Final answer:
The area of the rectangle is increasing at a rate of 200 cm²/s when the length is 40 cm and the width is 20 cm.
Step-by-step explanation:
When the length of a rectangle is increasing at a rate of 6 cm/s and its width is increasing at a rate of 2 cm/s, the rate at which the area of the rectangle increases can be found using the product of the rates of increase for the length and width. This is derived from the formula for the area of a rectangle (A = length × width). Given that the length (L) is 40 cm and the width (W) is 20 cm, the area (A) is L × W (40 cm × 20 cm = 800 cm²).
To find the rate of change of the area (×A/×t), we need to differentiate the area with respect to time. ×A/×t = L' × W + L × W', where L' represents the rate of change of the length and W' represents the rate of change of the width. Substituting the given rates and dimensions, we have ×A/×t = 6 cm/s × 20 cm + 40 cm × 2 cm/s, which simplifies to 120 cm²/s + 80 cm²/s, yielding a total increase in area of 200 cm²/s.