Rates of Change (1 Viewer)

[DcM]

help..

i cant do some questions in fitzpatrick 3u change of rates

Ex 25 a)

23. A metal sphere is dissolving in acid. It remains spherical and the rate at which its volume decreases is proportional to its surface area. Show that the radius of the sphere decreases at a constant rate.

24. A circular cylinder of height 6 cm and base radius 4 cm sits on a table with its axis vertical. A point source of light moves vertically upwards at a speed of 3 cm/s above the central axis of the cylinder, thus casting a circular shadow on the table. Find the rate at which the radius of the shadow is decreasing when the light is at a distance 4 cm above the top of the cylinder.

thanks!

 

[:: ck ::]

wtf daniel i thought u were up to like 26d!!!

anyways i got the answers

23. A metal sphere is dissolving in acid. It remains spherical and the rate at which its volume decreases is proportional to its surface area. Show that the radius of the sphere decreases at a constant rate.

Let dV/dt = -k S [as it is proportional to surface area] , where k is a constant

Now V=4/3 pi r ^3
dV/dr = 4 pi r ^2 = S
dr / dV = 1 / ( 4 pi r^2 ) = 1/S

dr / dt = dr / dV * dV / dt
= 1/s * (-kS)
= -k

.'. rate which radius of sphere decreases is a constant rate

 

[:: ck ::]

24. A circular cylinder of height 6 cm and base radius 4 cm sits on a table with its axis vertical. A point source of light moves vertically upwards at a speed of 3 cm/s above the central axis of the cylinder, thus casting a circular shadow on the table. Find the rate at which the radius of the shadow is decreasing when the light is at a distance 4 cm above the top of the cylinder.

Let x = length of shadow, y = height of the light source

Using similar triangles,
x / (x + 6)
= 8 / (8 + y)
xy = 48
x = 48/y
dx/dy = -48/y²

Now dr / dt = (1/2) dx / dt = (1/2) dx / dy * dy / dt
dr / dt = (1/2) * (-3) * 3
= (-4.5 cm/s)

.'. the shadow is decreasing at a rate of 4.5 cm/s

 

[acmilan]

I thought this exercise was the hardest of chapter 25

 

[NiKz*~]

25d q17 is pretty hard too =P

 

[:: ck ::]

hahaha i rkn nick... addison spent like 2 days doing q17 in class

crazy question >.<"