A 10-foot ladder rests against a wall. If the top of the ladder starts sliding down along the wall at a rate of 3 feet per second, how fast is the angle between the ladder and the ground changing when the bottom of the ladder is 6 feet away from the wall? Enter the magnitude of the velocity with proper units.

Question

BlackTom AI · Accepted Answer

We start by identifying the geometry and variables. Let the ladder length be L = 10 ft, the distance from the wall to the bottom be x, and the height of the top above the ground be y. The ladder forms a right triangle with x^2 + y^2 = 10^2. The angle θ is between the ladder and the ground, so we can relate x and y to θ via x = L cos θ = 10 cos θ and y = L sin θ = 10 sin θ.

Differentiate the relation y = 10 sin θ with respect to time t to connect dy/dt and dθ/dt:
dy/dt = 10 cos θ · dθ/dt.
We are told the top slides down along the wall at dy/dt = −3 ft/s (negative because y is decreasing).

When the bottom is 6 ft from the wall, x = 6, so cos θ = x/10 = 6/10 = 0.6.
Plug into the differentiated equation: −3 = 10 · (0.6) · dθ/dt = 6 · dθ/dt.
Thus dθ/dt = −3/6 = −0.5 rad/s.
The magnitude of the angular speed is |dθ/dt| = 0.5 rad/s. Since the top is sliding down, the angle θ is decreasing, which explains the negative sign in dθ/dt.

Note on the provided options: the given option set contains a placeholder option (often labeled as a generic "回答" or similar) rather than the correct numeric value. The correct magnitude is 0.5 radians per second.

查看解析

登录即可查看完整答案

类似问题

Solve the problem. Round your answer, if appropriate. A man 6 ft tall walks at a rate of 6 ft/sec away from a lamppost that is 18 ft high. At what rate is the length of his shadow changing when he is 30 ft away from the lamppost? (Do not round your answer)

一架飞机正在沿着抛物线移动。当平面穿过该点时，其坐标以 m/s 的速率减小。在那一刻，平面的坐标是_________________________________

A plane is moving along the parabola 𝑦 = 5 − ( 2 𝑥 + 1 ) 2 3 . As the plane passes through the point ( 2 , − 10 3 ) , its 𝑦 -coordinate is decreasing at a rate of 4.2 m/s. At that instant, the 𝑥 -coordinate of the plane is _________________________________

The volume 𝑉 of a cylinder of height 4 cm and radius 𝑟 is given by: 𝑉 = 4 𝜋 𝑟 2 What is the relationship between 𝑑 𝑉 𝑑 𝑡 and 𝑑 𝑟 𝑑 𝑡 ? (Hint: Differentiate both sides of 𝑉 = 4 𝜋 𝑟 2 with respect to 𝑡 .)

Water is flowing into a large spherical tank at a constant rate (e.g. 1 gallon per minute). Let 𝐻 ( 𝑡 ) be the height of the water in the tank. Which of the following is true about 𝑑 𝐻 𝑑 𝑡 ?

更多留学生实用工具

考试浏览器助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单