Question

If the heights of two cones are in the ratio 1:4 and their diameter are in the ratio 4:5 what is the ratio of their volumes?

a. 1:4

b. 4:25

c. 4:23

d. 1:6

Answer 1

The ratio of their volume of cone is b)4:25
hope you 'll understand.............

Answer 2

The ratios of the volumes is b. 4 : 25

Step-by-step explanation:

The volume of the cone with height h, radius of the circular base r is given as:

$V=\frac{\pi r^{2} h}{3}$

Given that the heights are in the ratio 1:4, if $h_1$, $h_2$ are the heights of two cones, then  

$h_1 : h_2 = 1 : 4$

Also, the diameter ratio is given as 4 : 5  

If $D_1$ and $D_2$ are the diameters of the cones, then $D_1 : D_2 = 4 : 5$

Diameter = 2 $\times$ radius

$D_1 : D_2 = 4 : 5$

$2r_1 : 2r_2 =  4 : 5$

$r_1 : r_2 =\frac{4}{2} : \frac{5}{2}=2 : \frac{5}{2}$

Now, Ratio of the volumes of two cones when volume of first cone is V1 and Volume of second cone is V2 is

$V 1 : V 2=\frac{\pi r_{1}^{2} h_{1}}{3} : \frac{\pi r_{2}^{2} h_{2}}{3}$

$=\frac{\frac{\pi r_{1}^{2} h_{1}}{3}}{\frac{\pi r_{2}^{2} h_{2}}{3}}$

$=\frac{\frac{r_{1}^{2} h_{1}}{3}}{\frac{r_{2}^{2} h_{2}}{3}}$

$=\frac{r_{1}^{2} h_{1}}{r_{2}^{2} h_{2}}$

$=\left(\frac{r 1}{r 2}\right)^{2} \times \frac{h 1}{h 2}$

$=\left(\frac{2}{5 / 2}\right)^{2} \times \frac{1}{4}$

$=\left(\frac{4}{5}\right)^{2} \times \frac{1}{4}$

$=\frac{16}{25} \times \frac{1}{4}$

$V 1 : V 2=\frac{4}{25}$