Basic Engineering Mathematics, Fifth Edition

Solving quadratic equations 109

Hence, the mass will reach a height of 16m after
0 .59s on the ascent and after 5.53s on the descent.

Problem 25. Ashedis4.0m long and 2.0mwide.

A concrete path of constant width is laid all the way

around the shed. If the area of the path is 9.50m^2 ,

calculate its width to the nearest centimetre

Figure 14.1 shows a plan view of the shed with its
surrounding path of widthtmetres.

t

2.0 m

4.0 m (4.0^12 t)

SHED

t

Figure 14.1

Area of path= 2 ( 2. 0 ×t)+ 2 t( 4. 0 + 2 t)

i.e. 9. 50 = 4. 0 t+ 8. 0 t+ 4 t^2

or 4 t^2 + 12. 0 t− 9. 50 = 0

Hence,

t=

−( 12. 0 )±

√

( 12. 0 )^2 − 4 ( 4 )(− 9. 50 )

2 ( 4 )

=

− 12. 0 ±

√

296. 0

8

=

− 12. 0 ± 17. 20465

8

i.e. t= 0 .6506m or− 3 .65058m.

Neglecting the negative result, which is meaningless,
the width of the path,t= 0 .651mor65cmcorrect to
the nearest centimetre.

Problem 26. If the total surface area of a solid

cone is 486.2cm^2 and its slant height is 15.3cm,

determine its base diameter.

From Chapter 27, page 245, the total surface areaAof
a solid cone is given byA=πrl+πr^2 ,wherelis the
slant height andrthe base radius.

IfA= 482 .2andl= 15 .3, then

482. 2 =πr( 15. 3 )+πr^2

i.e. πr^2 + 15. 3 πr− 482. 2 = 0

or r^2 + 15. 3 r−

482. 2

π

= 0

Using the quadratic formula,

r=

− 15. 3 ±

√[

( 15. 3 )^2 − 4

(

− 482. 2

π

)]

2

=

− 15. 3 ±

√

848. 0461

2

=

− 15. 3 ± 29. 12123

2

Hence, radiusr= 6 .9106cm (or− 22 .21cm, which is

meaningless and is thus ignored).

Thus, thediameter of the base= 2 r= 2 ( 6. 9106 )

=13.82cm.

Now try the following Practice Exercise

PracticeExercise 57 Practicalproblems

involving quadratic equations (answers on

page 346)

1. The angle a rotating shaft turns through int
   seconds is given byθ=ωt+

1

2

αt^2. Deter-

mine the time taken to complete 4 radians if

ωis 3.0 rad/s andαis 0.60rad/s^2.

2. The powerPdeveloped in an electrical cir-
   cuit is given byP= 10 I− 8 I^2 ,whereIis
   the current in amperes. Determine the current
   necessary to produce a power of 2.5 watts in
   the circuit.


3. The area of a triangle is 47.6cm^2 and its
   perpendicular height is 4.3cm more than its
   base length. Determine the length of the base
   correct to 3 significant figures.


4. The sag,l, in metres in a cable stretched
   between two supports, distancexmapart,is
   given byl=

12

x

+x. Determine the distance

between the supports when the sag is 20m.

5. TheaciddissociationconstantKaofethanoic
   acid is 1. 8 × 10 −^5 moldm−^3 for a particu-
   lar solution. Using the Ostwald dilution law,