Two ceramic cylinders have the following properties. Cylinder $\mathrm{A}$ has radius $\mathrm{R}$ and height $2 \mathrm{H}$, while Cylinder $B$ has radius $3 R$ and height $H / 4$. Their resistances are measured, and it is found that $R_{A} / R_{B}=2$. What is the ratio of the resistivities $\rho_{\mathrm{A}} / \rho_{\mathrm{B}}$ of the ceramics from which the cylinders are made?
A) $1 / 36$
B) $16 / 9$
C) $9 / 8$
D) $9 / 4$
E) 36
F) $1 / 18$

Question

Two  ceramic cylinders have the following properties. Cylinder $\mathrm{A}$ has radius $\mathrm{R}$ and height $2 \mathrm{H}$, while Cylinder $B$ has radius $3 R$ and height $H / 4$. Their resistances are measured, and it is found that $R_{A} / R_{B}=2$. What is the ratio of the resistivities $\rho_{\mathrm{A}} / \rho_{\mathrm{B}}$ of the ceramics from which the cylinders are made?
A) $1 / 36$
B) $16 / 9$
C) $9 / 8$
D) $9 / 4$
E) 36
F) $1 / 18$

Quizplus · Accepted Answer

The resistance of a cylinder is given by $R=\rho \frac{L}{A}$, where $\rho$ is the resistivity, $L$ is the length, and $A$ is the cross-sectional area. For cylinder A, $L=2H$ and $A=\pi R^2$, while for cylinder B, $L=H/4$ and $A=\pi (3R)^2$. Therefore,$$\frac{R_A}{R_B}=\frac{\rho_A (2H)}{\pi R^2}\cdot\frac{\pi (3R)^2}{\rho_B (H/4)}=\frac{24\rho_A}{\rho_B}$$Since $R_A/R_B=2$, we have $\rho_A/\rho_B=1/12$.

Two Ceramic Cylinders Have the Following Properties A\mathrm{A}A Has Radius R\mathrm{R}R

Two Ceramic Cylinders Have the Following Properties $\mathrm{A}$ Has Radius $\mathrm{R}$