Question

A 5 ft long man walks away from the foot of a 12(½) ft high lamp post at the rate of 3 mph. What will be the rate at which the shadow increases?

(a) 0mph

(b) 1mph

(c) 2mph

(d) 3mph

I got this question at a job interview.

My question is from Application of Derivative in chapter Application of Derivatives of Mathematics – Class 12

Answer 1

The correct option is (c) 2mph

Explanation: Let, AB be the lamp-post whose foot is A, and B is the source of light, and given (AB)’ = 12(½) ft.

Let MN denote the position of the man at time t where (MN)’ = 5ft.

Join BN and produce it to meet AM(produced) at P.

Then the length of man’s shadow= (MP)’

Assume, (AM)’ = x and (MP)’ = y. Then,

(PA)’ = (AM)’ + (MP)’ = x + y

And dx/dt = velocity of the man = 3

Clearly, triangles APB and MPN are similar.

Thus, (PM)’/(MN)’ = (PA)’/(AB)’

Or, y/5 = (x + y)/12(½)

Or, (25/2)y = 5x + 5y

Or, 3y = 2x

Or, y = (2/3)x

Thus, dy/dt = (2/3)(dx/dt)

As, dx/dt = 2,

= 2/3*3 = 2mph