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)

Question

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)

BlackTom AI · Accepted Answer

To begin, I’ll restate the setup and the quantities involved so we can reason through the options carefully.
- A lamppost of height 18 ft cast a shadow for a man who is 6 ft tall. The man is moving away from the lamppost at 6 ft/sec. Let x be the horizontal distance from the lamppost to the man, and let s be the length of the man’s shadow on the ground (from the man to the tip of the shadow).
- The question asks for ds/dt, the rate at which the shadow’s length is changing, when x = 30 ft.

Now we analyze each option one by one.
Option A: 3 ft/sec
- This matches the standard related-rates setup for this geometry. By similar triangles, 18/(x + s) = 6/s, which simplifies to s = x/2. Differentiating with respect to time gives ds/dt = (1/2) dx/dt. Since the man walks away at dx/dt = 6 ft/s, we get ds/dt = 3 ft/s. Therefore this option is correct mathematically for the given scenario.

Option B: 30 ft/sec
- This would imply the shadow length is changing ten times as fast as the man’s distance from the lamp. The relationship s = x/2 shows ds/dt is exactly half of dx/dt, not ten times as large. Unless dx/dt were 60 ft/s, ds/dt cannot be 30 ft/s. Hence this is incorrect.

Option C: ft/sec
- This option is missing a numeric value and only provides the units. Without a numeric rate, it does not answer the question. In the context of the problem, we must supply a specific rate, so this choice is invalid.

Option D: ft/sec
- Similar to Option C, this option only lists units without a numeric value. It fails to specify a concrete rate for ds/dt. Therefore it cannot be the correct answer.

Option E: (also) ft/sec
- Again, this is just a unit without a number. It does not specify the rate of shadow-length change, so it is not correct.

Summary:
- The valid, calculated rate is ds/dt = 3 ft/sec, derived from the geometric relationship between the lamp, the man, and the shadow, via similar triangles and differentiation.
- The other numeric options either contradict the derived relation ds/dt = (1/2) dx/dt, or are incomplete (units only) and do not provide a numeric rate.

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) 单项选择题

类似问题

A conical tank has H=1.80 m and R=0.60 m. The net inflow rate is Qnet=2.50×10^−4 m^3/s. At the instant the liquid height is h=0.90 m, what is the instantaneous rate dh/dt?

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 .)

更多留学生实用工具

智能学习助手

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