# Class 12 RD Sharma Solutions – Chapter 14 Differentials, Errors and Approximations – Exercise 14.1 | Set 1

Last Updated : 12 Dec, 2021

### Question 1: If y=sin x and x changes from Ï€/2 to 22/14, what is the approximate change in y?

Solution:

According to the given condition,

x = Ï€/2, and

x+â–³x = 22/14

â–³x = 22/14-x = 22/14 – Ï€/2

As, y = sin x

= cos x

= cos (Ï€/2) = 0

â–³y =  â–³x

â–³y = 0 â–³x

â–³y = 0 (22/14 – Ï€/2)

â–³y = 0

Hence, there will be no change in y.

### Question 2: The radius of a sphere shrinks from 10 to 9.8 cm. Find approximately the decrease in its volume?

Solution:

According to the given condition,

x = 10, and

Let â–³x be the error in the radius and â–³y be the error in the volume

x+â–³x = 9.8

â–³x = 9.8-x = 9.8-10 = -0.2

As, Volume of sphere =

= 4Ï€x2

= 4Ï€(10)2 = 400 Ï€

â–³y =  â–³x

â–³y = (400 Ï€) (-0.2)

â–³y = -80 Ï€

Hence, approximate decrease in its volume will be -80 Ï€ cm3

### Question 3: The circular metal plate expands under heating so that its radius increases by k%. Find the approximate increase in the area of the plate, if the radius of the plate before heating is 10 cm.

Solution:

According to the given condition,

x = 10, and

Let â–³x be the error in the radius and â–³y be the error in the surface area

â–³x/x Ã— 100 = k

â–³x = (k Ã— 10)/100 = k/10

As, Area of circular metal = Ï€x2

= Ï€(2x) = 2Ï€x

= 2Ï€(10) = 20 Ï€

â–³y =  â–³x

â–³y = (20 Ï€) (k/10)

â–³y = 2kÏ€

Hence, approximate increase in the area of the plate is 2kÏ€ cm2

### Question 4: Find the percentage error in calculating the surface area of a cubical box if an error of 1% is made in measuring the lengths of edges of the cube.

Solution:

According to the given condition,

Let â–³x be the error in the length and â–³y be the error in the surface area

Let’s take length as x

â–³x/x Ã— 100 = 1

â–³x = x/100

x+â–³x = x+(x/100)

As, surface area of the cube = 6x2

= 6(2x) = 12x

â–³y =  â–³x

â–³y = (12x) (x/100)

â–³y = 0.12 x2

So, â–³y/y = 0.12 x2/6 x2 = 0.02

Percentage change in y = â–³y/y Ã— 100 = 0.02 Ã— 100 = 2

Hence, the percentage error in calculating the surface area of a cubical box is 2%

### Question 5: If there is an error of 0.1% in the measurement of the radius of a sphere, find approximately the percentage error in the calculation of the volume of the sphere.

Solution:

According to the given condition,

As, Volume of sphere =

Let â–³x be the error in the radius and â–³y be the error in the volume

â–³x/x Ã— 100 = 0.1

â–³x/x = 1/1000

As, y =

= 4Ï€x2

dy = 4Ï€x2 dx

â–³y = (4Ï€x2) â–³x

Change in volume,

â–³y/y =

â–³y/y =

â–³y/y =  = 3(0.001) = 0.003

Percentage change in y = â–³y/y Ã— 100 = 0.003 Ã— 100 = 0.3

Hence, approximately the percentage error in the calculation of the volume of the sphere is 0.3%

### Question 6: The pressure p and the volume v of a gas are connected by the relation pv1.4 = constant. Find the percentage error in p corresponding to a decrease of 1/2% in v.

Solution:

According to the given condition,

= – 1/2%

pv1.4 = constant = k(say)

Taking log on both sides, we get

log(pv1.4) = log (k)

log(p)+log(v1.4) = log k

log(p) + 1.4 log(v) = log k

Differentiating wrt v, we get

Percentage change in p = â–³p/p Ã— 100 =  Ã— 100 = -1.4

= -1.4

= 0.7 %

Hence, percentage error in p is 0.7%.

### Question 7: The height of a cone increases by k%, its semi-vertical angle remaining the same. What is the approximate percentage increase?

Solution:

According to the given condition,

Let h be the height, y be the surface area. V be the volume, l be the slant height and r be the radius of the cone.

Let â–³h be the change in the height. â–³r be the change in the radius of base and â–³l be the change in slant height.

Semi-vertical angle remaining the same.

â–³h/h = â–³r/r = â–³l/l

and,

â–³h/h Ã— 100 = k

â–³h/h Ã— 100 = â–³r/r Ã— 100 = â–³l/l Ã— 100 = k

### (i) in total surface area, and

Solution:

Total surface area of the cone

y = Ï€rl + Ï€r2

Differentiating both the sides wrt r, we get

= Ï€l + Ï€r  + 2Ï€r

= Ï€l + Ï€r  + 2Ï€r

= Ï€l + Ï€l + 2Ï€r

= 2Ï€l + 2Ï€r = 2Ï€(r+l)

â–³y =  â–³r

â–³y = (2Ï€(r+l))

Percentage change in y = â–³y/y Ã— 100 =  Ã— 100

= 2k %

Hence, percentage increase in total surface area of cone 2k%.

### (ii) in the volume assuming that k is small?

Solution:

Volume of cone (y) =

Differentiating both the sides wrt h, we get

(r2 + h(2r )

(r2 + h(2r )

(r2 + 2r2)

= Ï€r2

â–³y =  â–³h

â–³y = (Ï€r2

Percentage change in y = â–³y/y Ã— 100 =  Ã— 100

= 3k %

Hence, percentage increase in the volume of cone 3k%.

