Question

Find what length of canvas $1.5m$ in width is required to make a conical tent $48m$ in diameter and $7m$ in height? Given that $10\% $ of canvas is used in stitching. Also find cost of canvas at a rate of $Rs.24$ per meter.

Answer 1

Hint: Here we are asked to find the length of the canvas to make a conical tent of given width and diameter and height. For this, we first need to find the curved surface area of cone since the shape of the tent is cone. Then to find the cost of the canvas for that we will use the surface area that we have found earlier and also add ten percent of canvas to it as it is given that ten percent is needed for stitching. And finally, to find the length of the canvas by dividing the area by width.

Complete step by step answer:
Given:
Width of canvas is \[1.5{\text{ }}m\]
Diameter of tent to be made is $48m$
Hence the radius is \[\dfrac{D}{2}\]
\[= \dfrac{{48}}{2}\]
\[= 24\]
Height \[ = {\text{ }}7m\]
Slant height \[ = \sqrt {{r^2} + {h^2}} \]
Given $r = 24m,h = 7m$
\[= \sqrt {{{(24)}^2} + {{(7)}^2}} \]
\[= \sqrt {576 + 49} \]
\[= \sqrt {625} \]
\[= 25m\]
From our previous knowledge we know that curved surface area of the cone is \[\pi rl\]
\[= \dfrac{{22}}{7} \times 24 \times 25\]
\[= \dfrac{{13200}}{7}{m^2}\]
Hence, we found the area of the canvas as \[\dfrac{{13200}}{7}{m^2}\]
Hence the amount of canvas that will be required for stitching is given as \[10\% \]
\[= \dfrac{{13200}}{7} \times \dfrac{{10}}{{100}}\]
\[= \dfrac{{1320}}{7}{m^2}\]
Total canvas that is required is: curved surface area $ + $ extra canvas need for stitching
\[= \dfrac{{13200}}{7} + \dfrac{{1320}}{7}\]
\[= \dfrac{{14520}}{7}{m^2}\]
Length of canvas can be found by
\[= \dfrac{{\dfrac{{14520}}{7}}}{{\dfrac{3}{2}}}\]
\[= \dfrac{{9680}}{7}\]
\[= 1382.86m\]
Therefore, the length of canvas in meters is $1382.86m$
Full cost:
\[= \dfrac{{9680}}{7} \times Rs24\]
\[= Rs.33,188.4\]
Altogether the cost of canvas is \[Rs.33,188.4\]

Additional information:
A cone is a 3-dimensional geometric form that tapers effortlessly from a flat base to a point called the apex or vertex. A cone is shaped by using a set of line segments, half of-traces, or strains connecting a commonplace thing, the apex, to all the factors on a base this is in a plane that doesn't contain the apex.

Note:
We don’t have to get confused with slant height and height both are different. To find the required meter of canvas we always think of area formula for cone we have curved surface area so we will find it using that formula. Since the curved surface formula need slant height, we need to find them by using the data we have. We know that slant height requires radius and height radius is found from the given diameter using this we have found the value of slant height.