Section 3.9 Related Rates 23. Use the fact that the distance(in meters) a dropped stone falls after

Section 3.9 Related Rates

23. Use the fact that the distance(in meters) a dropped stone falls after t seconds is d = 4.9t². A woman stands near the edge of a cliff and drops a stone over the edge. Exactly one second later she drops another stone. One second after that, how fast is the distance between the two stones changing?

Solution:

270 ¤ CHAPTER 3 DIFFERENTIATION RULES

22. Given 

 = −1 ms, find

 when  = 8 m. ²= ²+ 1 ⇒ 2

 = 2

 ⇒



 = 





 = −

. When  = 8,  =√65, so

 = −

√65

8 . Thus, the boat approaches the dock at

√65

8 ≈ 101 ms.

23. Let  be the distance (in meters) the first dropped stone has traveled, and let  be the distance (in meters) the stone dropped one second later has traveled. Let  be the time (in seconds) since the woman drops the second stone. Using  = 49², we have  = 49( + 1)²and  = 49². Let  be the distance between the stones. Then  =  −  and we have



 =

 −

 ⇒ 

 = 98( + 1) − 98 = 98 ms.

24. Given: Two men 10 m apart each drop a stone, the second one, one minute after the first. Let  be the distance (in meters) the first dropped stone has traveled, and let  be the distance (in meters) the second stone has traveled. Let  be the time (in seconds) since the man drops the second stone. Using  = 49², we have

 = 49( + 1)²and  = 49². Let  be the vertical distance between the stones. Then  =  −  ⇒



 =

 −

 ⇒ 

 = 98( + 1) − 98 = 98 ms.

By the Pythagorean Theorem, ² = 10²+ ². Differentiating with respect to , we obtain 2

 = 2

 ⇒ 

 =  ()

 . One second after the second stone is dropped,  = 1, so

 =  −  = 49(1 + 1)²− 49(1)²= 147 m, and  =

10²+ (147)²=√

31609, so

 =147 (98)

√31609  810 ms.

25. If  = the rate at which water is pumped in, then

 =  − 10,000, where

 =¹₃²is the volume at time . By similar triangles, 2= 

6 ⇒  =1 3 ⇒

 =¹₃₁

32

 =₂₇^³ ⇒ 

 =  9²

. When  = 200 cm,



 = 20 cmmin, so  − 10,000 = 

9(200)²(20) ⇒  = 10,000 +800,000

9  ≈ 289,253 cm³min.

26. The distance  of the particle to the origin is given by  =

²+ ², so ²= ²+ [2 sin(2)]² ⇒ 2

 = 2

 + 4 · 2 sin  2

cos  2

· 2



 ⇒ 

 = 

 + 2 sin  2

cos  2 

. When

( ) =

1 3 1

 ,  =

1 3

2

+ 1²=

10 9 = 1

3

√10, so1 3

√10

 = 1 3

√10 + 2 sin  6 cos 

6·√ 10 ⇒

1 3



 =1 3+ 2

1 2

1 2

√3



⇒ 

 = 1 +3√ 3  2 cms.

° 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.c

29. Gravel is being dumped from a conveyor belt at a rate of 30 ft³/min, and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10 ft high?

250 Chapter 3 Differentiation Rules

the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10 ft high?

30. A kite 100 ft above the ground moves horizontally at a speed of 8 ftys. At what rate is the angle between the string and the horizontal decreasing when 200 ft of string has been let out?

31. The sides of an equilateral triangle are increasing at a rate of 10 cmymin. At what rate is the area of the triangle increasing when the sides are 30 cm long?

32. How fast is the angle between the ladder and the ground changing in Example 2 when the bottom of the ladder is 6 ft from the wall?

33. The top of a ladder slides down a vertical wall at a rate of 0.15 mys. At the moment when the bottom of the ladder is 3 m from the wall, it slides away from the wall at a rate of 0.2 mys. How long is the ladder?

34. According to the model we used to solve Example 2, what happens as the top of the ladder approaches the ground? Is the model appropriate for small values of y?

35. If the minute hand of a clock has length r (in centimeters), find the rate at which it sweeps out area as a function of r.

36. A faucet is filling a hemispherical basin of diameter 60 cm with water at a rate of 2 Lymin. Find the rate at which the water is rising in the basin when it is half full. [Use the following facts: 1 L is 1000 cm³. The volume of the portion of a sphere with radius r from the bottom to a height h is V − 

(

rh²2¹₃h³

)

, as we will show in Chapter 6.]

