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Math Help - Area and Perimeter of Shaded Region (Answer Confirmation)

  1. #1
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    Area and Perimeter of Shaded Region (Answer Confirmation)

    Area and Perimeter of Shaded Region (Answer Confirmation)-picture.jpgIn attachment there are 8 question and i want to confirm my answers

    Q1)Area(Shaded)=41
    Perimeter(Shaded)=25

    Q2)Area(shaded)=114
    Perimeter(shaded)=6.44

    Q3)Area(shaded)=490
    Perimeter(shaded)=75

    Q4)Area(shaded)=88.9
    Perimeter(shaded)=

    Q5)Area(shaded)=55
    Perimeter(shaded)=14

    Q6)Area(shaded)=123
    Perimeter(shaded)=21

    Q7)Area(shaded)=14
    Perimeter(shaded)=15.5

    Q8)Area(shaded)=
    Perimeter(shaded)=37.5
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  2. #2
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    Hi haftakhan,

    For i, I got this:

    Area of big circle: 100 \pi
    Area of small circle: 36 \pi

    Area of shaded region: 100 \pi - 36 \pi \approx 201.06

    The perimeter of the circle with radius 10: 20 \pi \approx 62.83
    The perimeter of the circle with radius 6: 12\pi \approx 37.70
    The combined perimeter of the shaded region: \approx 100.53

    For ii, I got this:

    Area of circle: 100 \pi
    Area of Square: 14.14^2

    Area of shaded region: 100 \pi - 14.14^2 \approx 114.22

    The perimeter of the shaded region: 20\pi + 4(14.14) \approx 119.39


    For iii, the area looks good, but I don't see how you got 75 for the perimeter.


    I haven't looked at the others, but if you want to post them individually and show how you got your answers, that'd be good.
    Last edited by masters; August 17th 2010 at 12:07 PM. Reason: Thanks Soroban, I wasn't considering all the bounderies of the shaded regions.
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  3. #3
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    Hi haftakhan,
    I agree with masters show your work. I looked only at 8 where you had no area answer. My answer is area 72pi and 24pi for perimeter
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  4. #4
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    Hello, haftakhan!

    Sorry, only a few of your answers are correct.

    Are you sure of your area and perimeter formulas?
    The perimeter is the amount of fencing to enclose the shaded region.


    Click image for larger version. 

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    In attachment there are 8 question and i want to confirm my answers

    (1) Area = 41, Perimeter = 25

    We have two concentric circles with radius 6 and 10.

    The area of the shaded region is the difference of the two areas:
    . . A \:=\:\pi(10^2) - \pi (6^2) \;=\;64\pi \;\approx\;201.1

    The perimeter of the shaded region is the sum of the two perimeters:
    . . P \;=\;2\pi(10) + 2\pi(6) \;=\;32\pi \;\approx\;100.5




    (2) Area = 114, Perimeter = 6.44

    We have a circle with radius 10
    . . and an inscribed square with side 10\sqrt{2}

    \text{(Area)} \;=\; \text{(Area of circle)} - \text{(Area of square)}
    . . A \;=\;\pi(10^2) - (10\sqrt{2})^2 \;=\;100\pi - 200 \;\approx\;114.2 .
    You're right!

    \text{(Perimeter)} \;=\;\text{(Perimeter of circle)} + \text{(Perimeter of square)}
    . . P \;=\;2\pi(10) + 4(10\sqrt{2}) \;=\;20\pi + 40\sqrt{2} \;\approx\; 119.4


    Corrected my typo . . . Thanks, masters!



    (3) Area = 490. Perimeter = 75

    We have a circle with radius 10 inside a circle with radius 16.

    \text{(Area)} \;=\;\text{(Area of large circle)} - \text{(Area of smaller circle)}
    . . A \;=\;\pi(16^2) - \pi(10^2) \;=\;156\pi \;\approx\;490.1 .
    Correct!

    \text{(Perimeter)} \;=\;\text{(Sum of the two perimeters)}
    . . P \;=\;2\pi(16) + 2\pi(10) \;=\;52\pi \;\approx\;163.4




    (4) Area = 88.9, Perimeter = ?

    We have a quarter-circle with radius 40
    . . and an isosceles right triangle with sides 40.

    \text{Area} \;=\;\text{(Area of quarter-circle)} - \text{(Area of triangle)}
    . . A \;=\;\frac{1}{4}\pi(40^2) - \frac{1}{2}(40)(40) \;=\;400\pi - 800 \;\approx\;456.6

    \text{(Perimeter)} \;=\;\text{Perimeter of quarter-circle)} + \text{Length of hypotenuse)}
    . . P \;=\;\frac{1}{4}\cdot2\pi(40) + 40\sqrt{2} \;\approx\; 119.4




    (5) Area = 55, Perimeter = 14

    We have a 16-by-16 square with two semicircles inside with radius 8.

    \text{(Area)} \;=\;\text{(Area of square)} - \text{(Area of circle)}
    . . A \;=\;16^2 - \pi(8^2) \;=\;256 - 64\pi \;\approx\;54.9 .
    Yes!

    \text{(Perimeter)} \;=\;\text{(2 sides of the square}) + \text{(Circumference of circle)}
    . . P \;=\;2(16) + 2\pi(8) \:=\:32 + 16\pi \;\approx\;82.3
    Last edited by Soroban; August 17th 2010 at 01:18 PM.
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  5. #5
    A riddle wrapped in an enigma
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    Soroban,

    Check your perimeter final answer in (2).
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  6. #6
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    i have identified my mistakes thanks to your answers.


    Are the last three answers right ?
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  7. #7
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    Also i want to know why Perimeter of both are added to get the Perimeter of the shaded region ?
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  8. #8
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    Hi haftakhan,
    Show your work for questions 6,7,8 and I will help you if I think they are wrong


    bjh
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