How do I work out this Mathematics problem?

Question

Two cylinders are mathematically similar. The smaller cylinder has a height of 20cm, and the larger has a height of 30cm. I worked out that since the larger cylinder has a diameter of 12cm, then the smaller cylinder must have a diameter of 8cm. However, if it takes 5ml of paint to cover the small cylinder, how much paint would be needed to cover the large one? Also, both cylinders are made of the same material. The smaller one weighs 120g. How much does the larger one weigh?

Answer 1

You've got the ratios right to calculate the new dimensions of the smaller cylinder.

Now, to calculate how much paint (that's code word for SURFACE AREA) you need to use the formula for SA with the cylinder's dimensions.

When asking you to calculate WEIGHT, that also depends on surface area if the thing is hollow or volume if the cylinders are solid. Do you know whether they're supposed to be hollow or solid?

Let's pretend they are hollow, for a moment. If they are, then the only mass comes from their surface area. The smaller cylinder has a surface area of:
SA = 2 pi rsquared + 2 pi*r*h
SA = 2 pi (16) + 2 pi *4*20
SA = 32pi +160pi
SA = 192pi (you can convert pi to 3.14 if that's what you're doing in class right now) If you do, it's going to be roughly 603cm squared.

But, you know that it weighs 120 g, so that means 603 cm squared has a mass of 120g. If you divide 120 by 603, that means every cm squared has a mass of about 0.2g (all calculations approximate).

So now, you can figure out the surface area of the larger cylinder, then multiply that surface area by roughly 0.2 and you'll get its mass in grams.

The other thing you could do instead is work out both surface areas ahead of time and make a proportion like this:

SA of larger/SA of smaller = mass of larger/mass of smaller

and solve for the mass of the larger one.

This is, of course, assuming that the cylinders are hollow. If they're solid, you'll want to do pretty much the same thing, but using their volumes instead.

The first question with the paint will be done in the same way: you KNOW that it takes 5ml to cover the small one, so you can use the ratio of:

SA small/SA large = 5ml/?? to find out how much paint is required.

Answer 2

The area of a cylinder is 2*Area of the base + Area of the rectangle that wraps around it. You take 2*the base because there's a top and bottom lid.

Area of base = pi*r^2
Circumference = 2pi*r
Height = h
So, Area of the small cylinder = 2*pi*4^2 + 2pi*4*20 =
192pi cm^2
Area of the large cylinder = 2*pi*6^2 + 2pi*6*30 = 432pi cm^2

If it takes 5ml to cover 192pi cm^2, then it takes 432/192 *5ml = 11.25 ml to cover the larger cylinder

Next question:
Volume of a cylinder = pi*r^2*h
small cylinder = pi*4^2*20 = 320pi
large cylinder = pi*6^2*30 = 1080pi
If the small one weighs 120g, the large one weighs 1080/320 * 120 = 405g

In these problems, we leave pi alone since it will factor out in the end. No sense multiplying and dividing by pi.