### Question 8: Show that the relative error in computing the volume of a sphere, due to an error in measuring the radius, is approximately equal to the three times the relative error in the radius.

Solution:

According to the given condition,

Let â–³x be the error in the radius and â–³y be the error in volume.

Volume of cone (y) =

Differentiating both the sides wrt x, we get

(3x2)

= 4Ï€x2

â–³y =  â–³x

â–³y = (4Ï€x2) (â–³x)

â–³y/y =

â–³y/y =

Hence proved!!

### (i)

Solution:

Considering the function as

y = f(x) =

Taking x = 25, and

x+â–³x = 25.02

â–³x = 25.02-25 = 0.2

â–³y = dy =  dx

â–³y =  â–³x

â–³y =  (0.02) = 0.002

Hence,  = y+â–³y = 5 + 0.002 = 5.002

### (ii)

Solution:

Considering the function as

y = f(x) =

Taking x = 0.008, and

x+â–³x = 0.009

â–³x = 0.009-0.008 = 0.001

â–³y = dy =  dx

â–³y =  â–³x

â–³y =  (0.001) =  = 0.008333

Hence,  = y+â–³y = 0.2 + 0.008333 = 0.208333

### (iii)

Solution:

Considering the function as

y = f(x) =

Taking x = 0.008, and

x+â–³x = 0.007

â–³x = 0.007-0.008 = -0.001

= 0.2

â–³y = dy =  dx

â–³y =  â–³x

â–³y =  (-0.001) =  = -0.008333

Hence,  = y+â–³y = 0.2 + (-0.008333) = 0.191667

### (iv)

Solution:

Considering the function as

y = f(x) =

Taking x = 400, and

x+â–³x = 401

â–³x = 401-400 = 1

â–³y = dy =  dx

â–³y =  â–³x

â–³y =  (1) = 0.025

Hence,  = y+â–³y = 20 + 0.025 = 20.025

### (v)

Solution:

Considering the function as

y = f(x) =

Taking x = 16, and

x+â–³x = 15

â–³x = 15-16 = -1

â–³y = dy =  dx

â–³y =  â–³x

â–³y =  (-1) =  = -0.03125

Hence,  = y+â–³y = 0.2 + (-0.03125) = 1.96875

### (vi)

Solution:

Considering the function as

y = f(x) =

Taking x = 256, and

x+â–³x = 255

â–³x = 255-256 = -1

= 4

â–³y = dy =  dx

â–³y =  â–³x

â–³y =  (-1) =  = -0.003906

Hence,  = y+â–³y = 0.2 + (-0.003906) = 3.9961

### (vii)

Solution:

Considering the function as

y = f(x) =

Taking x = 2, and

x+â–³x = 2.002

â–³x = 2.002-2 = 0.002

â–³y = dy =  dx

â–³y =  â–³x

â–³y =  (0.002) = -0.0005

Hence,  = y+â–³y =  + (-0.005) = 0.2495

### (viii) loge 4.04, it being given that log10 4=0.6021 and log10 e=0.4343

Solution:

Considering the function as

y = f(x) = loge x

Taking x = 4, and

x+â–³x = 4.04

â–³x = 4-4.04 = 0.04

y = loge

= loge 4 =  = 1.386368

â–³y = dy =  dx

â–³y =  â–³x

â–³y =  (0.04) = 0.01

Hence, loge 4.04 = y+â–³y = 1.386368 + 0.01 = 1.396368

### (ix) loge 10.02, it being given that loge 10=2.3026

Solution:

Considering the function as

y = f(x) = loge x

Taking x = 10, and

x+â–³x = 10.02

â–³x = 10.02-10 = 0.02

y = loge x

= loge 10 = 2.3026

â–³y = dy =  dx

â–³y =  â–³x

â–³y =  (0.02) = 0.002

Hence, loge 10.02 = y+â–³y = 2.3026 + 0.002 = 2.3046

### (x) log10 10.1, it being given that log10 e=0.4343

Solution:

Considering the function as

y = f(x) = log10 x

Taking x = 10, and

x+â–³x = 10.1

â–³x = 10.1-10 = 0.1

y = log10 x =

= log10 10 = 1

â–³y = dy =  dx

â–³y =  â–³x

â–³y =  (0.1) = 0.004343

Hence, loge 10.1 = y+â–³y = 1 + 0.004343 = 1.004343

### (xi) cos 61Â°, it being given that sin 60Â°=0.86603 and 1Â°=0.01745 radian.

Solution:

Considering the function as

y = f(x) = cos x

Taking x = 60Â°, and

x+â–³x = 61Â°

â–³x = 61Â°-60Â° = 1Â° = 0.01745 radian

y = cos x

= cos 60Â° = 0.5

= – sin x

= – sin 60Â° = -0.86603

â–³y = dy =  dx

â–³y = (-0.86603) â–³x

â–³y = (-0.86603) (0.01745) = -0.01511

Hence, cos 61Â° = y+â–³y = 0.5 + (-0.01511) = 0.48489

### (xii)

Solution:

Considering the function as

y = f(x) =

Taking x = 25, and

x+â–³x = 25.1

â–³x = 25.1-25 = 0.1

â–³y = dy =  dx

â–³y =  â–³x

â–³y =  (0.1) =  = -0.0004

Hence,  = y+â–³y =  + (-0.0004) = 0.1996

### (xiii)

Solution:

Considering the function as

y = f(x) = sin x

Taking x = 22/7, and

x+â–³x = 22/14

â–³x = 22/14-22/7 = -22/14

sin (-22/14) = -1

y = sin x

= sin (22/7) = 0

= cos x

= cos (22/7)= -1

â–³y = dy =  dx

â–³y = (-1) â–³x

â–³y = (-1) (-1) = 1

Hence, sin(22/14) = 0+1 = 1

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