Page 67 - Mathematics Class - XII
P. 67
Viva-Voce
1. Explain rate of change of quantity.
d
Ans. For the function, y = f (x), (f (x)) represents the rate of change of y with respect to x.
dx
Thus for example if ‘s’ represents the distance and ‘t’ the time, then ds represents the rate of change of
distance with respect to time. dt
2. What is the area of the rectangle after 5 seconds if length is decreasing at 2 cm/sec and breadth is increasing
at 2 cm/sec?
Ans. Area of rectangle after 5 seconds = (a – 10) (b + 10) cm , where a and b are length and breadth of rectangle
2
respectively.
3. If length a of a rectangle is decreasing at the rate of 4 cm/sec and breadth b of rectangle is increasing at
the rate of 2 cm/sec, what is the rate of change of area if a = 4 and b = 5.
Ans. Let A be the area of rectangle,
Then, A = a × b
dA db da
a b = a (2) + (– 4) b = (4 × 2) + (– 4 × 5) = 8 – 20 = – 12 cm /sec,
2
dt dt dt
Hence the area of the rectangle decreases at the rate of 12 cm /sec.
2
4. Find the rate of change of the volume of sphere with respect to its radius r.
4 dV 4
r
Ans. Volume of sphere, V = pr 3 r 3 2 4 .
2
3 dr 3
5. What is the value of A′(t) at maximum or minimum area where A(t) is area of a rectangle?
Ans. A′(t) = 0, when area is maximum or minimum.
MCQs
1. The rate of change of the area of a circle with respect to its radius r at r = 6 cm is
a) 10 p b) 12 p c) 8 p d) 11 p
2. The total revenue in Rupees received from the sale of x units of a product is given by R (x) = 3 x + 36x + 5.
2
The marginal revenue, when x = 15 is
a) 116 b) 96 c) 90 d) 126
3. What will be the value of rate of change of periphery of a circle with respect to their radius when r = 5 cm
a) 5p cm /sec b) 2p cm /sec c) 25p cm /sec d) 10p cm /s
2
2
2
2
4. The sides of the equilateral triangle are going to be changed at the rate of 2 cm/sec. The rate, at which the
area of triangle is changed when its side is 10 cm, will be
a) 3 cm /sec b) 10 cm /sec c) 10 3 cm /sec d) 10 3 cm /sec
2
2
2
2
3
Answers: 1. b) 12 p 2. d) 126 3. d) 10p cm /s 4. c) 10 3 cm /sec
2
2
65
