Hi,

I'm stuck on this question completely, wondernig if you can help me out.

"The cake is designed in the shape of two touching circles with two tangents to the outer circles. The circles touch at the origin, whilst the tangents meet the y axis at (0,6).
The outer circle interesects the y axis at (0,4).
The equation of the inner circle is given by x^2+y^2-3y = 0 (^2 = squared).

Find the total "length" of the icing needed to decorate the cake (ignore thickness)."

It's a 9 mark question and I have no idea where to start

2. Originally Posted by enji333
Hi,

I'm stuck on this question completely, wondernig if you can help me out.

"The cake is designed in the shape of two touching circles with two tangents to the outer circles. The circles touch at the origin, whilst the tangents meet the y axis at (0,6).
The outer circle interesects the y axis at (0,4).
The equation of the inner circle is given by x^2+y^2-3y = 0 (^2 = squared).

Find the total "length" of the icing needed to decorate the cake (ignore thickness)."

It's a 9 mark question and I have no idea where to start
Can you provide an image? I cannot envision this cake.

3. Is there a picture given? I am having a bit of a hard time figuring out how these circles are supposed to be arranged.

4. Ok

Originally Posted by enji333
Hi,

I'm stuck on this question completely, wondernig if you can help me out.

"The cake is designed in the shape of two touching circles with two tangents to the outer circles. The circles touch at the origin, whilst the tangents meet the y axis at (0,6).
The outer circle interesects the y axis at (0,4).
The equation of the inner circle is given by x^2+y^2-3y = 0 (^2 = squared).

Find the total "length" of the icing needed to decorate the cake (ignore thickness)."

It's a 9 mark question and I have no idea where to start
They are concentric circles...draw them and use the info given to solve it!

5. That's the drawing of it.

I still am a bit confused about where to start though.. I know it's not the hardest question, but i'm getting thrown by the fact it's 9 marks and i have no idea where to start.

6. Does the icing go only along the edges? (where the lines are in the graph)

7. "The thick lines shown in the template indicates to the confectioner ... where to put the black icing".

i guess it's just working out the length of the black lines??

but i did wonder if it meant if the icing went inside the black lines at the start, that's possibly what threw me?

8. Okay, so this is asking for the length of all the black lines then.

So, the perimeters of the two circles, and the lengths of the tangent lines.

The inner circle is given by
$\displaystyle x^2+y^2-3y=0$

And we know that perimeter is $\displaystyle 2*r*\pi$

So we can find the value of r, the radius, by putting the circle into standard form of [tex](x-h)^2+(y-k)^2 = r^2:
$\displaystyle x^2+(y-1.5)^2 -\frac 94=0$

$\displaystyle x^2+(y-1.5)^2 =\frac 94$

So the radius is $\displaystyle \sqrt{\frac 94} = \frac 32 = 1.5$

Then we plug this into our perimeter formula:
$\displaystyle p = 2*r*\pi = 2*1.5*\pi = 3\pi$

------

Now we find the perimeter of the outer circle We know that it intersects the y-axis at (0,0) and (0,4) so we can see that it's diameter is 4 units long. Since 2*radius = diameter, we can plug this directly into the perimeter formula:

$\displaystyle p = 2*r*\pi = d*\pi = 4*\pi$

------

For the tangents, I'm not sure how to approach that. I think I could solve it with calculus, what level of math are you in? Have you been taught a better way to find where a tangent line touches a circle if you are given a point along that tangent line?

9. i've been taught by using pythagoras

10. Originally Posted by enji333
i've been taught by using pythagoras
Can you give the endpoints of one of the tangent lines then? (also, wouldn't mind seeing how you do it ^_^)

With that information, we could finish the problem.

11. I would do it like..

work out the radius of the smaller circle, then use that to find out the point where the red line touches the y axis. Then, since it is a tangent, work out the equation of the bigger circle, then work out the radius of that to find out the end point of the red line. With that, I could work out the distance of the bottom side of the triangle (triangle from red line to point (0,6)). Work out the length of the line from where the red line touches the y axis to point (0,6) and use pythagoras to find out the length of the hypotenuse.

I could do it like that right?

edit: i done that drawing in paint, so it might look a bit 'off'.

12. Originally Posted by enji333
I would do it like..

work out the radius of the smaller circle, then use that to find out the point where the red line touches the y axis. Then, since it is a tangent, work out the equation of the bigger circle, then work out the radius of that to find out the end point of the red line. With that, I could work out the distance of the bottom side of the triangle (triangle from red line to point (0,6)). Work out the length of the line from where the red line touches the y axis to point (0,6) and use pythagoras to find out the length of the hypotenuse.

I could do it like that right?

edit: i done that drawing in paint, so it might look a bit 'off'.
I may be confused, but if the tangent lines are taken along the large circle, then wouldn't they be the same no matter what size the small circle is (or even if it didn't exist at all)? I don't understand how it can be used to calculate a tangent line for the large circle.

13. then i'm not sure either.

The way i learned it was like this:

If A(0,4), and the circle equation X^2 + y^2 = 4 (^2 = squared, still havn't worked out how you do that yet??).

OA = 4 units.
OB = 2 units (radius of circle).

then by using pythagoras, AB^2 = 4^2 - 2^2

that's how I got taught to work out lengths of tangents.

can i apply that here somewhere?

14. Originally Posted by enji333
then i'm not sure either.

The way i learned it was like this:

If A(0,4), and the circle equation X^2 + y^2 = 4 (^2 = squared, still havn't worked out how you do that yet??).

OA = 4 units.
OB = 2 units (radius of circle).

then by using pythagoras, AB^2 = 4^2 - 2^2

that's how I got taught to work out lengths of tangents.

can i apply that here somewhere?
I don't think we can, sorry.

Since we don't know the angle AOB or the (x,y) coordinates of B, or the slope of AB, then I don't think we can figure that out.

Perhaps we are misunderstanding the problem? (What level of math are you in? I can try to solve it with calculus if you are at that level)

Edit: I just thought of an idea that might work using this method, but I have to go to class now, I'll take a look at it when I get back.

15. scottish education system is different from everywhere else

hmm, i've been doing stuff like differentiation, further differentiation, circles, further integration, optimisation, double angle formulae, vectors, logs, multiple angle equations..

in the circle topic, equations of a circle, using the distance formula, intersection at a line and a circle.........!

....could i work out the equation of the tangent, work out the equation of the outer circle, find the intersection of the tangent and the outer circle, and use the distance formula to work out the length?

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