37. Boyle’s Law states that when a sample of gas is compressed at a constant temperature, the pressure P and volume V satisfy the equation PV − C, where C is a constant. Suppose that at a certain instant the volume is 600 cm³, the pressure is 150 kPa, and the pressure is increasing at a rate of 20 kPaymin. At what rate is the volume decreasing at this instant?

38. When air expands adiabatically (without gaining or losing heat), its pressure P and volume V are related by the equation PV^1.4− C, where C is a constant. Suppose that at a certain instant the volume is 400 cm³ and the pressure is 80 kPa and is decreasing at a rate of 10 kPaymin. At what rate is the volume increasing at this instant?

; 21. The altitude of a triangle is increasing at a rate of 1 cmymin

while the area of the triangle is increasing at a rate of 2 cm²ymin. At what rate is the base of the triangle changing when the altitude is 10 cm and the area is 100 cm²?

22. A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled in at a rate of 1 mys, how fast is the boat approaching the dock when it is 8 m from the dock?

23. At noon, ship A is 100 km west of ship B. Ship A is sailing south at 35 kmyh and ship B is sailing north at 25 kmyh.

How fast is the distance between the ships changing at 4:00 pm?

24. A particle moves along the curve y − 2 sinsxy2d. As the particle passes through the point

(

¹₃, 1

)

, its x-coordinate increases at a rate of s10 cmys. How fast is the distance from the particle to the origin changing at this instant?

25. Water is leaking out of an inverted conical tank at a rate of 10,000 cm³ymin at the same time that water is being pumped into the tank at a constant rate. The tank has height 6 m and the diameter at the top is 4 m. If the water level is rising at a rate of 20 cmymin when the height of the water is 2 m, find the rate at which water is being pumped into the tank.

26. A trough is 10 ft long and its ends have the shape of isos- celes triangles that are 3 ft across at the top and have a height of 1 ft. If the trough is being filled with water at a rate of 12 ft³ymin, how fast is the water level rising when the water is 6 inches deep?

27. A water trough is 10 m long and a cross-section has the shape of an isosceles trapezoid that is 30 cm wide at the bottom, 80 cm wide at the top, and has height 50 cm. If the trough is being filled with water at the rate of 0.2 m³ymin, how fast is the water level rising when the water is 30 cm deep?

28. A swimming pool is 20 ft wide, 40 ft long, 3 ft deep at the shallow end, and 9 ft deep at its deepest point. A cross- section is shown in the figure. If the pool is being filled at a rate of 0.8 ft³ymin, how fast is the water level rising when the depth at the deepest point is 5 ft?

3 6

12 16 6

6

29. Gravel is being dumped from a conveyor belt at a rate of 30 ft³ymin, and its coarseness is such that it forms a pile in

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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

Solution:

SECTION 3.9 RELATED RATES ¤ 251 25.If  = the rate at which water is pumped in, then

 =  − 10,000, where

 =¹₃²is the volume at time . By similar triangles, 2 =

6 ⇒  =1 3 ⇒

 =¹₃₁

32

 =₂₇^³ ⇒ 

 = 9² 

. When  = 200 cm,



 = 20 cmmin, so  − 10,000 = 

9(200)²(20) ⇒  = 10,000 +800,000

9  ≈ 289,253 cm³min.

26.By similar triangles,100 50 = 

, so  = 2. The trough has volume

 =¹₂(6) = 3(2) = 6² ⇒ 12 =

 = 12

 ⇒ 

 = 1 10. When  = 03,

 = 1 10 · 03= 1

3mmin.

27. The figure is labeled in meters. The area  of a trapezoid is

1

2(base1+base2)(height), and the volume  of the 10-meter-long trough is 10.

Thus, the volume of the trapezoid with height  is  = (10)¹₂[03 + (03 + 2)].

By similar triangles, 

 = 025 05 = 1

2, so 2 =  ⇒  = 5(06 + ) = 3 + 5². Now

 = 





 ⇒ 02 = (3 + 10)

 ⇒ 

 = 02

3 + 10. When  = 03,



 = 02

3 + 10(03) =02

6 mmin = 1

30mmin or10

3 cmmin.

28.The figure is drawn without the top 1 meter.

