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    So we were set this group task in which we choose from 2 questions. We figured out the first one but the second one got us dev'd and my group is basically hopeless on solving this. Here it is , and help will be very appreciated :

    A farmer has a triangular field with sides of length 80 yards,150 yards and 170 yards. He finds that his sheep tend to hide in the corners and that makes it difficult for his dogs to control them. He decides to use the land inside the triangular field to make a circular field , but he wants to make the circular field as large as possible. What length of fence will be required to form the boundary of the new circular field , how many acres will it contain?

    (also pls show any workings out cos it will help us loads , ty)
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    (Original post by bea.nicole)
    So we were set this group task in which we choose from 2 questions. We figured out the first one but the second one got us dev'd and my group is basically hopeless on solving this. Here it is , and help will be very appreciated :

    A farmer has a triangular field with sides of length 80 yards,150 yards and 170 yards. He finds that his sheep tend to hide in the corners and that makes it difficult for his dogs to control them. He decides to use the land inside the triangular field to make a circular field , but he wants to make the circular field as large as possible. What length of fence will be required to form the boundary of the new circular field , how many acres will it contain?

    (also pls show any workings out cos it will help us loads , ty)
    If you use the advanced trig cosine rule to work out one of the angles, followed by the sine rule to work out another angle you can minus both answers from 180 to find the final missing angle. After you do this, you'll realise you have a right angled triangle. (I didn't know how I was approaching this problem until I realised that the yard is indeed a right angle triangular one!)You should get something like in this picture

    Now imagine that triangle with a circle inside. We need to work out what the circumference of this circle is to find out how much fence he needs. So, a circle can only touch a line at one point right? This means that if the circle is the largest one possible it will have to touch all the sides of the triangle and therefore all the sides are tangents to this circle

    A tangent and a radius meet at a right angle.

    So the radii of the circle can make three right angles with the sides where they meet

    The radii which touch the 80 side and the 150 yard side make a square as this is the right angle part of the triangle. Like this:

    [attach]5.02945502955503e+29[/attach]
    You know know some equations which we can use later:
    150 - r
    And also 80 - r

    If you look at the hypotenuse (170 yard side) you can see that you can make two triangles by drawing an imaginary line from the middle of the 61.9 angle to the centre of this large circle. The two triangles share a common base (r) and a common side (the imaginary middle line).

    That means that the side of the triangle on the right will also be 80 - r as they are congruent triangles. The same thing happens on the other end, You get this basically:
    [attach]5.02945502955503e+23[/attach]

    But you already know that the side labelled (80 - r) + (150 - r) is 170 yards.

    Therefore: 80 + 150 - 2r = 170
    Solve and you get r = 30
    So the diameter must be 30 x 2 = 60
    Multiply by pi and you get something like 188.5 yards to 1.dp

    To work out how many acres it contains you just work out the area of the circle now, which should be easy since you know that the radius is 30 yards. 2827.4 (again to 1.dp)

    Let me know if that's right hahaha, I tried
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    (Original post by PanickingYear12)
    If you use the advanced trig cosine rule to work out one of the angles, followed by the sine rule to work out another angle you can minus both answers from 180 to find the final missing angle. After you do this, you'll realise you have a right angled triangle. (I didn't know how I was approaching this problem until I realised that the yard is indeed a right angle triangular one!)You should get something like in this picture

    Now imagine that triangle with a circle inside. We need to work out what the circumference of this circle is to find out how much fence he needs. So, a circle can only touch a line at one point right? This means that if the circle is the largest one possible it will have to touch all the sides of the triangle and therefore all the sides are tangents to this circle

    A tangent and a radius meet at a right angle.

    So the radii of the circle can make three right angles with the sides where they meet

    The radii which touch the 80 side and the 150 yard side make a square as this is the right angle part of the triangle. Like this:

    [attach]5.02945502955503e+29[/attach]
    You know know some equations which we can use later:
    150 - r
    And also 80 - r

    If you look at the hypotenuse (170 yard side) you can see that you can make two triangles by drawing an imaginary line from the middle of the 61.9 angle to the centre of this large circle. The two triangles share a common base (r) and a common side (the imaginary middle line).

    That means that the side of the triangle on the right will also be 80 - r as they are congruent triangles. The same thing happens on the other end, You get this basically:
    [attach]5.02945502955503e+23[/attach]

    But you already know that the side labelled (80 - r) + (150 - r) is 170 yards.

    Therefore: 80 + 150 - 2r = 170
    Solve and you get r = 30
    So the diameter must be 30 x 2 = 60
    Multiply by pi and you get something like 188.5 yards to 1.dp

    To work out how many acres it contains you just work out the area of the circle now, which should be easy since you know that the radius is 30 yards. 2827.4 (again to 1.dp)

    Let me know if that's right hahaha, I tried

    Wow , thank you so much! This really helped us get a good grasp of it so thank you xx
 
 
 
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