A rectangle has an area of 42 ft2. Suppose that the width and length of the rectangle are changing, while the area stays fixed.

If the length is increasing at a rate of 2 ft/s, what is the rate of change of the width with respect to time when the length is 6 ft?

Question

A rectangle has an area of 42 ft2. Suppose that the width and length of the rectangle are changing, while the area stays fixed.

If the length is increasing at a rate of 2 ft/s, what is the rate of change of the width with respect to time when the length is 6 ft?

BlackTom AI · Accepted Answer

We start from the given area constraint: A = L × W = 42, which implies W = 42 / L when the area is fixed.

Next, differentiate with respect to time t: dA/dt = 0 = L(dW/dt) + W(dL/dt) since A is constant. Solve for dW/dt: dW/dt = -[W(dL/dt)] / L.

But W = 42 / L, so dW/dt = -(42/L) (dL/dt) / L = -(42/L^2) (dL/dt).

Given L = 6 ft and dL/dt = 2 ft/s, substitute: dW/dt = -(42/6^2) × 2 = -(42/36) × 2 = -(7/6) × 2 = -14/6 = -7/3 ft/s.

Now evaluate each option:

Option 1: "The width is decreasing at a rate of 7 3 ft/s." This option expresses a decreasing width with magnitude 7/3 ft/s, matching the computed value of -7/3 ft/s. The negative sign indicates a decrease, so the direction is correct, and the magnitude aligns with the calculation.

Option 2: "The width is increasing at a rate of 7 3 ft/s." This claims the width grows while the true rate is negative; the sign is incorrect, so this option misrepresents the direction of change.

Option 3: "The width is decreasing at a rate of 1 2 ft/s." Here the magnitude is 1/2 ft/s, which does not match the calculated 7/3 ft/s, making this incorrect.

Option 4: "The width is increasing at a rate of 1 2 ft/s." This states both the wrong direction and an incorrect magnitude, so it is false.

Option 5: "The width is not changing because the area is fixed." While the area being fixed constrains the relationship between L and W, it does not imply dW/dt = 0 on its own; if L changes with time, W must adjust to keep the product constant, so this statement is incorrect.

In summary, the analysis shows that the first option accurately describes the situation, with the width decreasing at 7/3 ft/s when L is 6 ft and dL/dt is 2 ft/s; the other options include incorrect direction, incorrect magnitude, or misunderstand the impact of a changing length on the width under a constant area.

A rectangle has an area of 42 ft2. Suppose that the width and length of the rectangle are changing, while the area stays fixed. If the length is increasing at a rate of 2 ft/s, what is the rate of change of the width with respect to time when the length is 6 ft?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?

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 .)

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