 =¹₂( + 3)(6) = 3( + 3)and, from similar triangles,



= 15 2 and

 =4

2, so  =  + 3 +  =3

4 + 3 + 2 = 3 +11

4 . ^{1.5 3 4}

2

Thus,  = 3

 6 +11

4



 = 18 +33²

4 and so 01 = 

 =

 18 +33

2



. When  = 1,



 = 01

18 + 1(332) = 1

345≈ 00029 mmin.

29.We are given that

 = 3m³min.  = ¹₃² = ¹₃

 2

2

 =³

12 ⇒



 =





 ⇒ 3 =² 4



 ⇒ 

 = 12

². When  = 3 m,



 = 12 3² = 4

3≈ 042 mmin.

° 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.c

30. A swimming pool is 5m wide, 10m long, 1m deep at the shallow end, and 3m deep at its deepest point. A cross- section is shown in the figure. If the pool is being filled at a rate of 0.1m³/min, how fast is the water level rising when the depth at the deepest point is 1m?

252 CHAPTER 3 Differentiation Rules

17. Two cars start moving from the same point. One travels south at 30 km/h and the other travels west at 72 km/h. At what rate is the distance between the cars increasing two hours later?

18. A spotlight on the ground shines on a wall 12 m away. If a man 2 m tall walks from the spotlight toward the building at a speed of 1.6 m/s, how fast is the le時thof his shadow on the building decreasing when he is 4 m from the building?

19. A man starts walking north at 1.2 m/s from a point P. Five minutes later a woman starts walking south at 1.6 m/s from a point 200 m due east of P. At what rate are the people moving apart 15 min after the woman starts walking?

20. A baseball diamond is a square with side 18 m. A batter hits the ball and runs toward first base with a speed of 7.5 m/s.

(a) At what rate is his distance from second base decreasing when he is halfway to first base?

(b) At what rate is his distance from third base increasing at the same moment?

21. The altitude of a triangle is increasing at a rate of 1 cm/min while the area of the triangle is increasing at a rate of 2 cm²/min. At what rate is the base of the triangle changing when the altitude is 10 cm and the area is 100 cm²? 22. A boat is pulled into a dock by a rope attached to the bow of

the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled in at a rate of 1 m/s, how fast is the boat approaching the dock when it is 8 m from the dock?

23-24 Use the fact that the distance (in meters) a dropped stone falls after t seconds is d = 4.9t²

23. A woman stands near the ed^2:0e of a cliff and droos a stone - ^~. - _ . . . . _..- ^_.~t"'

over the edge. Exactly one second later she drops another stone. One second after that, how fast is the distance between the two stones changing?

24. Two men stand 10 m apart on level ground near the edge of a

cli 旺 Oneman drops a stone and one second later the other man drops a stone. One second after that, how fast is the dis- tance between the two stones changing?

25. Water is leaking out of an inverted conical tank at a rate of 10,000 cm³/min at the same time that water is being pumped into the tank at a constant rate. The tank has height 6 m and the diameter at the top is 4 m. If the water level is rising at a rate of 20 cm/min when the height of the water is 2 m，且fin the ra孔at紀ea瓜twhich wa剖te叮r i心sbeing pumped into the tank.

26. A particle moves along the curve y = 2 sin(1Tx/2). As the particle passes through且已point (t^, 1), its x-coordinate increases at a rate of

.J

^{10 cm}/s. How fast is the distance from the particle to the origin changing at this instant?

27. A water trough is 10m long and a cross-section has the shape of an isosceles trapezoid that is 30 cm wide at the bottom, 80 cm wide at the top, and has height 50 cm. If the trough is being自 lledwith water at the rate ofO.2 m'/min, how fast is the water level rising when the water is 30 cm deep?

28. A trough is 6 m long and its ends have the shape of isos- celes triangles that are 1 m across at the top and have a height of 50 cm. If the trough is being filled with water at a rate of 1.2 m

3 /

m in, how fast is the water level rising when the water is 30 centimeters deep?

29. Gravel is being dumped from a conveyor belt at a rate of 3 叮'/min，and its coarseness is s叫1that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 3 m high?

30. A swimming pool is 5 m wide, 10m long, 1 m deep at the shallow end, and 3 m deep at its deepest point. A cross-section is shown in the figure. If the pool is being自 lledat a rate of 0.1 m '/min, how fast is the water level rising when the depth at the deepest point is I m?

1.5 3 4

31. The sides of an equilateral triangle are increasing at a rate of 10 cm/min. At what rate is the area of the tria時 leincreasing when the sides are 30 cm long?

