Question

The volume of the conical tent is $1232\ {{\text{m}}^{\text{3}}}$ and the area of its base is $154\ {{\text{m}}^{\text{2}}}$. Find length of the canvas required to cover this conical tent if the canvas is 2 m in width is:
(A) $270\ m$
(B) $272\ m$
(C) $276m$
(D) $275\ m$

Answer 1

Hint: Obtain the values of radius and height of the conical tent by using the area of the base and volume. Use the value of radius and height to calculate the lateral area of the conical tent.

Complete step by step solution:
We know from the question that the volume of the conical tent is $V=1232\ {{\text{m}}^{\text{3}}}$ and the area of base of the conical tent is ${{A}_{b}}=154\ {{\text{m}}^{2}}$.
We know that the base of the conical shape is a circle. Let the radius of the base of the conical tent be $r$, then the expression of the base conical tent is,
$\Rightarrow {{A}_{b}}=\pi {{r}^{2}}$
Now we substitute the value of ${{A}_{b}}$ as $154\ {{\text{m}}^{\text{2}}}$ in the above expression, we get,
$\Rightarrow 154\ {{\text{m}}^{\text{2}}}=\dfrac{22}{7}\times {{r}^{2}}$
$\Rightarrow \text{ }{{r}^{2}}=49\ {{\text{m}}^{\text{2}}}$
\[\Rightarrow \text{ }r=7\ \text{m}\]
Let the height of the conical tent be $h$ and it can be calculated by using the expression of the volume of the cone, which is,
$V=\dfrac{1}{3}\pi {{r}^{2}}h$
Now we substitute the values of $V$ as $1232\ {{\text{m}}^{3}}$ and $r$ as $7\ \text{m}$ in the above expression, we get,
\[1232\ {{\text{m}}^{\text{3}}}=\dfrac{1}{3}\times \dfrac{22}{7}\times {{\left( 7\ \text{m} \right)}^{2}}h\]
\[\Rightarrow h=\dfrac{1232\times 3\times 7}{7\times 22}\]
\[\Rightarrow h=24\ \text{m}\]
Now we let the slant height of the conical tent be $l$ and its expression is,
$l=\sqrt{{{r}^{2}}+{{h}^{2}}}$
Now we substitute the values of $h$ as $\text{24}\ \text{m}$ and $r$ as $7\ \text{m}$ in the above expression, we get,
\[\Rightarrow l=\sqrt{{{\left( 7\ \text{m} \right)}^{2}}+{{\left( 24\ \text{m} \right)}^{2}}}\]
\[\text{ }=\sqrt{49\ {{\text{m}}^{2}}+576\ {{\text{m}}^{\text{2}}}}\]
\[=\sqrt{625\ {{\text{m}}^{\text{2}}}}\]
After simplifying the above equation, the value of slant height is $25\ \text{m}$.
We know that the expression of the lateral surface of the cone is,
$S=\pi rl$
Now we substitute the values of $l$ as $25\ \text{m}$ and $r$ as $7\ \text{m}$ in the above expression, we get,
$S=\dfrac{22}{7}\times 7\times 25$
$=550\ {{\text{m}}^{\text{2}}}$
Now the area of the canvas needed to make the cover of the conical tent will be equal to the lateral surface area of the conical tent.
Let the area of the canvas be ${{A}_{c}}$ and it can be expressed as,
$A={{l}_{c}}\times {{w}_{c}}$
Here, the length of the canvas is ${{l}_{c}}$ and width of the canvas is ${{w}_{c}}$.
We know that, $A=S$. So we get,
$S={{l}_{c}}\times {{w}_{c}}$
Now we substitute the value of $S$ as $550\ {{\text{m}}^{\text{2}}}$ and ${{w}_{c}}$ as $2\ \text{m}$.
$550\ {{\text{m}}^{\text{2}}}={{l}_{c}}\times 2\ \text{m}$
${{l}_{c}}=\dfrac{550\ {{\text{m}}^{\text{2}}}}{2\ \text{m}}$
$=275\ \text{m}$
Hence, the length of the canvas required to cover this conical tent if the canvas is \[2\text{ }m\] in width is $275\ \text{m}$.

Option (D) is correct.

Note: A cone is a three-dimensional shape that has a circular base and the curved side meets up at a single point apex or vertex. The lateral surface area of the cone is the area of the side surface or lateral surface only, which is expressed as $S=\pi rl$, here $r$ is the radius and $l$ is the slant height.