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Math Help - Radian Homework Urgent!

  1. #1
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    Exclamation Radian Homework Urgent!

    Right my sister is stook on her radian homework, she tried to make an account on here but it wouldn't let her so I'm posting this for her, I would help her but we haven't studied radians yet!

    The questions are:

    1) The diagram shows the triangle OCD with OC=OD=17 and CD=30cm. The mid point of CD is M. With centre M, a semicircular arc A1 is drawn on CD as diameter. With centre O and radius 17cm, a circular arc A2 is drawn from C to D. The shaded region R is bounded by the arcs A1 and A2. Calculate, giving answers to 2 decimal places:

    (a) The area of the triangle OCD(she worked this out on her own lol) answer being 120cm^2
    (b) The angle COD in radians.
    (c) The area of the shaded region R.
    Diagram-


    2) The diagram shows a circle, centre O, of radius 6cm. The points A and B are on the circumference of the circle. The area of the shaded major sector is 80cm^2.
    Given that angle AOB=θ radians, where 0<θ<pi, calculate:
    (a) The value, the 3 decimal places, of θ.
    (b) The length in cm, the 2 decimal places, of the minor arc AB.
    Diagram-


    She has a few more questions that need answering so can I post them when we get answers for the others? I hate having to type them out

    Thanks in advance!
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  2. #2
    Senior Member ecMathGeek's Avatar
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    Quote Originally Posted by jonannekeke View Post
    Right my sister is stook on her radian homework, she tried to make an account on here but it wouldn't let her so I'm posting this for her, I would help her but we haven't studied radians yet!

    The questions are:

    1) The diagram shows the triangle OCD with OC=OD=17 and CD=30cm. The mid point of CD is M. With centre M, a semicircular arc A1 is drawn on CD as diameter. With centre O and radius 17cm, a circular arc A2 is drawn from C to D. The shaded region R is bounded by the arcs A1 and A2. Calculate, giving answers to 2 decimal places:

    (a) The area of the triangle OCD(she worked this out on her own lol) answer being 120cm^2
    (b) The angle COD in radians.
    (c) The area of the shaded region R.
    Diagram-


    2) The diagram shows a circle, centre O, of radius 6cm. The points A and B are on the circumference of the circle. The area of the shaded major sector is 80cm^2.
    Given that angle AOB=θ radians, where 0<θ<pi, calculate:
    (a) The value, the 3 decimal places, of θ.
    (b) The length in cm, the 2 decimal places, of the minor arc AB.
    Diagram-


    She has a few more questions that need answering so can I post them when we get answers for the others? I hate having to type them out

    Thanks in advance!
    (b) The angle COD in radians.
    Let's consider angle COD A, side CD a, side OD b and side CO c:
    CosA = (b^2 + c^2 - a^2)/(2bc)
    A = arccos[(b^2 + c^2 - a^2)/(2bc)]
    A = arccos[(17^2 + 17^2 - 30^2)/(2*17^2)]

    I don't have my calculator, so you need to find the value of that
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  3. #3
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    how do you convert it to radians?
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  4. #4
    Senior Member ecMathGeek's Avatar
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    Quote Originally Posted by jonannekeke View Post
    Right my sister is stook on her radian homework, she tried to make an account on here but it wouldn't let her so I'm posting this for her, I would help her but we haven't studied radians yet!

    The questions are:

    1) The diagram shows the triangle OCD with OC=OD=17 and CD=30cm. The mid point of CD is M. With centre M, a semicircular arc A1 is drawn on CD as diameter. With centre O and radius 17cm, a circular arc A2 is drawn from C to D. The shaded region R is bounded by the arcs A1 and A2. Calculate, giving answers to 2 decimal places:

    (a) The area of the triangle OCD(she worked this out on her own lol) answer being 120cm^2
    (b) The angle COD in radians.
    (c) The area of the shaded region R.
    Diagram-


    2) The diagram shows a circle, centre O, of radius 6cm. The points A and B are on the circumference of the circle. The area of the shaded major sector is 80cm^2.
    Given that angle AOB=θ radians, where 0<θ<pi, calculate:
    (a) The value, the 3 decimal places, of θ.
    (b) The length in cm, the 2 decimal places, of the minor arc AB.
    Diagram-


    She has a few more questions that need answering so can I post them when we get answers for the others? I hate having to type them out

    Thanks in advance!
    (c) The area of the shaded region R.
    For this, we need to find the area of the region bounded by the semicircle A1, add to that the area of triangle OCD, and then subtract the total area under A2.