Solution:

1

SECTION 3.9 RELATED RATES ¤ 271

27. The figure is labeled in meters. The area  of a trapezoid is

1

2(base1+base2)(height), and the volume  of the 10­meter­long trough is 10.

Thus, the volume of the trapezoid with height  is  = (10)¹₂[03 + (03 + 2)].

By similar triangles, 

 = 025 05 = 1

2, so 2 =  ⇒  = 5(06 + ) = 3 + 5². Now

 = 





 ⇒ 02 = (3 + 10)

 ⇒ 

 = 02

3 + 10. When  = 03,



 = 02

3 + 10(03)= 02

6 mmin = 1

30mmin or10

3 cmmin.

28.

29.

30.

31. The area  of an equilateral triangle with side  is given by  =¹₄√ 3 ².



 =¹₄√ 3 · 2

 = ¹₄√

3 · 2(30)(10) = 150√

3 cm²min.

32.

° 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.c

By similar triangles, _0.5¹ = _h^b, sob = 2h. The trough has volume V = ¹₂bh(6) = 3(2h)h = 6h² ⇒ 1.2 = ^dVdt = 12h^dh_dt ⇒ ^dhdt = _10h¹ . Whenh = 0.3, ^dh_dt =

1

10·0.3 =¹₃ m/min.

We are given that^dV_dt = 3 m³/min.V = ¹₃πr²h = ¹₃π ^h₂
2

h = ^πh₁₂³ ⇒ ^dVdt = ^dV_dh^dh_dt ⇒ 3 = ^πh₄^2dh_dt ⇒ ^dh_dt =_πh¹²2. Whenh = 3 m, ^dh_dt =₃¹²2π= _3π⁴ ≈ 0.4246 m/min.

The figure is drawn without the top 1 meter.

V =¹₂(b + 3)h(5) =⁵₂(b + 3)h and, from similar triangles,^x_h= ^1.5₂ =

3

4and ^y_h=⁴₂ = 2, so b = x + 3 + y = ^3h₄ + 3 + 2h = 3 +^11h₄ . Thus, V =⁵₂ 6 +^11h₄

h = 15h+⁵⁵₈h²and so0.1 = ^dV_dt = 15 +⁵⁵₄h
_dh

dt. Whenh = 1, ^dh_dt = =₅₇₅² ≈ 0.00348 m/min.

We are givendx/dt = 2 m/s. cot θ = ₅₀^x ⇒ x = 50 cot θ ⇒ ^dxdt =−50 csc²θ^dθ_dt ⇒

dθ

dt =−^sin50^2θ · 2. When y = 100, sin θ = 100⁵⁰ = ¹₂ ⇒ ^dθdt =−^(1/2)50² · 2 =

−₁₀₀¹ rad/s. The angle is decreasing at a rate of₁₀₀¹ rad/s.

0.1

(¹⁵⁺⁵⁵4)

44. Two carts, A and B, are connected by a rope 12m long that passes over a pulley P .(See the figure.) The point Q is on the floor 4m directly beneath P and between the carts. Cart A is being pulled away from Q at a speed of 0.5m/s. How fast is cart B moving toward Q at the instant when cart A is 3m from Q?

32. A kite 50 rn above the ground moves horizontally at a speed of 2 m/ s. At what rate is the angle between the string and the horizontal decreasing when 100 m of string has been let out?

33. A car is traveling north on a straight road at 20 m/ s and a drone is flying east at 6 m/s at an elevation of 25 m. At one instant the drone passes directly over the car. How fast is the distance between the drone and the car changing 5 seconds later?

34. If the minute hand of a clock has length r (in centimeters), find the rate at which it sweeps out area as a function of r.

35. How fast is the angle between the ladder and the ground b

changing in Example 2 when the bottom of the ladder is 3 m from the wall?

36. According to the model we used to solve Example 2, what happens as the top of the ladder approaches the ground? Is the model appropriate for small values of y?

37. Boyle's Law states that when a sample of gas is compressed at a constant temperature, the pressure P and volume V sat- isfy the equation PV = C, where C is a constant. Suppose that at a certain instant the volume is 600 cm³, the pressure is 150 kPa^, and the pressure is increasing at a rate of 20 kPa/min. At what rate is the volume decreasi 時 atthis instant?

回 到.

A faucet

is 創 ling

a hemispherical basin of diameter 60 cm with water at a rate of 2 L/min. Find the rate at which the water is rising in the basin when it is half full. [Use the fol- lowing facts: 1 L is 1000 cm³. The volume of the portion of a sphere with radius r from the bottom to a height h is

V= 竹 (rh

² - th³⁾, as we will show in Chapter 6.]

