WAEC 2019 · Paper 2 · Q11

A ladder $11\text{ m}$ long leans against a vertical wall at an angle of $75^\circ$ to the ground. The ladder is then pushed $0.2\text{ m}$ up the wall.

(a)
Illustrate the information in a diagram.
Model answer
Not to scale: the angles are opened out so both positions of the ladder can be seen.
A clear sketch is enough (it need not be to scale), but it must show every given fact: draw a vertical wall and level ground at right angles. The ladder $AB = 11$ m makes $75^\circ$ with the ground. After the top is pushed $0.2$ m up to $A_1$ , the ladder $A_1B_1$ is still 11 m long but steeper: its foot moves closer to the wall. Label the new angle $\theta$ ; it works out to about $80^\circ$ .
(b)
Find, correct to the nearest whole number, the: (i) new angle which the ladder makes with the ground; (ii) distance the foot of the ladder has moved from its original position.
Separate values with commas, e.g. 3, −2

Worked solution (try it first)

(a)

Draw the wall vertical and the ground horizontal, with the 11 m ladder making $75^\circ$ with the ground.
Then draw the new position, reaching 0.2 m higher up the wall, with its foot nearer the wall.

(b)(i)

At first the ladder reaches $11\sin 75^\circ \approx 10.625$ m up the wall.
Pushed 0.2 m up, it reaches $10.825$ m.
The ladder is still 11 m long: $\sin\theta = \frac{10.825}{11}$
$\approx 0.9841$ , so $\theta \approx 79.8^\circ$ , which is $80^\circ$ to the nearest whole number.

(ii)

The foot was $11\cos 75^\circ \approx 2.847$ m from the wall and is now $11\cos 79.8^\circ \approx 1.95$ m from it.
It moved $2.847 - 1.95 \approx 0.9$ m, which is 1 m to the nearest whole number.

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