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IB Mathematics: Applications and Interpretation HL · 12.10: Geometry & Trigonometry · 12.10.15

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Grazing goat problems: sector and triangle area Problem set B

Split the grazed region into triangles and sectors. Check that the calculator's angle mode matches the angles you use. Give answers to 3 significant figures unless a question says otherwise.

Each question matches the question with the same number in set A. Do the ones you got wrong there.

Difficulty: first skills applying them stretch

Skills

  1. A field is a right-angled triangle with sides 7 m, 24 m and 25 m. A cow, a horse and a donkey are each tied to a different corner by a 3 m rope. The ropes are too short to meet, and the animals cannot leave the field.

    24 m7 m25.0 mABC
    1. (a) Find the area each animal can graze.
    2. (b) Which animal grazes the most?

  2. A field is a right-angled triangle with sides 12 m, 16 m and 20 m. Three llamas are each tied to a different corner by a 5 m rope. The ropes are too short to meet, and the llamas cannot leave the field. Find the percentage of the field the llamas graze altogether.

    16 m12 m20.0 mABC

  3. A field is a rectangle 11 m by 4 m. A pony is tied by a 5.5 m rope to a post at one corner. Find the area of the field the pony can graze.

    11 m4 m5.5 mX

In context

  1. A garden is a right-angled triangle with sides 15 m, 20 m and 25 m. A llama is tied to the midpoint of the hypotenuse by a rope just long enough to reach both ends of it. A second llama is tied to the midpoint of the 20 m side by a 10 m rope. Neither llama can leave the garden.

    20 m15 m12.5 m12.5 m12.5 mABCM
    1. (a) Find the percentage of the garden the first llama can graze.
    2. (b) The hypotenuse cuts a segment off the circle of the second llama. Find the area the second llama can graze.

  2. A rectangular school field measures 14 m by 5 m. A tethered ram is tied by a 7 m rope to a post at one corner of the field. Find the percentage of the field the ram can graze, to the nearest integer.

    14 m5 m7 mX

  3. A fenced triangular pen CDE stands in a large field. A ram is tied by a 6.5 m rope at a point A on the outside of the fence CD. A is 3.5 m from the corner C, and CD is 11 m long. The angle of the pen at C is θ radians. The ram can graze 78 m² of grass. Find θ.

    6.5 mAC

Stretch

  1. A rectangular paddock is 13 m by 3 m. A horse is tied to a post at one corner by a rope of length a metres, with \(3 < a < 13\), so that it can graze exactly half of the paddock.

    1. (a) Show that \(a^2\arcsin\frac{3}{a} + 3\sqrt{a^2-9} = 39\).
    2. (b) Find the value of a.

  2. A rectangular paddock is 13 m by 5 m. A horse is tied to a post at one corner by a rope of length a metres, with \(5 < a < 13\). The horse can graze exactly 45% of the paddock. Find the value of a.

jmaths.xyz · AI HL · 12.10.15 Grazing goat problems: sector and triangle area