39. If two resistors with resistances RI and R2 are connected in parallel, as shown in the figure, then the total resistance R, measured in ohms (D), is given by

1 1 1

一=一一+一­

R RI R2

If R ^I and R2 are increasing at rates of 0.3 D/s and 0.2 D/s, respectively, how fast is R changing when RI = 80 D and R2 = 100 D?

40. When air expands adiabatically (without gaining or losing heat), its pressure P and volume V are related by the equa- tion PV^I4 = C, where C is a constant. Suppose that at a certain instant the volume is 400 cm³ and the pressure is 80 kPa and is decreasing at a rate of 10 kPa/min. At what rate is the volume increasing at this instant?

SECTION 3.9 Related Rates 253

41. Two straight roads diverge from an intersection at an angle of 60^0• Two cars leave the intersection at the same time, the first traveli 時 downone road at 60 km/h and the second traveling down the other road at 100 km/h. How fast is the distance between the cars changing after half an hour? [Hint:

Use the Law of Cosines (Formula 21 in Appendix D).]

42. Brain weight B as a function of body weight W in 且 shhas been modeled by the power function B = 0.007W^2月，where B and W are measured in grams. A model for body weight as a function of body length L (measured in centimeters) is W = 0.12L²⁵³. lf, over 10 million years, the average length of a certain species of fish evolved from 15 cm to 20 cm at a constant rate, how fast was this species' brain growing when its average length was 18 cm?

43. Two sides of a triangle have lengths 12 m and 15 m. The angle between them is increasing at a rate of 2 o/min. How fast is the length of the third side increasing when the angle between the sides of 白 xedlength is 60⁰? [Hint: Use the Law of Cosines (Formula 2] in Appendix D).]

44. Two carts, A and B, are connected by a rope 12 m long that passes over a pulley P. (See the figure.) The point Q is on the ftoor 4 m directly beneath P and between the carts. Cm1 A is being pulled away from Q at a speed of 0.5 m / s. How fast is cart B moving toward Q at the instant when cart A is 3 m from Q?

A B

Q

45. A television camera is positioned 1200 m from the base of a rocket launching pad. The angle of elevation of the camera has to change at the correct rate in order to keep the rocket in sight. Also, the mechanism for focusing the camera has to take into account the increasing distance from the camera to the rising rocket. Let's assume the rocket rises vertically and its speed is 200 m/s when it has risen 900 m.

(a) How fast is the distance from the television camera to the rocket changing at that moment?

(b) lf the television camera is always kept aimed at the rocket, how fast is the camera's angle of elevation changing at that same moment?

46. A lighthouse is located on a small island 3 km away from the nearest point P on a straight shoreline and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from P?

47. A plane fties horizontally at an altitude of 5 km and passes directly over a tracking telescope on the ground. When the

Solution:

274 ¤ CHAPTER 3 DIFFERENTIATION RULES 44.

45.

46.We are given that

 = 4(2) = 8radmin.  = 3 tan  ⇒



 = 3 sec²

. When  = 1, tan  =¹₃, so sec² = 1 +₁

3

2

=¹⁰₉

and 

 = 3₁₀

9

(8) =⁸⁰₃ ≈ 838 kmmin.

47.cot  = 

5 ⇒ − csc²

 = 1 5



 ⇒ − csc

3

2

− 6



= 1 5



 ⇒



 =5

6

 2

√3

2

=10

9kmmin [≈ 130 mih]

48.We are given that

 =2 rad

2 min = radmin By the Pythagorean Theorem, when

 = 6,  = 8, so sin  =₁₀⁶ and cos  = ₁₀⁸. From the figure, sin  =  10 ⇒

 = 10 sin , so

 = 10 cos 

 = 10

 8 10



 = 8mmin.

° 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.c

UsingQ for the origin, we are given ^dx_dt = 0.5 m/s and need to find^dy_dt whenx = 5.

Using the Pythagorean Theorem twice, we have√

x²+ 4²+p

y²+ 4² = 12, the total length of the rope. Differentiating with respect tot, we get

√ x x²+4²

dx dt +√ ^y

y²+4² dy

dt = 0, so^dy_dt =−^x

√y²+4² y√

x²+4² dx dt. Now, whenx =−3, 12 =p

(−3)²+ 4²+p

y²+ 4²= 5 +p

y²+ 4²⇔p

y²+ 4²= 7, and y =√

7²− 4² =√ 33.

