Exercise 9B
Question 1
State the relationship between the given variables as an equation, using k for the constant of variation.
(a) The volume V a gas at a fixed temperature varies inversely as the pressure P.
Sol :
V=volume of gas
P=pressure
K=temperature(constant)
$V\propto \frac{1}{P}$
∴$V=\frac{K}{P}$
(b) The current l in an electrical circuit of fixed voltage varies inversely as the resistance R.
Sol :
I=current
R=ressistance
K=voltage(constant)
$I \propto \frac{1}{R}$
∴$I=\frac{K}{R}$
(c) The height h of a cylinder of fixed volume varies inversely as the area A of the base.
Sol :
h=height
A=Area
K=volume(constant)
$h\propto \frac{1}{A}$
∴$h=\frac{K}{A}$
(d) The frequency f of an electromagnetic wave is inversely proportional to the length l of the wave.
Sol :
f=frequency
l=length of wave
k=constant
$f\propto \frac{1}{l}$
∴$f=\frac{K}{l}$
Question 2
Fill in the blanks in the following tables by determining first whether x and y very directly or inversely:
(a)
| x | 3 | 6 | ? | 27 | ? |
| y | 11 | 22 | 33 | ? | 880 |
Sol :
$\frac{y}{x}=\frac{11}{3}=k$
∴$\frac{y_4}{x_4}=k$
$x_4=33\times \frac{3}{11}$=9
$\frac{y_5}{x_5}=k$
$y_5=\frac{11}{5}\times 27$=99
$\frac{y_6}{x_6}=k$
$x_6=880\times \frac{3}{11}=240$
Ans 9,99,240 (Directly)
(b)
| x | 30 | 20 | 15 | ? | ? |
| y | 6 | 4 | ? | 2 | 1 |
(c)
| x | 2 | 3 | 4 | ? | 8 |
| y | 48 | ? | 24 | 16 | ? |
Sol :
$y \propto \frac{1}{x}$
xy=k=48×2=96
y2x2=k
$y_2=\frac{x}{x_2}$
$=\frac{96}{3}=32$
y4x4=k
$x_4=\frac{k}{y_1}$
$x_4=\frac{96}{16}=6$
y5x5=k
$y_5=\frac{k}{x_5}$
$x_4=\frac{96}{8}=12$
Ans 32, 6, 12 (Inversly)
(d)
| x | 1 | 5 | 10 | ? | ? |
| y | 125 | ? | 12.5 | 5 | 1 |
Sol :
$y\propto \frac{1}{x}$
∴xy=k=125×1=125
x2y2=k
x4y4=k
x5y5=k
Question 3
(a) If y varies inversely as x, and y = 9 when x = 2. find y when x = 3.
Sol :
| y | 9 | ? | ? |
| x | 2 | 12 | 3 |
$y \propto \frac{1}{x}$
∴yx=k
Now , y=9 , x=2
∴k=yx
or k=9×2=18
Again,
x=12
∴$y=\frac{18}{12}=\frac{3}{2}$
x=3
∴$y=\frac{k}{x}=\frac{18}{3}=6$
Ans $\frac{3}{2}$ ,3
(b) If u is inversely proportional to v, and is u = 12 when v = 3, find u when v = 9.
Sol :
$u \propto \frac{1}{v}$
∴uv=k
Now u=12 ,v=3 ∴k=12×3=36
(c) If c is inversely proportional to d, and if c = 18 when d = 2/3, find d when c = 6/7
Sol :
$c\propto \frac{1}{d}$ ∴cd=k
Now ,
c=18 ,$d=\frac{2}{3}$ ∴$k=18 \times \frac{2}{3}=12$
Again $c=\frac{6}{7}$ ∴$d=\frac{k}{c}12\times \frac{7}{6}=14$
(d) If m is inversely propotional to n, and if m = 0.02 when n = 5, find m when n = 0.2
Sol :
$m \propto \frac{1}{n}$ , ∴mn=k
Now ,m=0.02 , n=5 ∴k=0.02×5=0.10
mn=k
m×0.2=0.10
m=0.5
Question 4
Navin cycles to his school at an average speed of 12 km/hr. It takes him 20 minutes to reach the school. If he wants to reach his school in 15 minutes, what should be his average speed?
