Question

a staircase was being built which had an incline of 45° to the horizontal and started 13m away from the supporting wall. what was the length of the staircase?

Explanation:

Step1: Recall the cosine formula

In a right - triangle, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Here, $\theta = 45^{\circ}$, the adjacent side $a = 13$ m, and the hypotenuse $h$ is the length of the staircase. So, $\cos45^{\circ}=\frac{13}{h}$.

Step2: Solve for $h$

We know that $\cos45^{\circ}=\frac{\sqrt{2}}{2}$. Then, $h=\frac{13}{\cos45^{\circ}}$. Substitute $\cos45^{\circ}=\frac{\sqrt{2}}{2}$ into the formula: $h=\frac{13}{\frac{\sqrt{2}}{2}}=13\times\frac{2}{\sqrt{2}}$. Rationalize the denominator: $h = 13\sqrt{2}\approx13\times1.414 = 18.382$ m.

Answer:

The length of the staircase is approximately $18.4$ m.