An aircraft was to take off from a certain airport at 8 a.m but it was delayed by 30 minutes.To make up for the lost time,?

Let the normal speed be x km/ hour

Increased speed = ( x+250) km/ hour

TIME = DISTANCE / SPEED

1500 / x - 1500 / ( x+250) = 1/2

MULTIPLY BOTH SIDES OF EQUATION BY 2x ( x+250)

3000( x+250) - 3000x = x ( x+250)

3000x + 750000 -3000x = x^2 +250x

x^2 +250x = 750000

ADD 15625

x^2 +250x +15625 = 765625

( x + 125)^2 = 765625

Rejecting the negative , x= \/765625 -125 = 750 km/ hour

ANSWER b) 750 km/h


CHECK

1500/750 = 2 HOURS

1500/1000 = 1.5 HOURS

2 HOURS - 1.5HOURS = 0.5 HOURS

No you don't have to solve the quadratic. As crypto hints, make use of the options. (If options are not given, you can still make guesses until you find the correct answer.)

Test the easiest options first. If the usual speed is 750 or 1000 km/h (options b and d) then the journey times would be 1500/750 = 2 hours or 1500/1000 = 1.5 hours. The difference in times is 30 minutes and the difference in speed is 250 km/h, so we have stumbled on the correct answer straight away.

ANSWER : b) 750 km/h.


If options are not given, make guesses using numbers which are easy to calculate with :

Suppose the usual speed is v. The usual time is 1500/v and the new time is 1500/(v+250). The difference should be 30 minutes so
1500/v - 1500/(v+250) = 1/2
3000/v - 3000/(v+250) = 1.
Try v=250 :
LHS = 12 - 6 = 6 (much too big).
Try v=500 :
LHS = 6 - 4 = 2 (getting close).
Try v=750 :
LHS = 4 - 3 = 1 = RHS.
So original speed is 750 km/h.

b) You have to solve the quadratic.

You have

vt = (v + 250) (t - .5)

vt = vt -.5v+ 250t - 125

125 = 250t - .5v

since vt = 1500 you have t = 1500/v

125 = 250(1500/v) - .5v

And this will indeed lead to a 2nd degree equation that you can solve using the
quadratic formula

If you want a shortcut, you are given possible answers. Before
going ahead perhaps we can try one of them to see if it might work. 750
is a good one to try since it goes into 1500 evenly

125 = 250(1500/v) - .5v becomes

125 = 250 (2) - .5(750)
125 = 500 - 375

Looks like thats the answer.

speed * time = distance

st = 1500

(s + 250)(t - 30/60) = 1500
(s + 250)(t - .5) = 1500
st - .5s + 250t - 125 = 1500
1500 - .5s + 250t - 125 = 1500
250t - .5s = 125
.5(500t - s) = 125
500t - s = 250

t = 1500/s

500(1500/s) - s = 250
750000 - s^2 = 250s
0 = s^2 + 250s - 750000

s = 750, -1000

ignore negative solution

s = 750 kph