# Math Help - Calculate upper and lower bound gradients

1. ## Calculate upper and lower bound gradients

In triangle ABC, angle ABC=90 degrees
AB=5.3, correct to 2 sig figs
BC=4.8cm, correct to 2 sig figs
The base, AB of the triangle is horizontal

Calculate the lower and upper bounds of the line AC.

Never encountered before. I don't even know how to calculate the gradient. Or what the gradient is, only ever seen it in graphs. Is it a vector of some sort?

2. Do you know the Pythagoras' Theorem?

It says that for a right angled triangle, the sum of the squares of two shorter sides of a right angled triangle is equal to the square of the longest side (hypotenuse) of the triangle.

$AC^2 = AB^2 + BC^2$

Here, you need to find the minimum and maximum length of AC.

Let's start by the minimum.

AB is 5.3 to two significant figures. The mininum value of AB could be 5.25 (when rounded off to 2 sf give 5.3).
BC is 4.8 to 2 sf. The minimum value of BC is 4.75.

From those two values, $AC = \sqrt{5.25^2 + 4.75^2} = 7.1 cm$ (2sf)

Using the same logic, the maximum values of AB and BC are 5.34 and 4.84 respcetively, giving a length AC of $AC = \sqrt{5.34^2 + 4.84^2} = 7.2 cm$ (2sf)

3. It gives no indication whatsoever that it's a right-angled triangle

4. Of course it does

Did you see that the angle ABC = 90 degrees

5. How silly of me, thanks for pointing that out.

By upper and lower limits I thought it meant the angle of steepness not the length of the line.

If a question says (not particularly this one) x is .... to 2 dp and y is ... to 3dp then asks 'what would be an approximate degree of accuracy?' would I just write down what they originally said? Because that IS an approximate degree of accuracy :/

6. In physics, there is a whole chapter on errors and uncertainty. I don't know if these apply here, but I would have answered just like you.

A line joining the points (0, 0) and (1, 1) is drawn. Each coordinate is accurate to 0.05 unit. What is the how accurate is the length of the line.

Errors are stackable, and the calculated length of the line is square root of 2. The length is then $\sqrt2 \pm 2(0.05)$

The accuracy therefore come to 1 unit.

If it was about dp, the answer to this example would be to 1 dp.

Now, if both values were to be used to give a single value, for example, if x + y from your example was asked, then the final value will be to accurate to 2 dp. You cannot say to 3 dp because you are less accurate in one of the numbers.

I hope it helped.