CBSE Guess > Papers > Important Questions > Class XII > 2010 > Maths > Mathematics By Mr. M.P.Keshari

Application of derivatives

6.1. Rate of Change of Quantities.

Q.1. A point source of light along a straight road is at a height of ‘a’ metres. A boy ‘b’ metres in height is walking along the road. How fast is his shadow increasing if he is walking away from the light at the rate of c metres per minute?

Solution :

Fig.

Let lamp-post be AB and CD be the boy whose distance from lamp-post at any time t be x m, let CE = y m be its shadow. Then
dx/dt = c m/m.
As, ∆ BAE ~ ∆ DCE, AB/CD = AE/CE
=> a/b = (x + y)/y
=> ay = b(x + y)
=> (a – b) y = bx
=> (a – b)dy/dt = b dx/dt = bc
Therefore, dy/dt = bc/(a – b). [Ans.]

Q.2. The two equal sides of an isosceles triangle with fixed base b cm are decreasing at the rate of 3 cm/sec. How fast is the area decreasing when the two equal sides are equal to the base?

Solution :

Fig.

Let ABC be an isosceles triangle with AB = AC = a cm(say).
Given : da/dt = 3 cm/sec and BC = b.
AD is drawn perpendicular to BC => BD = DC = b/2.
Therefore, AD = √(a² – b²/4).
Let A = Area of ∆ ABC = 1/2.b√(a² – b²/4)
=> dA/dt = b/2.1/2(a² – b²/4)– 1/2.2a.da/dt
= (3ab/2). 1/√(a² – b²/4)
= (3b²/2).1/√(b² – b²/4) [when a = b]
=> dA/dt = (3b/2).(2/√3)
= (√3)b. [Ans.]

Paper By Mr. M.P.Keshari
Email Id : [email protected]
Ph No. : 09434150289