Question 7 of 82intermediate🔧 ApplyNumerical4 marks
The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When x = 8cm and y = 6cm, find the rates of change of
(a)
the perimeter, and
Answer
The perimeter is decreasing at the rate of 2 cm/min.
(b)
the area of the rectangle.
Answer
The area is increasing at the rate of 2 cm²/min.
Explanation
This question is similar to Example 4 in the textbook. The key insight is that since length is decreasing, dx/dt is negative (-5 cm/min), and since width is increasing, dy/dt is positive (+4 cm/min). For perimeter, we differentiate P = 2(x + y) with respect to time. For area, we use the product rule on A = x·y and substitute the given values of x and y along with their rates of change.