Calculus-Sec-3-4-Solution

3.4    Approximations and Related Rates           by SGLee - HSKim, YJLim

1-3. Use differential to approximate the following quantities.

1.

Let f(x)=(x+25)^(1/2). Then f'(x)=1/(2sqrt{x+25}).

Thus dy=f'(x)dx.

Let x=0 and dx=2.

sqrt(27) = f(0) + f'(0) 2

2.  (and )

In order to find an approximate of , we define a function

Then

Now we use differentials to approximate. An approximate of  can be found by using

. Set  and  at

. Therefore .

3.

In order to find an approximation of , we define a function . Then  .

Now we use differentials to approximate. Set x=0  and  at

`Find an approximation of  by using . (dy=f'(0) 0.05)`

There are Sage built in command  .n()  for finding such approximations.

4.  The height of a circular cone is the same as the radius of its circular bottom.

The height and radius were measured and found to be 4cm with a possible error in measurement of at most 0.04cm.

What is the relative error in using this value to compute the volume?

(Because )

5. Find an approximation of the difference between the surface areas of two spheres whose radii are 5cm and 5.05cm, respectively.

. Thus . Since we have .

6. The period of a pendulum is given by the formula , where L is the length of the pendulum measured in meters

and  is the gravitational constant. If the length of the pendulum is measured to be 3m with a possible error in mea-surement 1cm.

What is the approximate percentage error in calculating the period ?

,   ,

.

The approximate percentage error = (relative error)  100% = .

7. A ladder 10 meters long is leaning against a wall. If the foot of the ladder is being pulled away from the wall at 4m/s,

how fast is the top of the ladder sliding down the wall when the foot of the ladder is 8 meter away from the wall?

Let  be the distance between the foot of the ladder to the wall

and  be the distance between the top of the ladder to the bottom.

Then from the Pythagorean Theorem,

.

Since  ,  , so  .

Therefore, the speed of the ladder sliding down from the top is .

8. A ladder 10 meters long is leaning against a wall. If the top of the ladder is sliding down the wall at 3m/s,

how fast is the foot of the ladder being pulled away from the wall when the foot of the ladder is 6 meter away from the wall?

Since .

9.  A ladder 10m long is leaning against a wall. If the top of the ladder is sliding down the wall at 1m/s,

how fast is the angle between the top of the ladder and the wall changing when the foot of the ladder is 5m away from the wall?

when .  ,    .

Therefore, when the foot of ladder is 5m away, angle at the top of the ladder changes is .

10. Two cars start moving from the same point. One travels south at km/hr and the other travels west at km/hr.

How fast is the distance changing between the two cars?

Distance of travel south is , distance of travel west is , and the distance between the two travelers is  at time

.

11. Water is being pumped at a rate of 20 liters per minute into a tank shaped like a frustrum of a right circular cone.

The tank has an altitude of 8 meters and lower and upper radii of 2 and 4 meters, respectively.

How fast is the water level rising when the depth of the water is 3 meters?

.  Let .

If

Therefore the water level rises  when the depth of the water is 3 meters.

12. Water is being pumped at a rate of 20 liters per minute into a tank shaped like a hemisphere.

The tank has a radius of 8 meters. How fast is the water level rising when the depth of the water is 3 meters?

.

13. A snowball melts at a rate proportional to its surface area. Does the radius shrink at a constant rate?

If it melts to 1/4 its original volume in one hour, how long does it take to melt completely?

The rate of melting is described by . Since the snowball melts at a rate proportional to its surface area.

Notice that  so  is constant. Let the volume of first time be ,

after one hour,  .

The snowball melts completely at time ().