area of right circular cylinder

Question

A 15m tall pillar consists of two parts. The diameters of the base at the bottom and the top-most circular disc are in the ratio 4:3. If the total area of the curved surfaces of both the parts of the pillar is 110 m2. Find the radius of each portion of the pillar.

 

Two parts meaning a smaller diameter cylinder on top of a larger diameter cylinder and the height of bottom and top-most circular disc is 5m and 10m respectively.

Answer 1

So we've got two cylinders, one on top of another. There's a short fat one on the bottom that has a height of 5, and a tall narrow one on the top that has a height of 10. We don't know the radii yet.

 

The curved surface area of a cylinder is equal to the perimeter of the base times the height.

 

Since we have two cylinders, one wide and one narrow, lets call them A_wide and A_narrow

 

A_wide = 2*pi*r_wide*height_wide

 

A_narrow = 2*pi*r_narrow*height_narrow

 

From the question we know that:

 

height_wide = 5 m

height_narrow = 10 m

A_wide + A_narrow = 110 m^2

4/3*r_narrow = r_wide

 

So let's start filling in some variables.

 

A_wide = 2*pi*(4/3*r_narrow)*5

 

                 = 40/3*pi*r_narrow

 

A_narrow = 2*pi*r_narrow*10

 

                    = 20*pi*r_narrow

 

Let's add the two equations together.

 

A_wide + A_narrow = 40/3*pi*r_narrow + 20*pi*r_narrow

 

110 = 40/3*pi*r_narrow + 60/3*pi*r_narrow

 

110 = 100/3*pi*r_narrow

 

33/10pi = r_narrow

 

From earlier we remember that:

 

r_wide = 4/3*r_narrow

 

        = 4/3*33/10pi

 

r_wide = 44/10pi

 

So the uppermost pillar (the short narrow one) has a radius of 33/(10*pi) m.

The bottom pillar has a radius of 44/(10*pi) m.

 

Check your work:

 

r_narrow = 33/(10pi) * (4/3) = 44/(10pi) = r_wide. Ratios are good.

 

A_wide = 2*pi*(44/10pi)*5 = 44

A_narrow = 2*pi*(33/10pi)*10 = 66

 

66 + 44 = 110. Area checks out.

 

**

 

Are you sure the question didn't say that the wide diameter is taller and the narrow diameter is shorter? The numbers are way prettier that way.

 

You wind up with 3/pi and 4/pi in that scenario since the areas end up summing to 110/3*pi*r_narrow, which cancels beautifully with your total curved surface area of 110. Feel like 110 was intentionally chosen as the total area for that reason. Just wondering..