Question

a ladder is leaning against a wall so that it makes an angle of 31 degrees with the wall. if the base of the ladder is 6 feet from the bottom of the wall, how long is the ladder? be sure your calculator is in deg mode, and use the proper trig function on your calculator in your computation. round your answer to the nearest tenth of a foot, but do not include \ft\ or \feet\ with your response.

Explanation:

Step1: Identify the trigonometric relationship

We have a right triangle where the ladder is the hypotenuse (\(c\)), the distance from the base of the ladder to the wall is the side opposite the 31° angle (\(a = 6\) feet), and the angle between the ladder and the wall is 31°. So we use the sine function: \(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\), which becomes \(\sin(31^\circ)=\frac{6}{c}\).

Step2: Solve for the hypotenuse (ladder length)

Rearrange the formula to solve for \(c\): \(c = \frac{6}{\sin(31^\circ)}\).
Using a calculator in degree mode, \(\sin(31^\circ)\approx0.5150\).
Then \(c=\frac{6}{0.5150}\approx11.65\). Rounding to the nearest tenth gives \(11.7\).

Answer:

11.7