Question
The difference between outside and inside surface areas of cylindrical metallic pipe $14\ cm$ long is $44m^2$. If the pipe is made of $99cm^3$ of metal, find the outer and inner radii of the pipe.
✓
Answer
Let inner radius of pipe be $r_1$
Radius of outer cylinder be $r_2$
Length of cylinder $(h) = 14cm.$
Surface area of hollow cylinder $(r_2- r_1)=2\pi$
Given surface area of cylinder $= 44m^2$
$2\pi\text{Rh}-2\pi\text{r}\text{h}=44$
$2\pi\text{h}(\text{R}-\text{r})=44$
$2\times\frac{22}{7}\times14(\text{R}-\text{r})=44$
$\text{R}-\text{r}=\frac{44\times7}{2\times22\times14}$
$\text{R}-\text{r}=\frac{1}{2}\ ...(1)$
Now,
Volume of pipe $= 99cm^3$
$\pi\text{R}^2\text{h}-\pi_2\text{r}^2\text{h}=99$
$\pi\text{h}(\text{R}^2-\text{r}^2)=99$
$(\text{R}^2-\text{r}^2)=\frac{99}{\pi\times\text{h}}$
$(\text{R}+\text{r})(\text{R}-\text{r})=\frac{99\times7}{22\times14}$
$(\text{R}+\text{r})(\text{R}-\text{r})=\frac{99}{22\times2}$
Putting value of $\text{R}-\text{r}=\frac{1}{2}$
$(\text{R}+\text{r})\frac{1}{2}=\frac{99}{22\times2}$
$(\text{R}+\text{r})=\frac{9}{2}\ ...(2)$
Adding equation $(1)$ and $(2)$
$\text{R}-\text{r}=\frac{1}{2}$
$\text{R}+\text{r}=\frac{9}{2}$
$2\text{R}=5$
$\text{R}=2.5$
And $\text{r}=\frac{9}{2}-\frac{5}{2}$
$\text{r}=2\text{cm}$
Radius of outer cylinder be $r_2$
Length of cylinder $(h) = 14cm.$
Surface area of hollow cylinder $(r_2- r_1)=2\pi$
Given surface area of cylinder $= 44m^2$
$2\pi\text{Rh}-2\pi\text{r}\text{h}=44$
$2\pi\text{h}(\text{R}-\text{r})=44$
$2\times\frac{22}{7}\times14(\text{R}-\text{r})=44$
$\text{R}-\text{r}=\frac{44\times7}{2\times22\times14}$
$\text{R}-\text{r}=\frac{1}{2}\ ...(1)$
Now,
Volume of pipe $= 99cm^3$
$\pi\text{R}^2\text{h}-\pi_2\text{r}^2\text{h}=99$
$\pi\text{h}(\text{R}^2-\text{r}^2)=99$
$(\text{R}^2-\text{r}^2)=\frac{99}{\pi\times\text{h}}$
$(\text{R}+\text{r})(\text{R}-\text{r})=\frac{99\times7}{22\times14}$
$(\text{R}+\text{r})(\text{R}-\text{r})=\frac{99}{22\times2}$
Putting value of $\text{R}-\text{r}=\frac{1}{2}$
$(\text{R}+\text{r})\frac{1}{2}=\frac{99}{22\times2}$
$(\text{R}+\text{r})=\frac{9}{2}\ ...(2)$
Adding equation $(1)$ and $(2)$
$\text{R}-\text{r}=\frac{1}{2}$
$\text{R}+\text{r}=\frac{9}{2}$
$2\text{R}=5$
$\text{R}=2.5$
And $\text{r}=\frac{9}{2}-\frac{5}{2}$
$\text{r}=2\text{cm}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Find the value of $x$ in the following:
$\tan\text{x}=\sin45^\circ\cos45^\circ+\sin30^\circ$
→$\tan\text{x}=\sin45^\circ\cos45^\circ+\sin30^\circ$
For what value of $n,$ the $n^{th}$ terms of the arithmetic progressions $63, 65, 67, ...$ and $3, 10, 17, ...$ are equal?
→Prove that the points $(0, 0), (5, 5)$ and $(-5, 5)$ are the vertices of a right isosceles triangle.
→→
A solid is composed of a cylinder with hemispherical ends. If the length of the whole solid is $108\ cm$ and the diameter of the cylinder is $36\ cm,$ find the cost of polishing the surface at the rate of $7$ paise per $cm^2$. $ (\text{Use}\ \pi=3.1416)$
→If $\triangle\text{ABC}$ and $\triangle\text{AMP}$ are two right triangles, right angled at $B$ and $M$ respectively such that $\angle\text{MAP}=\angle\text{BAC.}$ Prove that
→- $\triangle\text{ABC}\sim\triangle\text{AMP}$
- $\frac{\text{CA}}{\text{PA}}=\frac{\text{BC}}{\text{MP}}$
The surface area of a sphere is the same as the curved surface area of a cone having the radius of the base as $120\ cm$ and height $160\ cm$. Find the radius of the sphere.
→Prove that $\frac{{\sin \theta - \cos \theta + 1}}{{\sin \theta + \cos \theta - 1}} = \frac{1}{{\sec \theta - \tan \theta }}$, using identity $\sec^2\theta=1+ \tan^2\theta$.
→If $1+\sqrt{2}$ is a root of a quadratic equation will rational coefficients, write its other root.
→From the top of a building $AB, 60m$ high, the angles of depression of the top and bottom of a vertical lamp post $CD$ are observed to be $30^\circ $ and $60^\circ $ respectively. Find
→- The horizontal distance between $AB$ and $CD.$
- The height of the lamp post.
- The difference between the heights of the building and the lamp post.