A kite is moving horizontally at a height of 151.5 metres. If the speed of the kite is 10 m/sec, how fast is the string being let out; when the kite is 250 m away from the boy who is flying the kite? The height of the boy is 1.5 m.

8 m/sec

The correct answer is Option (3) → 8 m/sec

Let AB be the boy and P be the position of the kite at any time $t$. Let $BM = x$ metres and $l$ be the length of the string at that time, then

$MP = 151.5 m, AB = 1.5 m$

$⇒NP = MP - MN = MP - AB = (151.5-1.5) m$

$= 150 m$.

From figure, $l^2 = x^2 + 150^2$.

Differentiating w.r.t. t, we get

$2l.\frac{dl}{dt}=2x\frac{dx}{dt}$ but $\frac{dx}{dt}=10\,m/sec$  (given)

$⇒\frac{dl}{dt}=\left(\frac{x}{l}×10\right)\,m/sec$

Now, when $l = 250\, m, 250^2 = x^2 + 150^2⇒ x^2 = 40000⇒ x=200\, m$

When $l=250\,m,\frac{dl}{dt}=\left(\frac{200}{250}×10\right)m/sec=8\,m/sec$

Hence, the string is being let out at the speed of 8 m/sec when the kite is 250 m away from the boy.