Math help please?

Number 1

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  • 8 months ago
    Favourite answer

    The yellow surface area is:

    yellow = (7 * 6)/2

    yellow = 21 cm²

    The blue surface area is:

    blue = (π.r²) * (α/360) → where r is the radius of the circle, in your case: r = OA = 7

    blue = (49π) * (α/360) → where: tan(α) = AB/OA = 6/7 → α ≈ 40.6 °

    blue ≈ 49π * (40.6/360)

    blue ≈ 17.36 cm²

    The surface area of the shaded region is:

    = yellow - blue

    ≈ 21 - 17.36

    ≈ 3.6386 cm²

    To be more precise:

    blue = (49π) * (α/360) → where: tan(α) = AB/OA = 6/7

    blue = 49π * atan(6/7) / 360

    blue = (49/360).π.atan(6/7)

    The surface area of the shaded region is:

    = yellow - blue

    = 21 - (49/360).π.atan(6/7)

    = [7560 - 49π.atan(6/7)]/360

    Attachment image
  • 1 week ago

     The figure shows a right-angled triangle OAB. 

     AOC is a minor sector enclosed in the triangle. 

     If OA = 7 cm, AB = 6 cm, 

     calculate the area and perimeter 

     of the shaded region.

     The area of the triangle OAB = 21 cm^2

     OB = √85 = 9.22

     CB = 2.22

     Angle BOA = 40.60°

     The area of the minor sector AOC:

     49 pi (40.6/360) = 17.3608... cm^2

     The area of the shaded region = 3.6302... cm^2

     Length of arc AC = 14 pi (40.6/360) = 4.96 cm

     The perimeter of the shaded region 

     = (6 + 2.22 + 4.96) cm

     = 13.18 cm

                                                                                                              

  • 8 months ago

    Let x is angle in vertex O

    x = arc tan (6/7) = 0.70863 rad

    Now length of arc AC is

    AC= OA * x = 7 * 0.70863 = 4.96

    Use Pythagorean Theorem to find

    OB = sqrt (6^2+7^2) = sqrt (85) = 9.22

    BC= OB-OC=9.22-7 = 2.22

    Perimeter of shaded region ABC is

     AB+BC+AC = 6 + 2.22 + 4.96 = 13.18

    Area of sector AOC relates to area of circle with radius OA the same as angle x relates to 2Pi (full circle). Hence the area of AOC is

    (x/(2Pi))*(OA^2 * Pi) = x * OA^2 / 2 = 0.70863 * 7^2 / 2 = 17.36

    Area of triangle OAB is

    OA*AB / 2 = 7 * 6 / 2 = 21

    Area of shaded region is difference between the two

    21 - 17.36 = 3.64

     

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