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.
Answer & explanation
Correct answer: option 3
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.