Question

use the pythagorean theorem to find $ell$. pythagorean theorem: $a^{2}+b^{2}=c^{2}$

Explanation:

Step1: Identify the right - angled triangle sides

The height of the cone is \(a = 3\) ft and the radius (half of the base diameter) is \(b = 2\) ft. We want to find the slant height \(\ell\) which is \(c\) in the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\).

Step2: Substitute values into the Pythagorean theorem

Substitute \(a = 3\) and \(b = 2\) into \(a^{2}+b^{2}=c^{2}\), we get \(3^{2}+2^{2}=c^{2}\), so \(c^{2}=9 + 4\).

Step3: Calculate the value of \(c^{2}\)

\(9+4=13\), so \(c^{2}=13\).

Step4: Solve for \(c\) (which is \(\ell\))

Since \(c=\ell\) and \(c^{2}=13\), then \(\ell=\sqrt{13}\) ft.

Answer:

\(\sqrt{13}\)