Question

1. apply when a stepladder is placed as shown, is it possible for the lower ends of the ladder to be 12 feet apart? explain.

Explanation:

Step1: Identify the triangle type

The stepladder forms an isosceles triangle with two sides of 12 ft and the included angle of \(58^\circ\). We can use the Law of Cosines to find the length of the base (distance between the lower ends), which is \(c\) in the Law of Cosines formula \(c^{2}=a^{2}+b^{2}-2ab\cos(C)\), where \(a = b=12\) ft and \(C = 58^\circ\).

Step2: Apply the Law of Cosines

Substitute \(a = 12\), \(b = 12\), and \(C=58^\circ\) into the formula:
\[

$$\begin{align*} c^{2}&=12^{2}+12^{2}-2\times12\times12\times\cos(58^\circ)\\ c^{2}&=144 + 144- 288\times\cos(58^\circ)\\ \cos(58^\circ)&\approx0.5299\\ c^{2}&=288-288\times0.5299\\ c^{2}&=288\times(1 - 0.5299)\\ c^{2}&=288\times0.4701\\ c^{2}&\approx135.3888\\ c&\approx\sqrt{135.3888}\\ c&\approx11.64 \end{align*}$$

\]

Step3: Compare with 12 ft

We calculated the base length \(c\approx11.64\) ft, which is less than 12 ft. So, it is not possible for the lower ends to be 12 feet apart because the maximum possible distance (based on the given side lengths and included angle) is approximately 11.64 feet.

Answer:

No, it is not possible. Using the Law of Cosines on the isosceles triangle with sides \(a = b=12\) ft and included angle \(58^\circ\), the distance between the lower ends is calculated as approximately \(11.64\) ft, which is less than \(12\) ft.