A logger is loading logs onto his log lorry. The maximum load of the lorry is 2550 kilograms. The lorry is not safe to drive if it is carrying more than the maximum load. The logger loads the last log onto the lorry and the total mass of logs on the lorry is 2850 kilograms to the nearest 50 kilograms. The logger takes two logs off the lorry the mass of the two logs is 330 kilograms to the nearest 10 kilograms. Is it now safe for the logger to drive his lorry? Show your working.
So this question is a question that’s gonna involve upper and lower bounds. And that’s because we can see the things around it to certain values. So for instance, the logs on the lorry are rounded to the nearest 50 kilograms. So first of all, what I’m going to do is investigate the bounds for the total mass of the logs, which I’m gonna call capital 𝑀.
To demonstrate the bounds for the total mass of the logs, we’ve got a number line drawn here to help us visualize it. Well, we can see that anything including 2825 and above it up to 2850 would round up to 2850. It would not round down to 2800 if we’re rounding to the nearest kick, 50 kilograms. And then if we look at the other end of the line we’ve drawn, so that’s the pink line, we can see that anything up to 2875 kilograms and above 2850 kilograms would round down to 2850 kilograms. It would not round up to 2900 kilograms.
One thing to notice about our visual representation is that there’s a colored dot above 2825 and an open dot above 2875. And that’s because the colored dot means “and including.” So we’re including 2825. However, the open dot would mean “not including,” so everything right up to 2875, but not including itself. So therefore, we can say for the bounds for the total mass of logs capital 𝑀 are 𝑀 is greater than or equal to 2825 kilograms but less than 2875 kilograms.
So now if we take a look at the bounds for the mass of logs, well we know that the logs, so the two logs that the logger takes off the lorry, well they weigh 330 kilograms to the nearest 10 kilograms. So again, I’ve used our visual representation. So we can see that anything above 325 kilograms up to 330 kilograms is gonna round up to 330 kilograms and that’s including 325 kilograms. And anything down from 335 kilograms will round down to 330 kilograms.
So therefore, we can say that the bounds for the mass of logs, so the mass of the two logs that are taken off the lorry, are 𝑀 the mass is greater than or equal to 325 but less than 335 kilograms. So now that we have the bounds of the mass of logs on the lorry and the bounds for the two logs that are taken off the lorry, we can work out whether it’s safe to drive.
And to do that, we need to work out the bounds for the load 𝐿 after logs are removed. Well, we can say that the minimum bound is gonna be 2490 and that’s because we’ve got the minimum original load minus the maximum amount for the two logs that were taken off because that will give us the smallest possible load left on the lorry. So that’s gonna be 2825 minus 335.
Well, five take away five is zero and then we borrow from the eight. And we’ve got 12 take away three which is nine. Then, we’ve got seven take away three which is four. And then we got two take away zero. So we got 2490 and that’s our lower bound. Well, our upper bound is going to be 2550. And this time, we’ve got this by subtracting the lowest possible weight of the two logs that were removed from the highest possible weight of the load that was already on the lorry. And when we do that, we get five minus five which is zero, seven minus two which is five, eight minus three which is five, and two minus zero which gives us two. So we’ve got 2550.
Well, have we answered the question: we found the bounds for the load after logs were removed or no? And this is key. We need to see what the question is actually asking. The question is asking, “is it now safe for the logger to drive his lorry?” Well, yes it is safe as a maximum possible load left on the lorry is 2550 kilograms. And we saw that that was the upper bound for the load after logs are being removed.
So therefore, when looking back to the question, we know it’s safe because it says that the maximum load of the lorry is 2550 kilograms and the lorry is not safe to drive if it is any more than that maximum load. And we can see that ours won’t be.