Question

1. a ladder is placed against a vertical wall. to be safe, it must be inclined at between 70° and 80° to the ground. (a) is the ladder safe? (b) calculate the length of the ladder.

Explanation:

Step1: Find the angle of the ladder with the ground

We know that $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Here, the opposite side to the angle the ladder makes with the ground is the height on the wall ($h = 5.8$m) and the adjacent side is the distance from the wall on the ground ($d=1.5$m). So, $\tan\theta=\frac{5.8}{1.5}\approx3.867$. Then $\theta=\arctan(3.867)$. Using a calculator, $\theta\approx75.4^{\circ}$.

Step2: Check if the ladder is safe

Since the safe - angle range is between $70^{\circ}$ and $80^{\circ}$, and $70^{\circ}<75.4^{\circ}<80^{\circ}$, the ladder is safe.

Step3: Calculate the length of the ladder

Using the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $a = 1.5$m and $b = 5.8$m, and $c$ is the length of the ladder. So $c=\sqrt{1.5^{2}+5.8^{2}}=\sqrt{2.25 + 33.64}=\sqrt{35.89}\approx5.99$m.

Answer:

(a) Yes, the ladder is safe.
(b) The length of the ladder is approximately $5.99$m.