Answer to Question #290743 in Math for Vin Rana

Question

Two cylindrical wood are similar in shape and have diameters 9mm and 27mm.

(a) find the ratio of their masses.

(b) if the mass of the bigger wood is 30kg,find the mass of the smaller wood.

Answer 1

(a)

$$V=\pi r^2 h$$

If these two cylinders are similar

$$r_1/r_2=h_1/h_2, \rho_1=\rho_2$$

Then

$$\dfrac{V_1}{V_2}=\dfrac{\pi r_1^2h_1 }{\pi r_2^2h_2}=(\dfrac{r_1}{r_2})^3=(\dfrac{d_1}{d_2})^3$$

$$\dfrac{m_1}{m_2}=\dfrac{\rho_1V_1}{\rho_2V_2}=\dfrac{V_1}{V_2}=(\dfrac{d_1}{d_2})^3$$

$$=(\dfrac{9\ mm}{27\ mm})^3=\dfrac{1}{27}$$

(b)

$$m_1=\dfrac{1}{27}m_2=\dfrac{1}{27}(30\ kg)=\dfrac{10}{9}\ kg$$

The mass of the smaller woods is $\dfrac{10}{9}$ kg.