Question

8 formula 0.5 points a ladder is leaning against a wall so that it makes an angle of 27 degrees with the wall. if the base of the ladder is 4 feet from the bottom of the wall, how long is the ladder?
image of ladder and wall
be sure your calculator is in deg mode, and use the proper trig function on your calculator in the computation.
round your answer to the nearest tenth of a foot, but do not include \ft\ with your response.
answer
blank
9 formula 0.5 points an observer stands on the bank of a river, and looks directly across the river to a tree on the opposite bank. the angle of elevation from the feet of the observer to the top of the tree is 48 degrees.
if the tree is 23 feet tall, how wide is the river?

Explanation:

Problem 8

Step1: Identify the trigonometric relationship

We have a right triangle where the ladder is the hypotenuse, the distance from the wall to the base of the ladder (4 feet) is the side opposite the 27° angle (between the ladder and the wall), and we need to find the length of the ladder (hypotenuse \( c \)). Using the sine function: \( \sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}} \), so \( \sin(27^\circ)=\frac{4}{c} \).

Step2: Solve for \( c \)

Rearrange the formula: \( c = \frac{4}{\sin(27^\circ)} \). Calculate \( \sin(27^\circ) \) (in degree mode) ≈ 0.4540. Then \( c=\frac{4}{0.4540}\approx8.8 \).

Step1: Identify the trigonometric relationship

We have a right triangle where the tree height (23 feet) is the opposite side to the 48° angle of elevation, and the river width is the adjacent side (\( x \)) to the angle. Using the tangent function: \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \), so \( \tan(48^\circ)=\frac{23}{x} \).

Step2: Solve for \( x \)

Rearrange the formula: \( x = \frac{23}{\tan(48^\circ)} \). Calculate \( \tan(48^\circ) \) (in degree mode) ≈ 1.1106. Then \( x=\frac{23}{1.1106}\approx20.7 \).

Answer:

8.8

Problem 9