    Here goes:
    The area of the semicircle A1 is 1/2*pi*r^2, where r is 15 (half of 30).
    A_A1 = 1/2*pi(15^2) = 225pi/2 cm^2
    The area of triangle OCD is 120 cm^2
    Notice that A2 bounds a 'slice' of a circle bounded by the lines OC and OD. the area of this region is equal to the angle COD times the radius of the circle, 17. Once you find the radius, from step 2, multiply that by 17 to find this area.
    Last, take the combined areas (225pi/2 + 120) and subtract the area of the above region to find the area of R.
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  5. #5
    Senior Member ecMathGeek's Avatar
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    Quote Originally Posted by jonannekeke View Post
    how do you convert it to radians?
    If your calculator is set in radians mode, it automatically gives you the answer in radians. Otherwise, if you are getting an answer in degrees, to convert to radians:

    (angle)*pi/180 = the angle in radians
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  6. #6
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    how do you set your calculator to radian mode? she has a casio one btw
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  7. #7
    Senior Member ecMathGeek's Avatar
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    Quote Originally Posted by jonannekeke View Post
    how do you set your calculator to radian mode? she has a casio one btw
    My friend is setting up an account. He should be able to help you with this one. (I don't know casio)

    [Edit] My friend is lame. lol
    He's no help. But I asked someone else.

    Just press [Mode] until you see an option to change the angle mode from degrees to radians (or vise versa). That should hopefully help.
    Last edited by ecMathGeek; March 22nd 2007 at 03:55 PM.
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  8. #8
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    oh yea, I remember having to do that once lol

    could you help with the second question too?
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  9. #9
    Senior Member ecMathGeek's Avatar
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    Quote Originally Posted by ecMathGeek View Post
    (c) The area of the shaded region R.
    For this, we need to find the area of the region bounded by the semicircle A1, add to that the area of triangle OCD, and then subtract the total area under A2.

    Here goes:
    The area of the semicircle A1 is 1/2*pi*r^2, where r is 15 (half of 30).
    A_A1 = 1/2*pi(15^2) = 225pi/2 cm^2
    The area of triangle OCD is 120 cm^2
    Notice that A2 bounds a 'slice' of a circle bounded by the lines OC and OD. the area of this region is equal to the angle COD times the radius of the circle, 17. Once you find the radius, from step 2, multiply that by 17 to find this area.
    Last, take the combined areas (225pi/2 + 120) and subtract the area of the above region to find the area of R.
    I made a big error! OOPS

    That's not the area

    The actual area is based on the area of a circle pi*r^2, but this needs to be multiplied by the ratio of the (angle)/(2pi). So the actual area of this region is: A/2*(17)^2

    Last, take the combined areas (225pi/2 + 120) and subtract the area of the above region to find the area of R.
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  10. #10
    Senior Member ecMathGeek's Avatar
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    Quote Originally Posted by jonannekeke View Post
    2) The diagram shows a circle, centre O, of radius 6cm. The points A and B are on the circumference of the circle. The area of the shaded major sector is 80cm^2.
    Given that angle AOB=θ radians, where 0<θ<pi, calculate:
    (a) The value, the 3 decimal places, of θ.
    (b) The length in cm, the 2 decimal places, of the minor arc AB.
    Diagram-
    (a) The value, the 3 decimal places, of θ.
    The area of that region is equal to its (angle)/2*r^2
    The angle of that region is 2pi - θ
    The area of that region is 80 cm^2
    (Area) = (angle)/2*r^2
    (angle) = 2(Area)/r^2
    2pi - θ = 2(80)/(6^2)
    -θ = 160/36 - 2pi
    θ = 2pi - 160/36

    (b) The length in cm, the 2 decimal places, of the minor arc AB.
    The arclength of a circle is based on its circumference 2pi*r, but is multiplied by the ratio of the (angle)/(2pi). So arclength is (angle)*r
    (Minor_Arc AB) = (2pi - θ)*r = (2pi - θ)*17

    Once you find θ, plug it in to the above equation to find the arc length.
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