an electrician leans an extension ladder against the outside wall of a house so that it reaches an electric…

an electrician leans an extension ladder against the outside wall of a house so that it reaches an electric box 24 feet up. the ladder makes an angle of 70° with the ground. find the length of the ladder. round your answer to the nearest hundredth of a foot if necessary.

an electrician leans an extension ladder against the outside wall of a house so that it reaches an electric box 24 feet up. the ladder makes an angle of 70° with the ground. find the length of the ladder. round your answer to the nearest hundredth of a foot if necessary.

Answer

Explanation:

Step1: Identify trigonometric relation

We have a right - triangle where the height (opposite side to the given angle) is 24 feet and the length of the ladder is the hypotenuse $x$. We use the sine function since $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. Here, $\theta = 70^{\circ}$ and the opposite side $y = 24$ and hypotenuse is $x$. So, $\sin70^{\circ}=\frac{24}{x}$.

Step2: Solve for $x$

We can re - arrange the equation $\sin70^{\circ}=\frac{24}{x}$ to get $x=\frac{24}{\sin70^{\circ}}$. Since $\sin70^{\circ}\approx0.9397$, then $x=\frac{24}{0.9397}\approx25.54$.

Answer:

$25.54$ feet