Someone please help me with the questions in the attached pdf. I know you need to use the distance formula for question 1a, but after that, my composure and confidence in myself to do the rest of the questions falls apart.
Even just some explanation on how to get started on Questions 1a-f would be much appreciated. If someone can go through some possible methods for approaching the remaining questions, that would be even more awesome!
Part aThe length \(AX\) is the hypotenuse of a right-angled triangle \((\triangle ACX)\), so we have \[d=\sqrt{150^2+(200-x)^2}=\sqrt{x^2-400x+62500}\]
Part bThe time taken is given by the distance divided by the speed. Thus, \[T_1=\frac{d}{1}=\sqrt{x^2-400x+62500}\]
Part cThe total time is given by the time taken to travel \(AX\) plus the time taken to travel \(BX\). Using the same idea as in
part b, can you write an expression for the time taken to travel \(BX\) in terms of \(x\)?
Spoiler
\[T(x)=\sqrt{x^2-400x+62500}+\frac{x}{2}\]
Part dWhat is a useful domain of \(x\)? Hint: is there much point travelling to the right when at point \(A\)?
Part e.iNow that your have \(T(x)\), use you CAS.
Part e.iiThis is a standard optimisation problem. What is the value of \(\dfrac{dT}{dx}\) at the point where \(T\) is minimum?
Part fNow that you have the value of \(x\) for which \(T\) is minimum, what is the minimum value of \(T\)?
Part gThis question tells you what to do. Pick values slightly to the left and slightly to the right of the critical value found in
part e.ii. Determine the values of \(\dfrac{dT}{dx}\) at these values. What do these tell you about the nature of the stationary point found?
Part h\(x=200\)
Part i\(x=0\)
Part j\(x=100\)
Part k.iUse your CAS to sketch it. Remember that \(T\) has a domain.
Part k.iiUse the graph you just sketched.
Okay, I'm going to leave it here.
Part L onwards requires a bit of thought. Make a start with what I've given.