Question

a pole of length l is carried horizontally around a corner where a 3 - ft - wide hallway meets a 4 - ft - wide hallway, as shown in the figure on the right. for 0 < θ < π/2, find the relationship between l and θ at the moment when the pole simultaneously touches both walls and the corner p. estimate θ when l = 10 ft. identify the relationship between l and θ when the pole simultaneously touches both walls and the corner p. choose the correct answer below. a. l(θ)=3 sin θ + 4 cos θ b. l(θ)=3 csc θ + 4 sec θ c. l(θ)=3 cos θ + 4 sin θ d. l(θ)=3 sec θ + 4 csc θ

Explanation:

Step1: Use right - triangle trigonometry

Let's consider two right - triangles formed at the corner. If the angle between the pole and the wall is $\theta$, and the widths of the hallways are $a = 3$ ft and $b = 4$ ft. The length of the pole $L$ can be expressed in terms of $\theta$ using the sum of the lengths of the two segments of the pole in the two hallways.
The length of the part of the pole in the 3 - ft hallway is $L_1=3\csc\theta$ and the length of the part of the pole in the 4 - ft hallway is $L_2 = 4\sec\theta$.

Step2: Find the total length formula

The total length of the pole $L$ is the sum of these two lengths. So, $L(\theta)=3\csc\theta + 4\sec\theta$.

Answer:

B. $L(\theta)=3\csc\theta + 4\sec\theta$