Math Is Fun Forum

This time, I was confident:

24. Under a certain enlargement (3,4) → (0,-2) and (-5,2) → (4,-1). Find the image of (-3,-8) under this enlargement.

Now, I'm a little rusty, so I looked up the form of the matrix for the "scaling" transformation and found:

Gives a scaling by k in all directions.

I tried this, but my answer was wrong. Then I thought: 'hey! maybe the scaling isn't the same in both directions,' so I tried instead:

Still wrong. Then I realised: 'oh! the centre may not be (0,0),' so I tried:

Still wrong. Then I thought: 'well, what if the x and y coordinates of the centre of enlargement are different?' So that was when I had my last try:

Running the numbers gives me:

But:

Does not give the right answer.

So my question for the good folks at mathisfun is: who's wrong - me, or the book?

Hey, me again.

I can't decide whether this is something I've never done, or something I've forgotten, so let's just dive straight in with the question.

22. A plane intersects a right-circular cone and is parallel to the base of the cone. If V is the vertex of the cone and VA the altitude, then let B be the point where the plane cuts VA. It it given that VB:BA = 3:2. Are the following statements true or false?
(i) The ratio of the volume of the small cone (the upper part of the section) to the volume of the whole cone is 8:27.
(ii) The ratio of the surface area of the small cone to the surface area of the whole cone is 9:25.
(iii) The ratio of the volume of the small cone to the volume of the frustum of the cone (the lower part of the section) is 27:125.

Now, I dare not even try to draw this, but I'm pretty confident I do know what the question means. You've got your standard cone, where the "top" (here, I believe, referred to as the 'vertex', though I believe 'apex' is also used) of the cone is directly above the centre of the base. The top of the cone is labelled V and we extend an imaginary line from the top of the cone to the centre of the base, which we'll call A. This line is VA - the "altitude" (i.e. the height of the cone, i.e. the distance from the base of the cone to the point at the top). Now, at a point B along this line a plane cuts the cone into two parts. So we're left with a "small cone" and a "frustum". Now, the ratio of the height of the small cone (VB) to the height of the frustum (BA) is 3:2. This means, then, that the height of the small cone (VB) is 3/5 that of the whole cone (VA) - am I right?

Now, what else do we know. Well, we know the volume of a cone is:

Where r is the radius of the base and h the height of the cone. So we know that for the whole cone h is VA and we know that for the small cone is 3/5VA.

What I don't know is what either of the radii are and I don't know how to work out in what proportion the radii are, which I imagine is important.

I'm also not really sure how to proceed on the basis of this information. What I jotted down is this:

Which I can kind of see makes (i) false (book says it is indeed false), but I don't really understand why - I couldn't explain it to anybody if they asked. I then looked at the next one and, yeah, apparently it's true, but I don't really know why it's true. One of the things that's stalling me is the fact that I'm not sure whether I'm supposed to worry about the radii and - if I am - how I can work out what ratio they're in

Hey ho, I've come up against a question on something that was never actually on my syllabus, so I don't really know where to begin. I googled around looking for things on modular arithmetic and the modulo operation, but I can't apply it to this question, because I can't really parse the question, I don't understand what it wants.

Here it is:

21. Write down all the solutions of 3x(3x + 4) = 0
(i) in arithmetic modulo six; (ii) in arithmetic modulo five.

Obviously the "basic" solutions are x = 0, x = -4/3, but I don't really know what to do with those numbers

The last question on integration by substitution in my books asks for a proof. I think I've got it, but obviously the proof's not given in the back, so I could be talking absolute nonsense. I was wondering, therefore, if someone could give it a look over and tell me if I'm right in my approach.

Here's the question:

My approach was as follows: