I need help with these cylinders..?

Question

Given 2 cylinders with :
ratio of the
circumference - 3/4
Lateral Face - 9/16
the diameter of the base - 3/4
volume - 27/64

a. If the Surface Area of the small cylinder is 243cm^3, what's the Surface Area of the larger cylinder?

b. If the volume of the larger is 144cm^3, wat's the volume of the smaller cylinder?

I can't seem to solve this so pls help me by giving me ur solution. that can rili help me a lot. Thank you guys!

Answer 1

Is there a hidden assumption here that the cylinders are similar? i.e. given the ratio of their circumferences is 3/4, all corresponding lengths are in the ratio 3/4. Of course, that has to be true of the circumference and the radius, but in this case it's also true of the heights. I'll use lower case r, h for the radius and height of the smaller cylinder, and R, H for the larger one.
The lateral face areas are
2πrh and 2πRH
Now r = (3/4)R (same ratio as diameters, since radius is just half the diameter), and so
2πrh = 2π(3/4)Rh.
Also we're told the smaller area is 9/16 of the larger one, and so
2π(3/4)Rh = (9/16)*2πRH
which leads to
(3/4)h = (9/16)H when we cancel 2πR.
Also divide both sides by 3/4 and we find
h = (3/4)H
which shows the cylinders are similar -- i.e. every pair of corresponding lengths are in the ratio 3/4.

So that means every pair of corresponding areas are in the ratio 9/16 (from squaring 3/4) and volumes are in the ratio 27/64.

But that's just checking up -- answering the question is easier than that:

Using a and A for the areas,
a/A = 9/16
A/a = 16/9
A/243 = 16/9 [by the way, it's cm^2 for area, not cm^3]
A = 243*16/9
... = 432
Larger surface area is 432 cm^2

Volumes:
v/V = 27/64
v/144 = 27/64
v = 144*27/64
...= 60.75
smaller volume is 60.75 cm^3

Answer 2

As far as I can see there is nothing tricky here. The smaller number in each of these ratios refers to the smaller of the cylinders and the larger to the larger.

a. The ratio of the surface areas is 9/16 (I assume you are not including the top and bottom of the cylinders) so:

9/16 = 243/A where A is the area of the lateral face of the larger cylinder
A = 16*243/9 = 432

b. The ratio of the volumes is 27/64 so:
27/64 = V/144 where V is the volume of the smaller cylinder
V = 144*27/64 = 60.75

Circumference is 2piR where R is the radius of the cylinder. So if the ratio of the circumferences is 3/4 then the ratio of the radii is also 3/4 and so for the diameters since the diameter is just twice the radius.
The area of the lateral face is just the circumference times the height of the cylinder or 2piRH. So the ratio of the areas for two cylinders is just rh/RH. And since r/R is 3/4 (see above) then the ratio of the areas is 3h/4H. Since this is given to be 9/16, we have 9/16 = 3h/4H and the ratio of the heights of the cylinders is 3/4.
Volume is just the area of the base times the height or piR^2H so the ratio of two volumes is r^2h/R^2H. Rewrite this as (r/R)^2(h/H) and substitute from the others to get (3/4)^2(3/4) and the ratio of volumes is 27/64 - just like the problem states.

Answer 3

Consider this:

The ratios of the linear measurements, that is the straight line

lengths of radii and height are in the ratio of 3:4. The ratio of

areas is in the ratio of the squares of corresponding sides,

that is in this case, in the ratio of 3^2 to 4^2, which is 9:16,

and the ratio of the volumes of similar shapes is in the ratio of

the cubes of corresponding sides, in this case in the ratio 3^3

to 4^3, that is 27 :64. So, to answer your question,

a] S.A. of smaller cylinder is 243. Let the S.A. of the larger

cylinder be X, so,

9/16 = 243/X, so

9X = 243 x 16, so,

9X = 3888, so

X = 3888/9, so

X = 432 sq cm.

b] Let the volume of the larger cylinder be V, so

144/V = 27/64,so

64V = 144 x 27, so

64V = 3888, so

V = 3888/64, so

V = 60.75. Hope this helps, Twiggy.