Leaving cert question

confused_help

Hi we were set work to do over the Summer but I am stuck on a problem. The surface area of a sphere is 4pir^2 and its volume is 4/3pir^3, where r is the radius of the sphere. Show that if we have two spheres of radii r1 and r2, respectively, then the ration of the surface areas can be written as A1/A2 = (V1/V2)^2/3. Sorry its hard to write maths symbols on keyboard

theLuggage

Have you started subbing the area and volume formulae into a1/a2 = (v1/v2)^2/3? Thats where to start anyway and go from there. Hope that helps you to get started.

confused_help

Yes, I tried filling it in and using indices rules but still cannot prove it.

Michael Collins

OK let's say the first sphere has radius [latex] r_1 [/latex], and the second sphere has radius [latex] r_2 [/latex]. Now, using the formula you gave, the volumes of each of these are

[latex] \displaystyle V_1 = \frac{4}{3}\pi r_1^3 \hbox{ , for sphere 1}[/latex]

[latex] \displaystyle V_2 = \frac{4}{3}\pi r_2^3 \hbox{ , for sphere 2.} [/latex]

Similarly, again using your formula, the surface areas are

[latex] \displaystyle A_1 = 4\pi r_1^2 \hbox{ , for sphere 1}[/latex]

[latex] \displaystyle A_2 = 4\pi r_2^2 \hbox{ , for sphere 2.} [/latex]

So first of all, have a go at getting [latex] \displaystyle \frac{A_1}{A_2} [/latex], by dividing the above two formula for surface area, and cancelling the terms common to top and bottom.

Now get [latex] \displaystyle \frac{V_1}{V_2} [/latex], by dividing the top two formula, and again cancelling.

When you have these, post both results back here and we can continue from there.

confused_help

Got it. Thanks!