Increasing Speed

[anaxcrosswords]

My maths is appalling, so can someone help me out with this?
I’m on a motorway driving at 70mph. The satnav shows my arrival time as 8:31. For whatever reason, it’s vital that I arrive at 8:30, so I increase my speed to 80mph (this is theoretical!).
How far must I travel at that speed to make up the 1 minute? I’m sure this is far easier to calculate than I think, but I’m having a mental block.

[bibblebub]

9⅓ miles

[New Judge]

It depends what the time is now and the distance you have left to travel.

If you have 100 yards left you might struggle. If you've 100 miles the extra speed required will be minimal.

[Mick-the-Miller]

The Satnav says if you proceed at 70 mph you will arrive at 8.31 but you need to arrive at 8.30 - so you have to work out how far short you would be which is the distance travelled in 1 minute at 70 mph - 70/60 = 1.16 miles.

Then you work out the difference in distance travelled at 70 mph and 80 mph - 80/60 - 70/60 = 1.33 - 1.16 = 0.17.

You then work out how long take you to cover 1.16 at the increased rate of 0.17 mph = 1.16/0.17 = 6.84 minutes.

Makes sense to me!

[New Judge]

But then I didn't read the question properly !!

Still (he says to save his face) if you've only got 100 yards to go..... :-)

[MallyJ]

depends on the speed limit .... if it is 70mph then be a little late..... leave earlier next time.......!!!!

[anaxcrosswords]

It was only theoretical, MallyJ. The main reason I posted was that, from my own experience, stopping at traffic lights a few times can add several minutes to a journey time - but, obviously, the difference between 0mph and, say, 40mph is far greater than adding 10mph to your speed.
You often hear of speeding motorists giving the excuse that they were running late, but it struck me that you'd have to travel at crazy speeds to make up whatever you lose by a few instances of standing still; ie speeding really isn't worth it.

[Gizmonster]

Mick's answer isn't quite right:

Let s = distance of trip.
Let t = total time of original journey, in hours (at 70mph)
Let x = time spent travelling at 80mph in 2nd journey, in hours.

At a constant 70mph:

s = 70t .... equation A

In 2nd journey:

s = 70(t - 1/60 - x) + 80x
s = 70t - 70/60 -70x + 80x .... equation B

Combining equations A and B:

70t = 70t - 7/6 + 10x
10x = 7/6

x = 7/60 = 7 minutes at 80mph
....... and you'll travel 80 x 7/60 = 9 1/3 miles (as bibblebub said)

Choosing an arbitrary value for s, to check, say s = 100km

for 1st journey:

time taken for journey = 100/70 = 85 5/7 minutes

time taken for 2nd journey:

(90 2/3)/70 + (9 1/3)/80 = 84 5/7 minutes, which is exactly 1 minute less than the than original journey at a constant 70mph.

So, eventually .... the answer is 7 minutes at 80mph and you'll travel 9 1/3 miles in this 7 minutes :)

[TWR]

No matter how bad your maths Anax, this is the trouble with a lot of people, leaving it until the last Minute, why set off sooner, that way you get there safer.

[fiction-factory]

The formula speed = distance divided by time will always solve these problems.