3U Maths Question Thread

[MEL7401]

Thx kiwiberry!
I'll focus on them

[RuiAce]

Thx kiwiberry!
I'll focus on them

And indeed.

IRRESPECTIVE of the situation past papers should always be prioritised for that reason.

[RuiAce]

Hey can i get some help with this question (I know its really easy dont judge pls xd but i just forgot everything about locus)

A is a point where the circle with eq x^2 + y^2 = 16 cuts the x-axis. Find the locus of the midpoints of all chords of this circle that contain the point A.

Cheers

[kiwiberry]

Could I get some help on this question:

A cylindrical water tank of radius 20m is partly filled with water. A sphere of radius 2m is lowered into the water at the constant rate of 0.3m/s. Find the rate at which the water level is rising when the sphere is half submerged.

Thanks

Edit: The answer's 1/330 m/sec btw!

[jamonwindeyer]

Could I get some help on this question:

A cylindrical water tank of radius 20m is partly filled with water. A sphere of radius 2m is lowered into the water at the constant rate of 0.3m/s. Find the rate at which the water level is rising when the sphere is half submerged.

Thanks

This is a really cool question - I'm trying to decide whether it can be done without knowing the formula for a volume of a hemisphere! Even with that, it's tough, because the volume displacement of the water is dependent on how far into the water is submerged, but how far into the water is submerged is also dependent on how much water has been displaced

Sooo actually you'd relate the volume of water displaced by the sphere to the height of the water, and then the height of the water back to volume again.

Afraid to do the math until I decide whether to use volume of a hemisphere or not :/

[Shadowxo]

Jamon : we know volume of sphere (4/3 π r³), so volume of hemisphere can be deduced by halving it

[RuiAce]

Jamon : we know volume of sphere (4/3 π r³), so volume of hemisphere can be deduced by halving it

The problem isn't the actual formula. The problem is whether or not we're allowed to assume this in the HSC despite its triviality.

[jamonwindeyer]

The problem isn't the actual formula. The problem is whether or not we're allowed to assume this in the HSC despite its triviality.

Nah nah, you could definitely assume that, I suppose hemisphere is the wrong word (though we will need it) - I mean more the volume of a spherical cap - I think to do this we'll need a formula linking the volume of a spherical cap to the height of said cap (and the radius of the sphere), which with a Google is apparently:



I can't think of a way to do it without this formula, and even with it I'm not sure about the approach. I'll have a proper think and attempt at it tomorrow, I'm thinking we'll need to define \(\frac{dH}{dt}=0.3+A\), where \(A\) is some function of the volume of the sphere in the water at any given time. Play with derivatives a bit from there?

[Dragomistress]

May someone help me?

Question:

Moderator edit [Aaron]: Posts merged, unnecessary double post.

[kiwiberry]

Nah nah, you could definitely assume that, I suppose hemisphere is the wrong word (though we will need it) - I mean more the volume of a spherical cap - I think to do this we'll need a formula linking the volume of a spherical cap to the height of said cap (and the radius of the sphere), which with a Google is apparently:



I can't think of a way to do it without this formula, and even with it I'm not sure about the approach. I'll have a proper think and attempt at it tomorrow, I'm thinking we'll need to define \(\frac{dH}{dt}=0.3+A\), where \(A\) is some function of the volume of the sphere in the water at any given time. Play with derivatives a bit from there?

Hey Jamon, I asked my teacher today and he said not to bother with this question, so don't worry about it!! IThanks though

[jamonwindeyer]

Hey Jamon, I asked my teacher today and he said not to bother with this question, so don't worry about it!! IThanks though

Thank God I scribbled a rough attempt at the math last night and it was making me nervous aha no worries kiwi!

[hanaacdr]

Could i please get sone help on part i and ii plz
Thank you

[RuiAce]

Could i please get sone help on part i and ii plz
Thank you
(Image removed from quote.)

Hint for first part: Consider the following theorems.

1. Alternate segment theorem
2. Three right angles in that diagram, two of which are angle AMQ and angle ALQ

[jamonwindeyer]

May someone help me?

Question:

Moderator edit [Aaron]: Posts merged, unnecessary double post.

Hey JK! For the first one, you need to find the equation of each median. The median is the line joining a vertex of a triangle to the midpoint of the opposite side. The easiest way to do this will be with the two point formula.

I'll use the point A to do one example! We need the midpoint of the opposing side BC:



So we use that point with the point A in the two point formula:



I hope I didn't make a math error up there  so you find the equation of your three medians, just like above, then use simultaneous equations to find the POI. Note that you can actually be clever - You only need two medians to find the POI, you can guarantee that the third will pass through the same point as well

As for the second question, it is taking the medians from the last question (but as intervals, not lines) and asking you to divide them in a given ratio. That's with the formula on your reference sheet, \(\left(\frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n}\right)\) - The coordinates of the point that does so will match the POI you found in the previous question. Thus, you will prove the result given in your solution

If you get stuck, post your working and we'll check you are on the right track and help you continue!

[jamonwindeyer]

Could i please get sone help on part i and ii plz
Thank you
(Image removed from quote.)

I'll add another hint too - Assign \(\angle LAQ=\alpha\) (or some other pronumeral). Then start labelling other angles as \(\alpha\) or \(90-\alpha\) - You should start to see the picture coming together

As a general rule, when in doubt, assign letters to shit