cbse chapter 4 quadratic equations word problems / time - work problems
exercise 4.3
9. Two water taps can fill a tank in [ 9+(3/8) ] hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank
Let x = time taken by tap of larger diameter to fill the tank (in hours)
given that the tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately
so time taken by tap of smaller diameter to fill the tank =(x+10) hours
together they take [ 9+(3/8) ] hours or (75/8) hours by converting the mixed fraction
remeber to use reciprocals in time /work problems
[1/x] +[1/(x+10)] = 1/{75/8}
[1/x] +[1/(x+10)] =[8/75]
multiply each term with 75x(x+10)
75(x+10) +75x =8x(x+10)
75x+750 +75x =8(x^2)+80x
8(x^2) -70x -750=0 dividing by 2
4(x^2) -35x -375=0
we search for two numbers
whose sum is (-35) and
product =4*(-375) =(-1500)
the numbers are (-60) and 25
use this to split the middle term
4(x^2) -35x -375=0
4(x^2) -60x +25x -375=0
4x[x-15]+25[x-15]=0
[x-15][4x+25] =0
x=15 or x={(-25) /4} which is rejected
x=15 hours for the tap of larger diameter.
x+10 =15+10=25hours for the tap of smaller diameter.
10. An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore (without taking into consideration the time they stop atintermediate stations). If the average speed of the express train is 11km/h more than that of the passenger train, find the average speed of the two trains.
x =speed of passenger train (in hours) (x+11) =speed express train (in hours) distance = 132 km time = distance/speed
t1= 132/x [time for passenger train]
t2= 132/(x+11) [time for express train]
given express train takes 1 hour less than a passenger train to travel 132 km
so
t1 -t2 =1
using the above values
[132/x] -[132/(x+11)] =1
multiply x(x+11)
132(x+11)-132x = 1x(x+11)
1452= (x^2) + 11 x
(x^2) +11x -1452 =0
to factorise search for two numbers whose product is (-1452) and sum is 11
numbers are 44 and (-33)
so (x+44)(x-33)=0
x=-44(which is rejected
x=33 km/hr
speed of passenger train =x = 33 km/hrspeed of express train =(x+11)=33+11 =44 km/hr
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If the equations x^2 -ax+b=0 and x^2-ex+f =0 have a root in common and the second equation has equal roots show that ae =2(b+f)
solving a quadratic equation by completing the square
solve 2x² -4x -7 = 0
