A rectangular plate is expanding. Its length x is increasing at the rate 1 cm/sec and its width y is decreasing at the rate 0.5 cm/sec. At the moment when x=4 and y=3, find the rate of change of (1) its area (2) its perimeter (3) its diagonal.
Answers
given, length of rectangular plate is increasing at the rate, dx/dt = 1 cm/s
width of plate is decreasing at the rate ,dy/dt = -0.5cm/s [ negative sign shows decreasing rate ]
(1) area of rectangle, A = x × y
differentiating both sides with respect to time,
⇒dA/dt = d(xy)/dt = x(dy/dt) + y(dx/dt)
at x = 4 and y = 3
⇒ rate of change in area of plate, dA/dt = 4 × (-0.5) + 3 × (1) = -2 + 3 = 1 cm²/s
(2) perimeter of rectangle , P = 2(x + y)
differentiating both sides with respect to time,
⇒dP/dt = 2(dx/dt + dy/dt)
rate of change in perimeter of plate, dP/dt = 2(1 - 0.5) = 1 cm/s
(3) diagonal of rectangle, D = √(x² + y²)
differentiating both sides with respect to time,
⇒dD/dt = 1/2√(x² + y²) × {2x (dx/dt) + 2y(dy/dt)}
⇒dD/dt = {x (dx/dt) + y(dy/dt)}/√(x² + y²)
at x = 4 and y = 3
rate of change in diagonal of plate, dD/dt = {4 × 1 + 3 × (-0.5)}/√(4² + 3²}
= {4 - 1.5}/5 = 0.5 cm/s