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
"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?
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
And we know that perimeter is
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:
So the radius is
Then we plug this into our perimeter formula:
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:
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?
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'.
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?
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.
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?