Grade 9 Geometry math question

[bladder23]

So the surface area of a cone is solved by this equation

pr2+prs (pi x radius squared) + (pi x radius x slant)

Now here is the question:


By what factor does the total surface area increase if the radius is tripled?


I would appreciate any help. Thanks

 

[oynaz]

Here is a tip: Isolate the radius in the equation. That should get you started (Actually, you have nearly solved the problem that way).

 

[Kyteland]

I don't know if there is a way to isolate the radius, because the slant is a function of the radius.

The area of a cone is pi*r^2 + pi*r*sqrt(r^2+h^2)
If you triple the radius you end up with 9*pi*r^2 + 3*pi*r*sqrt(9*r^2+h^2)

You can't reduce that ratio much

pi*r^2 + pi*r*sqrt(r^2+h^2) : 9*pi*r^2 + 3*pi*r*sqrt(9*r^2+h^2)
r^2 + r*sqrt(r^2+h^2) : 9*r^2 + 3*r*sqrt(9*r^2+h^2) // remove a factor of pi
r + sqrt(r^2+h^2) : 9*r + 3*sqrt(9*r^2+h^2) // remove a factor of r

That's about the best you can do unless you have more information.

 

[zerocool1]

Originally posted by: Kyteland
I don't know if there is a way to isolate the radius, because the slant is a function of the radius.

The area of a cone is pi*r^2 + pi*r*sqrt(r^2+h^2)
If you triple the radius you end up with 9*pi*r^2 + 3*pi*r*sqrt(9*r^2+h^2)

You can't reduce that ratio much

pi*r^2 + pi*r*sqrt(r^2+h^2) : 9*pi*r^2 + 3*pi*r*sqrt(9*r^2+h^2)
r^2 + r*sqrt(r^2+h^2) : 9*r^2 + 3*r*sqrt(9*r^2+h^2) // remove a factor of pi
r + sqrt(r^2+h^2) : 9*r + 3*sqrt(9*r^2+h^2) // remove a factor of r

That's about the best you can do unless you have more information.

how is slant a factor of the radius? assuming a right triangle nothing else matters. ceteris paribus...

 

[Jschmuck2]

...yes...

 

[Kyteland]

Originally posted by: zerocool1
how is slant a factor of the radius? assuming a right triangle nothing else matters. ceteris paribus...

The slant value is a function of both the height and the radius. In a cone there are only two surfaces. The surface area is determined by adding the two together. The base is easy, it's just a circle with radius r. That's the pi*r^2 term.

The other piece is the upper part of the cone. if you cut a straight line from the tip of the cone to the base and unfold it you end up with a partial circle, like the shape of pacman. The formula for the area of that shape is r*l/2, where r is the radius l is the length of the partial circumference. In the case of a right cone, that works out to be pi*r*sqrt(r^2+h^2). sqrt(r^2+h^2) is the hypotenuse of the right triangle formed by the base and the height of the cone.

So the formula for the surface area of a right cone works out to pi*r^2 + pi*r*sqrt(r^2+h^2), and theres no way to isolate the r term in that equation.

 

[bladder23]

I answered the question with an answer of

"A factor of 6"

Would I be correct then?

edit: oh noes, i think am wrong. the answer is 3

 

[Kyteland]

The answer is neither 6 nor 3. It is:

(r,h) = (r + sqrt(r^2+h^2)) / (9*r + 3*sqrt(9*r^2+h^2))
so
(1,1) = 7.657
(3,5) = 6.555
(3,1) = 8.790

Edit: unless there was more information you didn't include in your original description.