Basic Engineering Mathematics, Fifth Edition

Transposing formulae 89

Rearranging gives

√(

f+p

f−p

)

=

D

d

Squaring both sides gives

(

f+p

f−p

)

=

D^2

d^2

Cross-multiplying,i.e. multiplyingeach term

byd^2 (f−p),gives

d^2 (f+p)=D^2 (f−p)

Removing brackets gives d^2 f+d^2 p=D^2 f−D^2 p

Rearranging, to obtain terms inpon the LHS gives

d^2 p+D^2 p=D^2 f−d^2 f

Factorizing gives p(d^2 +D^2 )=f(D^2 −d^2 )

Dividing both sides by(d^2 +D^2 )gives

p=

f(D^2 −d^2 )

(d^2 +D^2 )

Now try the following Practice Exercise

PracticeExercise 48 Further transposing

formulae (answers on page 345)

Make the symbol indicated the subject of each of

the formulae shown in problems 1 to 7 and express

each in its simplest form.

1. y=

a^2 m−a^2 n

x

(a)

2. M=π(R^4 −r^4 )(R)


3. x+y=

r

3 +r

(r)

4. m=

μL

L+rCR

(L)

5. a^2 =

b^2 −c^2

b^2

(b)

6.

x

y

=

1 +r^2

1 −r^2

(r)

7.

p

q

=

√(

a+ 2 b

a− 2 b

)

(b)

8. A formula for the focal length,f,ofaconvex
   lens is

1

f

=

1

u

+

1

v

. Transpose the formula to
makevthe subject and evaluatevwhenf= 5
andu=6.
9. Thequantityofheat,Q,isgivenbytheformula
Q=mc(t 2 −t 1 ).Maket 2 the subject of the
formula and evaluatet 2 whenm=10,t 1 =15,
c=4andQ=1600.
10. The velocity,v, of water in a pipe appears

in the formula h=

0. 03 Lv^2

2 dg

. Express v
as the subject of the formula and evalu-
atevwhenh= 0. 712 ,L= 150 ,d= 0 .30 and
g= 9. 81
11. The sag,S, at the centre of a wire is given

by the formulaS=

√(

3 d(l−d)

8

)

.Makel

the subject of the formula and evaluatelwhen

d= 1 .75 andS= 0 .80.

12. In an electrical alternating current cir-
    cuit the impedance Z is given by

Z=

√√

√

√

{

R^2 +

(

ωL−

1

ωC

) 2 }

. Transpose

the formula to makeCthe subject and hence

evaluateCwhenZ= 130 ,R= 120 ,ω= 314

andL= 0. 32

13. An approximate relationship between the
    number of teeth,T, on a milling cutter, the
    diameter of cutter,D, and the depth of cut,d,
    is given byT=

12. 5 D

D+ 4 d

. Determine the value
ofDwhenT=10 andd=4mm.
14. Makeλ, the wavelength ofX-rays, the subject

of the following formula:

μ

ρ

=

CZ^4

√

λ^5 n

a