1) Water is running into a conical vessel, 15cm deep and 5cm in radius at the rate of 0.1cm^{3}/s. When the water is 6cm deep find at what rate

(i) water level is raising (ii) the water surface area is is increasing (iii) the wetted surface of the vessel is increasing (iv) slant height is increasing

2) A man 1.5m tall moves away from a pole 9.5m tall at a rate of 3m/s. Find the rate at which elevation of top of the pole is changing when the man is 6m away from the pole?

let ABC be the conical vessel. such that its radius OC = 5 cm and depth OA = 15 cm

let V be the volume of water in the vessel at any time t. let DE = r cm and AD = h cm.

therefore ..............(1)

now triangles ADE and AOC are similar triangles.

therefore

given: .............(3)

(i)

from eq(1) and (2):

now differentiating the equation wrt t :

....................(4)

(ii)

let water surface area be A.

differentiating wrt t :

.................(5)

(iii)

let the wetted surface of the vessel be S.

where be the slant height of water cone.

................(6)

differentiating wrt t :

(iv)

let the slant height be

differentiating wrt t:

hope this helps you.

cheers!!

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