Question

question #4
reference q.14152
a ladder 5.3 m long is inclined at an angle of 72° to the ground.
a. how far up the wall, to the nearest tenth of a metre, does the ladder reach?
b. use trigonometry to determine, to the nearest tenth of a metre, the distance between the bottom of the ladder and the bottom of the wall.
c. use the answer to a) and the pythagorean theorem to determine, to the nearest tenth of a metre, the distance between the bottom of the ladder and the bottom of the wall.
d. explain why the answers to b) and c) are different.

Explanation:

Step1: Find height on wall (a)

We know the length of the ladder (hypotenuse $c = 5.3$ m) and the angle with the ground $\theta=72^{\circ}$. Using the sine - function $\sin\theta=\frac{opposite}{hypotenuse}$. Let the height up the wall be $a$. So $a = c\sin\theta$. Substituting values, $a = 5.3\times\sin(72^{\circ})$. Since $\sin(72^{\circ})\approx0.9511$, then $a = 5.3\times0.9511\approx5.0$ m.

Step2: Find distance from wall (b)

Using the cosine - function $\cos\theta=\frac{adjacent}{hypotenuse}$. Let the distance from the bottom of the ladder to the wall be $b$. So $b = c\cos\theta$. Substituting $c = 5.3$ m and $\theta = 72^{\circ}$, and since $\cos(72^{\circ})\approx0.3090$, then $b=5.3\times0.3090\approx1.6$ m.

Step3: Find distance from wall using Pythagorean theorem (c)

From part (a), $a\approx5.0$ m and $c = 5.3$ m. By the Pythagorean theorem $c^{2}=a^{2}+b^{2}$, so $b=\sqrt{c^{2}-a^{2}}$. Substituting $c = 5.3$ and $a\approx5.0$, we have $b=\sqrt{5.3^{2}-5.0^{2}}=\sqrt{(5.3 + 5.0)(5.3 - 5.0)}=\sqrt{10.3\times0.3}=\sqrt{3.09}\approx1.8$ m.

Step4: Explain difference between b and c

In part (b), we used trigonometry directly with the given angle and hypotenuse. In part (c), we used the Pythagorean theorem with a rounded - off value of $a$ from part (a). The rounding in part (a) led to a different result in part (c) compared to the direct trigonometric calculation in part (b).

Answer:

a. $5.0$ m
b. $1.6$ m
c. $1.8$ m
d. The difference is due to rounding in part (a) when using the Pythagorean theorem in part (c) compared to the direct trigonometric calculation in part (b).