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?Single choice

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