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Math Help - Please help me with some Geometry! (Areas)

  1. #1
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    Exclamation Please help me with some Geometry! (Areas)

    Could someone walk me through these problems, step by step?

    1. Use figure 1 below. A ring of grass with an area of 314 yd^2 surrounds a circular flower bed. Find the width x of the ring of grass.

    2. Use figure 2 below. Bob cuts four congruent squares from the corners of a 30 in. by 50 in. rectangular piece of cardboard so that it can be folded to make a box. Find the side length s of the squares, given that the area of the bottom of the box is 200 in^2.

    3. Use figure 3 below. Billy has 80 m of fencing materials to enclose three sides of a rectangular garden. She will use the side of her garage as a border for the fourth side. Find the width x of the garden if its area is too be 700 m^2.



    If anyone could help me out, that would be great!
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by miyosuke View Post
    Could someone walk me through these problems, step by step?

    1. Use figure 1 below. A ring of grass with an area of 314 yd^2 surrounds a circular flower bed. Find the width x of the ring of grass.

    2. Use figure 2 below. Bob cuts four congruent squares from the corners of a 30 in. by 50 in. rectangular piece of cardboard so that it can be folded to make a box. Find the side length s of the squares, given that the area of the bottom of the box is 200 in^2.

    3. Use figure 3 below. Billy has 80 m of fencing materials to enclose three sides of a rectangular garden. She will use the side of her garage as a border for the fourth side. Find the width x of the garden if its area is too be 700 m^2.



    If anyone could help me out, that would be great!
    1. (Area of the ring of grass) + (Area of the flower bed) = Area of the big circle whose radius is 10+x

    so, A_{big} = \pi (10+x)^2 = 314 + 10^2 \pi, solve for x..

    2. (30 - 2s)(50-2s) = 200, solve for s

    3. using the same variables, you would form a system of 2 equation in 2 unknowns..
    (a) xy = 700
    (b) 2x + y = 80
    you can solve it..
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  3. #3
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    earboth's Avatar
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    Quote Originally Posted by miyosuke View Post
    Could someone walk me through these problems, step by step?

    1. Use figure 1 below. A ring of grass with an area of 314 yd^2 surrounds a circular flower bed. Find the width x of the ring of grass.
    ...
    Hi,

    the area of grass is calculated as a difference of 2 concentric circles:
    R = radius of the greater circle containing the flower bed and the grass: R = 10 +x
    r = radius of the flower bed: r = 10

    Area of the grass; A = (10+x)^2 \cdot \pi - 10^2 \cdot \pi= 314\ yd^2 = 100 \cdot \pi \ yd^2 . Now expand the bracket, divide the equation by \pi and you'll get a quadratic equation in x:
    x^2+20x-100=0~\iff~x=-10\pm \sqrt{200} x must be a positive number. So reject the negative solution. I've got x \approx 4.14\ yd

    Quote Originally Posted by miyosuke View Post
    ...
    2. Use figure 2 below. Bob cuts four congruent squares from the corners of a 30 in. by 50 in. rectangular piece of cardboard so that it can be folded to make a box. Find the side length s of the squares, given that the area of the bottom of the box is 200 in^2.
    ...
    The base area can be calculated by:
    A = (50-2s)(30-2s)=200~\iff~1500-160s+4s^2=200~,~0 \leq s\leq 15. You'll get a quadratic equation in s:

    4s^2-160s+1300=0. Solve for s. One solution doesn't satisfy the conditions. I've got s = 5(4-\sqrt{3})\approx 11.3397


    Quote Originally Posted by miyosuke View Post
    ...

    3. Use figure 3 below. Billy has 80 m of fencing materials to enclose three sides of a rectangular garden. She will use the side of her garage as a border for the fourth side. Find the width x of the garden if its area is too be 700 m^2.



    If anyone could help me out, that would be great!
    The area of the rectangle is:

    A=x \cdot y = 700~,~0 \leq x \leq 40
    The fence is 80 m long:
    f = 2x + y = 80~\implies~y=80-2x . Plug in this term of y into the first equation:

    700=x \cdot (80-2x)~\iff~700=-2x^2 + 80x. Solve this quadratic equation for x and check the condition for x. I've got a valid result: x = 5(4-\sqrt{2})\approx 12.93\ m
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