Answer

Verified

46.2k+ views

Hint: Here, we will use the formula of circumference \[2\pi r\] to find the radius of the circle. Then we will find the slant height \[l = \sqrt {{h^2} + {r^2}} \] to find the surface area of the canonical tent \[\pi rl\]. Then the surface area is divided by the width for the length of required canvas.

Given that the height \[h\] is 24 m.

Let the radius of the base of a conical tent is \[r\] meters and the slant height be \[l\] meters.

We know that the circumference of a circle is \[2\pi r\], where \[r\] is the radius of the circle.

Since we are given that the circumference of the conical tent is 44 m.

\[ \Rightarrow 2\pi r = 44\]

Dividing the above equation by \[2\pi \] on each of the sides, we get

\[

\Rightarrow \dfrac{{2\pi r}}{{2\pi }} = \dfrac{{44}}{{2\pi }} \\

\Rightarrow r = \dfrac{{44}}{{2\pi }} \\

\]

Substituting the value of \[\pi \] in the above equation and simplify it, we get

\[

\Rightarrow r = \dfrac{{44}}{{2 \times \dfrac{{22}}{7}}} \\

\Rightarrow r = \dfrac{{44 \times 7}}{{2 \times 22}} \\

\Rightarrow r = 7{\text{ m}} \\

\]

We will now find the slant height \[l\] by substituting the values of \[h\] and \[r\] in the equation \[l = \sqrt {{h^2} + {r^2}} \].

\[

\Rightarrow l = \sqrt {{7^2} + {{24}^2}} \\

\Rightarrow l = \sqrt {49 + 576} \\

\Rightarrow l = \sqrt {625} \\

\Rightarrow l = 25{\text{ m}} \\

\]

Substituting the above values of \[r\] and \[l\] in the surface area \[\pi rl\], we get

\[

{\text{Surface area}} = \pi rl \\

= \dfrac{{22}}{7} \times 7 \times 25 \\

= 22 \times 25 \\

= 550{\text{ }}{{\text{m}}^2} \\

\]

We know that the area of canvas is equal to the surface area of the cone.

Thus, the area of the canvas is 550 m\[^2\].

Using the width of the canvas is 2 m, we will now find the length of the canvas by dividing the area of canvas by 2.

\[

{\text{Length of canvas}} = \dfrac{{{\text{Area of canvas}}}}{2} \\

= \dfrac{{550}}{2} \\

= 275{\text{ m}} \\

\]

Thus, the length of canvas used in making the tent is 275 m.

Hence, the option B is correct.

Note: In this question, students should be familiar with formulae of circumference of a circle, \[2\pi r\] and the surface area of cone, \[\pi rl\]. We will use the equality of area of canvas and the surface area of the cone for the correct solution. We have to divide the area of the canvas by the width of the canvas to find the length of the canvas.

__Complete step-by-step solution__Given that the height \[h\] is 24 m.

Let the radius of the base of a conical tent is \[r\] meters and the slant height be \[l\] meters.

We know that the circumference of a circle is \[2\pi r\], where \[r\] is the radius of the circle.

Since we are given that the circumference of the conical tent is 44 m.

\[ \Rightarrow 2\pi r = 44\]

Dividing the above equation by \[2\pi \] on each of the sides, we get

\[

\Rightarrow \dfrac{{2\pi r}}{{2\pi }} = \dfrac{{44}}{{2\pi }} \\

\Rightarrow r = \dfrac{{44}}{{2\pi }} \\

\]

Substituting the value of \[\pi \] in the above equation and simplify it, we get

\[

\Rightarrow r = \dfrac{{44}}{{2 \times \dfrac{{22}}{7}}} \\

\Rightarrow r = \dfrac{{44 \times 7}}{{2 \times 22}} \\

\Rightarrow r = 7{\text{ m}} \\

\]

We will now find the slant height \[l\] by substituting the values of \[h\] and \[r\] in the equation \[l = \sqrt {{h^2} + {r^2}} \].

\[

\Rightarrow l = \sqrt {{7^2} + {{24}^2}} \\

\Rightarrow l = \sqrt {49 + 576} \\

\Rightarrow l = \sqrt {625} \\

\Rightarrow l = 25{\text{ m}} \\

\]

Substituting the above values of \[r\] and \[l\] in the surface area \[\pi rl\], we get

\[

{\text{Surface area}} = \pi rl \\

= \dfrac{{22}}{7} \times 7 \times 25 \\

= 22 \times 25 \\

= 550{\text{ }}{{\text{m}}^2} \\

\]

We know that the area of canvas is equal to the surface area of the cone.

Thus, the area of the canvas is 550 m\[^2\].

Using the width of the canvas is 2 m, we will now find the length of the canvas by dividing the area of canvas by 2.

\[

{\text{Length of canvas}} = \dfrac{{{\text{Area of canvas}}}}{2} \\

= \dfrac{{550}}{2} \\

= 275{\text{ m}} \\

\]

Thus, the length of canvas used in making the tent is 275 m.

Hence, the option B is correct.

Note: In this question, students should be familiar with formulae of circumference of a circle, \[2\pi r\] and the surface area of cone, \[\pi rl\]. We will use the equality of area of canvas and the surface area of the cone for the correct solution. We have to divide the area of the canvas by the width of the canvas to find the length of the canvas.

Recently Updated Pages

A hollow sphere of mass M and radius R is rotating class 1 physics JEE_Main

Two radioactive nuclei P and Q in a given sample decay class 1 physics JEE_Main

Let gx 1 + x x and fx left beginarray20c 1x 0 0x 0 class 12 maths JEE_Main

The number of ways in which 5 boys and 3 girls can-class-12-maths-JEE_Main

Find dfracddxleft left sin x rightlog x right A left class 12 maths JEE_Main

Distance of the point x1y1z1from the line fracx x2l class 12 maths JEE_Main

Other Pages

If the mass of the bob of a simple pendulum increases class 11 physics JEE_Main

Explain the construction and working of a GeigerMuller class 12 physics JEE_Main

Excluding stoppages the speed of a bus is 54 kmph and class 11 maths JEE_Main

when an object Is placed at a distance of 60 cm from class 12 physics JEE_Main

Electric field due to uniformly charged sphere class 12 physics JEE_Main

Two identical particles each having a charge of 20 class 12 physics JEE_Main