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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 A

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.

Difficulty: first skills applying them stretch

Skills

  1. A field is a right-angled triangle with sides 8 m, 15 m and 17 m. A goat, a sheep and a pony are each tied to a different corner by a 3.5 m rope. The ropes are too short to meet, and the animals cannot leave the field.

    15 m8 m17.0 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 10 m, 24 m and 26 m. Three goats are each tied to a different corner by a 4.5 m rope. The ropes are too short to meet, and the goats cannot leave the field. Find the percentage of the field the goats graze altogether.

    24 m10 m26.0 mABC

  3. A field is a rectangle 14 m by 5 m. A goat is tied by a 6.5 m rope to a post at one corner. Find the area of the field the goat can graze.

    14 m5 m6.5 mX

In context

  1. A paddock is a right-angled triangle with sides 6 m, 8 m and 10 m. A donkey is tied to the midpoint of the hypotenuse by a rope just long enough to reach both ends of it. A second donkey is tied to the midpoint of the 8 m side by a 4 m rope. Neither donkey can leave the paddock.

    8 m6 m5.00 m5.00 m5.00 mABCM
    1. (a) Find the percentage of the paddock the first donkey can graze.
    2. (b) The hypotenuse cuts a segment off the circle of the second donkey. Find the area the second donkey can graze.

  2. A rectangular lawn measures 15 m by 6 m. A goat is tied by an 8 m rope to a post at one corner of the lawn. Find the percentage of the lawn the goat can graze, to the nearest integer.

    15 m6 m8 mX

  3. A fenced triangular pond CDE stands in a large field. A goat is tied by an 8 m rope at a point A on the outside of the fence CD. A is 5 m from the corner C, and CD is 14 m long. The angle of the pond at C is θ radians. The goat can graze 110 m² of grass. Find θ.

    8 mAC

Stretch

  1. A rectangular field is 12 m by 5 m. A goat is tied to a post at one corner by a rope of length a metres, with \(5 < a < 12\), so that it can graze exactly half of the field.

    1. (a) Show that \(a^2\arcsin\frac{5}{a} + 5\sqrt{a^2-25} = 60\).
    2. (b) Find the value of a.

  2. A rectangular field is 14 m by 6 m. A goat is tied to a post at one corner by a rope of length a metres, with \(6 < a < 14\). The goat can graze exactly 55% of the field. Find the value of a.

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