Chapter 6 Application of Derivatives Ex 6.1 Solutions
Question - 11 : - A particle moves along thecurve
. Find the points on thecurve at which the y-coordinateis changing 8 times as fast as the x-coordinate.
Answer - 11 : -
The equation of the curve is given as:
The rate of change of the position of theparticle with respect to time (t) isgiven by,
When the y-coordinate of the particlechanges 8 times as fast as the
x-coordinatei.e.,
, we have:
When
When
Hence, the points requiredon the curve are (4, 11) and 
Question - 12 : - The radius of an air bubbleis increasing at the rate of 
cm/s. At what rate is the volume of the bubbleincreasing when the radius is 1 cm?
Answer - 12 : -
The air bubble is in the shape of a sphere.
Now, the volume of an air bubble (V) withradius (r) is given by,The rate of change of volume (V) withrespect to time (t) is given by,
It is given that
Therefore, when r = 1 cm,
Hence, the rate at which the volume of thebubble increases is 2π cm3/s.
Find the rate of change of its volumewith respect to x.
Answer - 13 : -
The volume of a sphere (V) with radius(r) is given by,
It is given that:
Diameter
Hence, the rate of change of volume withrespect to x is as
Question - 14 : - Sand is pouring from a pipe at the rate of12 cm3/s. The falling sand forms a cone on the ground in such away that the height of the cone is always one-sixth of the radius of the base.How fast is the height of the sand cone increasing when the height is 4 cm?
Answer - 14 : -
The volume of a cone (V) withradius (r) and height (h) is given by,
It is given that,
The rate of change of volume with respect totime (t) is given by,
[By chain rule]
It is also given that
Therefore, when h = 4 cm,we have:
Hence, when the height ofthe sand cone is 4 cm, its height is increasing at the rate of
Question - 15 : - The total cost C (x)in Rupees associated with the production of x units of an itemis given by
Find the marginal cost when 17 units are produced.
Answer - 15 : -
Marginal cost is the rate of change of total cost withrespect to output.
∴Marginal cost (MC)
When x = 17, MC = 0.021 (172) −0.006 (17) + 15
= 0.021(289) − 0.006(17) + 15
= 6.069 − 0.102 + 15
= 20.967
Hence, when 17 units are produced, the marginal cost isRs. 20.967.
The total revenue in Rupees received from thesale of x units of a product is given by
Find the marginal revenue when x =7.
Answer - 16 : -
Marginal revenue is the rate of change of total revenuewith respect to the number of units sold.
∴Marginal Revenue (MR) 
= 13(2x) + 26 = 26x + 26
When x = 7,
MR = 26(7) + 26 = 182 + 26 = 208
Hence, the required marginal revenue is Rs 208.
The rate of change of the area of a circlewith respect to its radius r at r = 6 cm is
(A) 10π (B) 12π (C) 8π (D) 11π
Answer - 17 : -
The area of a circle (A) withradius (r) is given by,
Therefore, the rate of change of the areawith respect to its radius r is
∴When r = 6 cm,
Hence, the required rate of change of thearea of a circle is 12π cm2/s.
The correct answer is B.
Question - 18 : - The total revenue in Rupees received fromthe sale of x units of a product is given by
. The marginal revenue, when 
is
(A) 116 (B) 96 (C) 90 (D) 126
Answer - 18 : -
Marginal revenue is the rate of change of total revenuewith respect to the number of units sold.
∴Marginal Revenue (MR)= 3(2x) + 36 = 6x + 36
∴When x = 15,
MR = 6(15) + 36 = 90 + 36 = 126
Hence, the required marginal revenue is Rs 126.
The correct answer is D.