Applications of Calculus (1 Viewer)

[FDownes]

I'm not quite sure how to approach this question, could someone help me out here? It asks;

A certaing brand of medicine tablet is in the shape of a sphere with diameter 5mm. Ther rate at which the pill dissolves is dr/dt = -k, where r is the radius of the sphere at time t hours, and k is a positive constant.

a) Show that r = 5/2 - kt

b) If the pill dissolves completely in 12 hours, find k.

 

[munchiecrunchie]

a) dr/dt = -k

therefore, r = integral (-k) dt
= -kt + C

since d = 5 when t = 0

ie when t = 0, r =5/2

therefore, C = 5/2

therefore r = -kt + 5/2

b) (this bit i'm not too sure about, somebody check this)

when t = 12, r = 0 (since dissolved)

*substitute into equation*

0 = 5/2 - 12k
12k = 5/2

therefore, k = 5/24

 

[FDownes]

Ooooh... I don't know why I didn't make the jump from diameter to radius. Thanks.

 

[FDownes]

Got another one;

Twenty-five wallabies are released on Wombat Island and the population is observed over the next six years. It is found that the rate of increase in the wallaby population is given by dP/dt = 12t - 3t², and the population itself is given by P = 25 + 6t² - t³. When does the population increase most rapidly?

 

[lolokay]

Got another one;

Twenty-five wallabies are released on Wombat Island and the population is observed over the next six years. It is found that the rate of increase in the wallaby population is given by dP/dt = 12t - 3t², and the population itself is given by P = 25 + 6t² - t³. When does the population increase most rapidly?

when t=2 ?
dt/dtt = 12 - 6t
12 - 6t = 0 at max/min rate of increase
t=2
when 12 - 6t > 0
t<2,
when 12 - 6t < 0
t>2
so it is the max increase

I don't really know calculus though

 

[munchiecrunchie]

Got another one;

Twenty-five wallabies are released on Wombat Island and the population is observed over the next six years. It is found that the rate of increase in the wallaby population is given by dP/dt = 12t - 3t², and the population itself is given by P = 25 + 6t² - t³. When does the population increase most rapidly?

this is probably a better way of setting the answer out:

population increases most rapidly at point of inflextion, ie, when d^2P/dt^2 = 0

d^2P/dt^2, ie the second derivative = 12 - 6t

12 - 6t = 0
6t = 12
therefore, t = 2

check for concavity change

when t = 3, 2nd derivative = -6 . . . . . . . < 0
when t =1, 2nd derivative = 6 . . . . . . . > 0

therefore concavity changes

therefore greatest rate of change when t = 2

 

[FDownes]

It's a bit late, but thanks for your help. Now I've got a new problem;

A ball is projected horizontally from a cliff of height 245m, and reaches the ground at a horizontal distance of 350m from the foot of the tower. Determine the intitial velocity V, and the velocity (direction and magnitude) on striking the ground (Take g = 10ms^-1).

All I can figure out is that it will take 7 seconds for the ball to hit the ground, but I'm stumped beyond that.

 

[lyounamu]

It's a bit late, but thanks for your help. Now I've got a new problem;

A ball is projected horizontally from a cliff of height 245m, and reaches the ground at a horizontal distance of 350m from the foot of the tower. Determine the intitial velocity V, and the velocity (direction and magnitude) on striking the ground (Take g = 10ms^-1).

All I can figure out is that it will take 7 seconds for the ball to hit the ground, but I'm stumped beyond that.

It's 7 seconds.

So x = Vt

When t = 7, x = 350
350 = 7V
V =50

 

[vds700]

It's 7 seconds.

So x = Vt

When t = 7, x = 350
350 = 7V
V =50

ok so Namu's done the first part, so ill do the 2nd part.

x. = 50
y. = -10t when t = 7, y. = -70

Draw up a right angled triangle withthe horizontal vcctor = 50 and the vertical vector = -70. The line joining these is the overall velocity.

Uving pythagoras' theorem,
v^2 = 2500 + 4900
v = 86.02 m/s
and to find the angle, tanA = 5/7, A = 54 degrees.

If you cant undersatnd what i have done, i can draw u a diagram and post it.

Hope this helps

 

[lyounamu]

ok so Namu's done the first part, so ill do the 2nd part.

x. = 50
y. = -10t when t = 7, y. = -70

Draw up a right angled triangle withthe horizontal vcctor = 50 and the vertical vector = -70. The line joining these is the overall velocity.

Uving pythagoras' theorem,
v^2 = 2500 + 4900
v = 86.02 m/s
and to find the angle, tanA = 5/7, A = 54 degrees.

If you cant undersatnd what i have done, i can draw u a diagram and post it.

Hope this helps

Didn't read the question properly. Didn't even know that there was a second part.

lol