Final answer:
The rate at which the area of the rectangle increases when the length is 20 cm and the width is 25 cm is 275 cm²/s. This rate is found by applying the product rule of differentiation, given the rates of change of both the length and width.
Step-by-step explanation:
The student is asking about the rate at which the area of a rectangle is increasing when the length and width are increasing at constant rates. This is an application of derivatives in Calculus, where we use rates of changes to find the derivative of a function. In this case, the area A of the rectangle is the function, with A = length × width. Using the given rates of change, where the length increases by 7 cm/s and the width increases by 5 cm/s, we can apply the product rule of differentiation to find the rate at which the area increases:
dA/dt = (d(length)/dt) × width + length × (d(width)/dt)
Plugging in the values when the length is 20 cm and the width is 25 cm, we get:
dA/dt = 7 cm/s × 25 cm + 20 cm × 5 cm/s
dA/dt = 175 cm²/s + 100 cm²/s
dA/dt = 275 cm²/s
Thus, the correct answer is B) 275cm²/s.