Sol :
Speed=s=12 km/hr , time=t=20min
∴$s\propto \frac{1}{t}$ ∴st=k
s1=12 , t1=20 , k=12×20=240
t2=15min ∴$S_2=\frac{k}{t_2}=\frac{240}{15}=16$km/hr
Question 5
The time needed to travel from one place to another is inversely proportional to the speed. A person travelling 72 km/hr can go from Dehradun to Lucknow in 10 hours. How fast must the person travel to make the trip in 9 hours?
Sol :
$T \propto \frac{1}{S}$ ∴TS=K
S1=72km/hr ,T1=10 h ∴K=S1T1=72×10=720
∴T1=9 h , $S_1=\frac{K}{T_1}=\frac{720}{9}=80$km/h
Question 6
28 pumps can empty a reservoir in 18 hours. In how many hours can 42 such pumps do the same work?
Sol :
$P\propto \frac{1}{T}$ ∴PT=K
P1=28 ,T1=18 h ∴K=P1T1=18×28=504
∴P2=42 , $T_2=\frac{K}{P_2}=\frac{504}{42}=12$h
Question 7
A stock of food grains is enough for 400 persons in 9 days. How long will the same stock last for 300 persons?
Sol :
$M\propto \frac{1}{D}$ MD=K
∴M1=400 per D1=9 ∴K=M1D1=400×9=3600
∴M2=300 per $D_2=\frac{K}{M_2}=\frac{3600}{300}=12$days
Question 8
A contractor, who had a workforce of 630 persons, undertook to complete a portion of a stadium in 14 months. He was asked to complete the job in 9 months. How many extra persons had he to employ?
Sol :
$M\propto \frac{1}{D}$ ∴MD=K
∴M1=630 per D1=14 months ∴K=M1D1=630×14=8820
∴D2=9 months $M_2=\frac{K}{D_2}=\frac{8820}{9}=980$days
∴Extra person=980-630=350
Question 9
Working 4 hours a day, Savita can type a manuscript in 15 days. How many hours a day should she work so as to finish the work in 10 days?
Sol :
$T\propto \frac{1}{D}$ ∴TD=K
∴T1=4 hour D1=15 days ∴K=T1D1=4×15=60
D2=10 days $T_2=\frac{k}{D_2}=\frac{60}{10}$=6 hour
Question 10
A train moving at a speed of 60 km/hr covers a certain distance in 7.5 hours. What should be the speed of the train to cover the same distance in 6 hours?
Sol :
$S \propto \frac{1}{T}$ ∴ST=K
∴S1=60 km/hour T1=7.5 hour ∴K=S1T1=60×7.5=450
T2=6 hour $S_2=\frac{450}{6}$=75 km/hour
Question 11
A garrison of 800 men had provisions for 39 days, However a reinforcement of 500 men arrived. For how many days will the food last now?
Sol :
$M \propto \frac{1}{D}$ ∴DM=K
∴M1=800 min D1=39 days ∴K=M1D1=800×39=31200
M2=500 min $D_2=\frac{31200}{500}$=62.4 days
∴The food last now=(62.4-39)=23.4≈24days
Question 12
A beseieged town has provisions to last for 3 weeks. Its population is 22400. How many people must be sent away in order that the provisions may last for 7 weeks?
Sol :
$P \propto \frac{1}{W}$ ∴PW=K
∴P1=22400 min W1=3 days ∴K=P1W1=22400×3=67200
W2=7 $P_2=\frac{67200}{7}$=9600
∴Number of provisions=22400-9600
=12800
Question 13
A hostel had rations for 60 days for 500 students. After 12 days, 300 more students joined the hostel. How long will the remaining rations last?
Sol :
(60-12)=48 days for (500+300)=800
| students | 500 | 800 |
| days | 48 | x |
∴$x=\frac{48\times 500}{800}$
=30 days
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