Question

If a cone is cut into two parts by a horizontal plane through the mid-point of its axis,the ratio of the volumes of upper and lower part is

• A. 1:2
• B. 2:1
• C. 1:7
• D. 1:8

Answer 1

The correct option is C

1:7

$\because △PDC\sim △PBA\left(AAaxiom\right)and{O}^{\prime }ismidpointofPO\therefore O^{\prime }C\parallel OA$

$and{O}^{\prime }C=\frac{1}{2}OA\Rightarrow r_2=\frac{1}{2}{r}_{1}andP{O}^{\prime }=\frac{1}{2}PO\Rightarrow h_2=\frac{1}{2}{h}_{1}Let{V}_{1}and{V}_{2}bethevolumesofconePBAandPDCrespectively{V}_1=\frac{1}{3}\pi r_1^2{h}_1{V}_2=\frac{1}{3}\pi r_2^2{h}_2=\frac{1}{3}\pi {\left(\frac{r_1}{2}\right)}^2\left(\frac{h_1}{2}\right)=\frac{1}{3}\pi \frac{r_1^2{h}_1}{4\times 2}=\frac{1}{8\times 3}\pi r_1^2{h}_1\therefore VolumeofDCAB=\frac{1}{3}\pi r_1^2{h}_1-\frac{1}{24}\pi r_1^2{h}_1=\frac{7}{24}\pi r_1^2{h}_1\therefore Ratiobetweenupperpartandlowerpart=\frac{1}{24}\pi r_1^2{h}_1:\frac{7}{24}\pi r_1^2{h}_1=\frac{1}{24}:\frac{7}{24}=1:7$