Question

Can someone please help me in detail if two spherical pieces of cookie dough have radii of 3cm and 5cm. The pieces are combined to form one large spherical piece of dough. What is the approx radius of the new sphere of dough?
Ratio of two volume cylinders (Geometry)

If two similar cylinders have heights of 75cm and 25cm. What is the ratio of the volume of the larger cylinder to the volume of the smaller cylinder? Can someone please explain in detail. Thank you!

Answer 1

Volume of a sphere:
$V = \frac{4}{3}\pi r^3$
Combining dough is like combining volume. So we get
$V_3=\frac{4}{3}\pi r_3^3=\frac{4}{3}\pi r_1^3 + \frac{4}{3}\pi r_2^3 = \frac{4}{3}\pi (r_1^3+r_2^3)$
Where $V_3$ is the combined volume of dough of radii $r_1$ and $r_2$
The new radius is therefore
$r_3^3=r_1^3+r_2^3$
Substituting values we get
$r_3^3=3^3+5^3=27+125=152$
$r_3=\sqrt[3] 152 \approx 5.34$

So the new dough is 5.34 cm, just a bit larger.

Cylinders:
$V=\pi r^2 h$
So volume is directly proportional to height, if radius is constant. So ratios stay the same. 75/25=3 and the volume is also 3 times larger (ratio 3:1).

Answer 2

The ratio of the lengths is 75 : 25 or 3 : 1 

The ratio of and area is the square of this: ie 9 : 1 or 75^2 : 25^2
(In working out the area the radius is squared)

The ratio of the volumes is the cube of this: ie 27 : 1 or 75^3 : 25^3
(In working out the volume the radius is cubed)

Hopefully this explains it