# Racing Car Driver

## The Scenario

You are an up and coming racing car driver, looking to land your first big sponsorship deal. If you can put in a good performance in the next race, a major sponsor will sign with you and you may be able to move up into the championship division. The pressure’s on – you’ll need to get every aspect of the race just right in order to have a chance.

## Math Scenario 1: Fuel Consumption

You are entered in a 250 km race, with 5 km laps. Your car can hold enough fuel for about 60 km of racing at top speed. Calculate how many pit stops you’ll need to make.

*#pit stops = distance / distance per tank of fuel*

*#pit stops = 250 / 60*

*#pit stops = 4.167*

Because you get the first 60 km of the race “for free” without a fuel stop, you need to round this number **down** to the nearest whole number.

So you’ll be able to complete the race with 4 pit stops.

## Math Scenario 2: Tyre Change

Halfway through the race, you have to make the decision about whether to change tyres at your pit stop. If you don’t change tyres, your car’s performance will slowly get worse and worse over the remaining 125 km of the race. But with your small, unskilled pit crew, changing tyres can take an extra 10 minutes!

Without changing tyres, your average speed will drop by 1 km/hr per minute of racing. We need to work out whether changing tyres is worthwhile. Assume your average speed for the first half of the race was 150 km/hr.

*Speed = 150 km/hr – 1t*

Where *t* is the time spent racing from the halfway point onwards in minutes

We can plot a graph of speed versus time for the remainder of the race:

The distance travelled by the car will be equal to the area under this graph, which represents speed multiplied by time. The area under the curve is made up of two shapes: a rectangle at the bottom, and a triangle above that.

If we say *t* is the time spent finishing the race, the distance travelled will equal (remember to divide by 60 where necessary to get from minutes to hours):

*d = rectangle area + triangle area*

*d = (150 – t) * t / 60 + 0.5 * t * t / 60*

*d = 2.5t – 0.01667t^2 + 0.008333t^2*

*d = 2.5t – 0.008333t^2*

But we also know that the remaining distance is 125 km, so we can substitute this in:

*125 =2.5t – 0.008333t^2*

*0.008333t^2 – 2.5t + 125 = 0*

And then use the quadratic formula to solve for t:

t = 236.6, 63.40

So without changing tyres, the car will take 63.4 minutes to finish the race.

We can now compare this to the time taken if you did take the time to change tyres:

*Time to finish with tyre change = tyre change duration + time to cover distance*

*Time to finish with tyre change = tyre change duration + distance / speed*

*Time to finish with tyre change = 10 + 60 * 125 / 150*

*Time to finish with tyre change = 60 minutes*

So even with the extra 10 minutes for a tyre change, it’s still worthwhile, as the extra speed kept for the rest of the race means you’ll finish faster overall.

## Math Scenario 3: Straight Line Speed versus Corner Speed

You have the option for your next race of making some modifications to the car. One set of options gives you better straight line speed, the other better speed through the corners.

Your next race is on the Sydney circuit, which contains 65% straights and 35% corners.

With the straight line speed setup, you average 200 km/hr on straights and 70 km/hr on corners.

With the cornering setup, you average 185 km/hr on straights and 90 km/hr on corners.

Which setup should you go with?

**Straight line speed setup:**

*Total time per circuit = time spent on straights + time spent on corners*

Let’s call the length of one lap *“l”*:

*Total time per circuit = 0.65l / 200 + 0.35l / 70*

*Total time per circuit = 0.00825l*

**Cornering speed setup:**

*Total time per circuit = time spent on straights + time spent on corners*

*Total time per circuit = 0.65l / 185 + 0.35l / 90*

*Total time per circuit = 0.0074l*

The cornering speed setup results in a significantly shorter total lap time.

## Real Life Example – Mining Truck Tyres

One of the big expenses for operating huge mining trucks is the cost of repairing and replacing the tyres on the trucks. Mining tyres can cost more than $100,000 **per tyre**, which means that maximizing the lifetime of the tyre can save millions of dollars over a fleet of trucks.

Let’s say you have the opportunity to license an onboard driving assistance software program that will make human drivers more efficient at driving, meaning the tyres will last 1.05 times as long as they currently do.

If you have 200 trucks across your mining fleet (each with four tyres), and your average replacement period for a tyre is 2 years at $50,000 per tyre, calculate whether the annual $500000 license cost for the software is worth it.

*Current replacement costs / annum = #trucks * tyres changed per truck per year * tyre replacement cost*

*Current replacement costs / annum = 200 * 2 * 50000*

*Current replacement costs / annum = $50000000*

That’s 50 million dollars per year.

The software program will make these tyres last 1.05 times as long:

*Cost saving = current cost per annum – current cost per annum / increased tyre lifetime ratio*

*Cost saving = 50000000 – 50000000 / 1.05*

*Cost saving = $2380952*

Using the software will only save about a quarter of a millon dollars per year, less than the $5000000 / year software license, so it’s not a good deal. You should try to negotiate a better deal with the software provider.