Mathematical Upper and Lower Bounds?

Question

Hey. I've recently started the topic of Bounds and Reciprocals and i need some help and advice. In my book it gives the questions of:

1. x and y are measured as 2.32m and 0.45m to the nearest 0.01m.

a) Find the upper and lower bounds of x and y

b) If z = x + 1/y, find the max and min possible values of z.

Also there is a question of the maximum and minimum values of calculations.

1. A floor is measured as being 5.3m x 4.2m to the nearest 10cm.

For this i tried to use the method i use for solving the upper and lower bounds but it doesn't work. For example for the 5.3m i would use 5.3 + or - divided by 1/2 then multiplied by 10. But this doesn't work. Can someone please help.

All comments are very much appreciated. Many thanks!

Answer 1

1. x and y are measured as 2.32m and 0.45m to the nearest 0.01m.

Means:

2.31m ≤ x ≤ 2.33m

0.44 m ≤ y ≤ 0.46 m

To find the max. for z

z = 2.33 + 1/.44 ( note we had to use the minimum y since the expression for z called for 1/y.)

When divided this is approximately:4.60(27) repeating.

or 5063/1100

For the second problem the upper bound for x is:

5.35 m and y = 4.25m

If you need to find the upper bound of anything related to x and y use those two numbers, such as the area which is 5.35X4.25

Answer 2

This looks like GCSE maths..

1a) Well, 0.01m = 1 centimetre. Now because it's to the nearest centimetre, they want you to find the values half a centimetre either side of it, because that would mean it would be rounded up or down to the same figure as the one in the question. So..

2.32m = 232 centimetres, and that figure has been rounded to the nearest centimetre. So the upper bound would be 232.5 centimetre - even though that would strictly be rounded up, it is so close to the actual answer that we just say 232.5. The lower bound is 231.5 - this would be rounded up to the nearest centimetre as 232. If we wanted the answers in metres, it would be 2.325 for the higher bound and 2.315 for the lower.

It's the same with y. 0.45m = 45 centimetres. The upper bound is 45.5, and the lower bound is 44.5. Or in metres, 0.455 and 0.445

b) Quite a hard question this one. First it asks for the MINIMUM value. So, we'll use the lower bound for x, and the UPPER bound for y because we're dividing by it - the bigger you divide by, the smaller the number. We'll convert the numbers back to metres for this question, which would just mean dividing our answers from before by 100. So, plugging the numbers in we get:

z = 2.315 + 1/0.455

and if you do that on a calculator you'll get 4.513m.

For the upper bound of z, we want to take the upper bound of x and the lower bound of y. Plugging this in we get:

z = 2.325 + 1/0.445

= 4.572m.

For the second question, we use exactly the same principals. 10cm = 0.1m, and it would be easier to do it in metres this time as we're only dealing with 2 d.p. So, the upper bound of 5.3 = 5.35 and the lower bound = 5.25. The upper bound of 4.2 = 4.25 and the lower bound = 4.15. This is because if they were rounded to the nearest tenth of a metre, they would still produce the same results (theoretically 4.25 would become 4.3, but it is so close to the answer that we just say 4.25 and GCSE mark schemes tell you to write 4.25). So if we want to find the biggest and smallest possible area, we do:

lower bound x * lower bound y <== lower bound

upper bound x * upper bound y <== upper bound

5.25m * 4.15m = 21.79m²

5.35 * 4.25 = 22.74m²

And those are your answers. Hope that helps, good luck in your GCSE.