone that is again 60cm high but will fit a head circumference of 65cm? Support your conclusion with mathematics.
    7) If the slant height has to be 60.5cm and the circumference 65cm, what height should the cone be?
    8) To produce a cone of the dimensions required in question 7, what angle needs to be cut out of the remaining major sector?
    9) How many more cones of these dimensions could you make from the remainging sector?

    Thanks heaps!!,
    Molly xx
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  2. #2
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    Quote Originally Posted by molz009 View Post
    ...the hat must be made in two parts:
    1) An open-ended cone for the hat part and
    2) An annulus for the brim.

    ...
    I've attached a sketch of the cone of the hat.

    You already know:

    c = 50 cm therefore r = \frac{50}{2\pi}\ cm
    For the brim you need a circle with the radius R = r + 10

    h = 60 cm therefore the length of the slanted line is s = \sqrt{r^2+h^2}

    The circumference of the head is c = 50 cm. The angle of the sector which is used to make the curved surface of the cone is \alpha = \frac cs \cdot \frac{360^\circ}{2 \pi}

    The total surface area is

    A = \frac12 \cdot c \cdot s+\pi(R^2 - r^2)

    With all those formulae you can do (most of) the questions.

    If you need additional informations, please ask again.
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