So whenx =−3, ^dydt =−^(√−3)(7)₃₃₍₅₎(−0.5) ≈ −0.37 m/s.

So cartB is moving towards Q at about 0.37 m/s.

(a) By the Pythagorean Theorem,1200²+ y²= l². Differentiating with respect to t, we obtain 2y^dy_dt = 2l^dl_dt. We know that^dy_dt = 200 m/s, so wheny = 900 m, l =√

1200²+ 900² =√2, 250, 000 = 1500 m and_dt^dl =^y_l^dy_dt = ₁₅₀₀⁹⁰⁰(200) = ⁶⁰⁰₅ = 120 m/s.

(b) Heretan θ = ₁₂₀₀^y ⇒ _dt^d(tan θ) = _dt^d ₁₂₀₀^y

⇒ sec²θ^dθ_dt = ₁₂₀₀^{1 dy}_dt ⇒ ^dθ_dt = ^cos₁₂₀₀^2θdy_dt. Wheny = 900 m, ^dy_dt = 200 m/s, l = 1500 and cos θ =¹²⁰⁰_l =¹²⁰⁰₁₅₀₀ =⁴₅, so^dθ_dt = ^(4/5)₁₂₀₀²(200) = 0.10 ˙6 rad/s.

− −

53. Suppose that the volume V of a rolling snowball increases so that dV /dt is proportional to the surface area of the snowball at time t. Show that the radius r increases at a constant rate, that is, dr/dt is constant.

Solution:

276 ¤ CHAPTER 3 DIFFERENTIATION RULES

one­twelfth of the circle, that is,^2₁₂ = ^₆ radians. We use () to find  at 1:00:  =

80 − 64 cos^6 =

80 − 32√ 3.

Substituting, we get 2

 = 64 sin 6



−11

6



⇒ 

 = 641 2

−^116

 2

80 − 32√

3 = − 88

3

80 − 32√

3≈ −186.

So at 1:00, the distance between the tips of the hands is decreasing at a rate of 186 mmh ≈ 0005 mms.

53.The volume of the snowball is given by  =4

3³, so

 =4

3 · 3²

 = 4² 

. Since the volume is proportional to the surface area , with  = 4², we also have

 =  · 4²for some constant . Equating the two expressions for 



gives 4²

 =  · 4² ⇒ 

 = , that is,  is constant.

3.10 Linear Approximations and Differentials

1.  () = ³− ²+ 3 ⇒ ⁰() = 3²− 2, so (−2) = −9 and ⁰(−2) = 16. Thus,

() =  (−2) + ⁰(−2)( − (−2)) = −9 + 16( + 2) = 16 + 23.

2.  () = ^3 ⇒ ⁰() = 3^3, so (0) = 1 and ⁰(0) = 3. Thus, () = (0) + ⁰(0)( − 0) = 1 + 3( − 0) = 3 + 1.

3.  () =√³

 ⇒ ⁰() = 1 3√³

², so (8) = 2 and ⁰(8) = 1 12. Thus,

() =  (8) + ⁰(8)( − 8) = 2 +12¹( − 8) = 12¹ +⁴₃. 4.  () = cos 2 ⇒ ⁰() = −2 sin 2, so 

6

= ¹₂ and ⁰ 6

= −√ 3. Thus,

() = _

6

+ ⁰_

6

 −^6

= ¹₂−√ 3

 −^6

= −√

3  + (√

3 )6 +¹₂.

5.  () =√

1 −  ⇒ ⁰() = −1 2√

1 − , so (0) = 1 and ⁰(0) = −¹2. Therefore,

√1 −  = () ≈ (0) + ⁰(0)( − 0) = 1 +

−¹2

( − 0) = 1 −¹2.

So√ 09 =√

1 − 01 ≈ 1 −¹2(01) = 095 and√

099 =√

1 − 001 ≈ 1 −¹2(001) = 0995.

6. () =√³

1 +  = (1 + )^13 ⇒ ⁰() =¹₃(1 + )^−23, so (0) = 1 and

⁰(0) = ¹₃. Therefore,√³

1 +  = () ≈ (0) + ⁰(0)( − 0) = 1 +¹3.

So√³

095 = ³

1 + (−005) ≈ 1 +¹3(−005) = 0983, and√³

11 = √³

1 + 01 ≈ 1 +¹3(01) = 103.

° 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.c

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