Math question help

Hi guys, so I get i) and also for ii) I get up to M=(x+a)^3(p+B) and then the expansion bit is just weird - I don't really get it.
BTW I have attached the question and the mark scheme.
Can someone please help?

Not sure what's so confusing

We have $\displaystyle (x+\alpha)^3 = x^3+3 \alpha x^2 + \underbrace{3 \alpha^2 x + \alpha^3}_{(1)}$ and we ignore (1) because that's what the question tells us to do, "ignoring any terms which involve powers or products of the error terms $\alpha$ and $\beta$"

Then we just expand $(x^3+3 \alpha x^2)(\rho +\beta) = \rho x^3 + 3 \alpha \rho x^2 + \beta x^3 + \alpha \beta x^2$ and we just ignore the last term because that's a product of two error terms.

Not sure what's so confusing

We have $\displaystyle (x+\alpha)^3 = x^3+3 \alpha x^2 + \underbrace{3 \alpha^2 x + \alpha^3}_{(1)}$ and we ignore (1) because that's what the question tells us to do, "ignoring any terms which involve powers or products of the error terms $\alpha$ and $\beta$"

Then we just expand $(x^3+3 \alpha x^2)(\rho +\beta) = \rho x^3 + 3 \alpha \rho x^2 + \beta x^3 + \alpha \beta x^2$ and we just ignore the last term because that's a product of two error terms.

I get everything from this question apart from the (You could check your answer by direct calculation) and the mark scheme answer “Check: new mass=(3.01)^3(10.02)..” bit

What do they mean by direct calculation? Sorry for my ignorance

Why have they used M =(x+a)^3(p+B) here to get the new mass 273.254? Part ii) says to “ignore any terms which involve powers or products of the error terms a and B” so why use M=(x+a)^3(p+B) which has error terms when fully expanded out?

Also, what is the point in getting the new mass? Surely the question is asking for the extra mass.. and then supposedly “checking” you have got the correct extra mass which I don’t know how you do.

So for the “You could check your answer by direct calculation”, I was just assuming..


And then sub in the x,a,p,B values into one of these 3 equations in (2)....

And then find the difference between (2) and (1) to get the extra mass

Could you please help me here? Massively appreciate your help so far with other questions

Directly calculate the new mass rather than approximate it, so use $(x+\alpha)^3(\rho + \beta)$ with your calc and get the value.



Because that gives you the exact mass, and you want to compare it with $\rho x^3$ to see that indeed you have an extra $\approx 3.24$ units.



Here just note that $\underbrace{(x+\alpha)^3(\rho + \beta)}_{\text{exact incl. error}} \approx \underbrace{\rho x^3}_{\text{exact, no error}} + \beta x^3 + 3\alpha \rho x^2$

Then subtracting the (exact, no error) from (exact incl. error) gives you the extra mass, which is specifically $(x+\alpha)^3(\rho + \beta) - \rho x^3 \approx \beta x^3 + 3\alpha \rho x^2$