Two cylinders have equal volume.Their heights are in the ratio 1:2.Find ratio of their radii? (9 std maths)

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Answer 1 · 2017-12-21T12:31:26

Solve for one height value equaling a multiple of the other, then substitute it into the formula for a cylinder's volume and use algebra to obtain

$\frac{r_{1}}{r_{2}} = \sqrt{2}$ $r_1/r_2 = sqrt(2)$ .

We have that the formula for a cylinder's volume is

$V_{cylinder} = π r^{2} h$ $V_("cylinder") = pi r^2 h$

Where $r$ $r$ is the radius and $h$ $h$ is the height. The given height ratio is

$\frac{h_{1}}{h_{2}} = \frac{1}{2}$ $h_1/h_2 = 1/2$

Where $h_{1}$ $h_1$ represents the height of the first cylinder, and $h_{2}$ $h_2$ represents that of second cylinder. We could solve for

$h_{1} = \frac{h_{2}}{2}$ $h_1 = (h_2)/2$

or

$h_{2} = 2 h_{1}$ $h_2 = 2 h_1$

both of which we can use, if we were to consider the ratio between their volumes being equal:

$\frac{π {(r_{1})}_{1}^{2} h_{1}}{π {(r_{2})}_{2}^{2} h_{2}} = 1$ $(pi (r_1)^2 h_1)/(pi (r_2)^2 h_2) = 1$

We could multiply the volume of the second cylinder to both sides to get

$π {(r_{1})}_{1}^{2} h_{1} = π {(r_{2})}_{2}^{2} h_{2}$ $pi (r_1)^2 h_1 = pi (r_2)^2 h_2$

Now, let's see, what can we do? We can substitute $h_{2} = 2 h_{1}$ $h_2 = 2 h_1$ :

$π {(r_{1})}_{1}^{2} h_{1} = π {(r_{2})}_{2}^{2} 2 h_{1}$ $pi (r_1)^2 h_1 = pi (r_2)^2 2 h_1$

It seems that $π$ $pi$ and $h_{1}$ $h_1$ both cancel out:

${(r_{1})}_{1}^{2} = 2 {(r_{2})}_{2}^{2}$ $(r_1)^2 = 2(r_2)^2$

Let's divide by ${(r_{2})}_{2}^{2}$ $(r_2)^2$ :

$\frac{{(r_{1})}_{1}^{2}}{{(r_{2})}_{2}^{2}} = 2$ $((r_1)^2)/((r_2)^2) = 2$

And take the square root:

$\frac{r_{1}}{r_{2}} = \sqrt{2}$ $r_1/r_2 = sqrt(2)$

We have just solved for the ratio between the radii, $\sqrt{2}$ $sqrt(2)$ .