The radii of the ends of a bucket \[30cm\] high is \[21cm\] and \[7cm\]. Find its capacity in litres and the amount of sheet required to make this bucket.
A. \[20.02{\text{ liters ; 3067}}c{m^2}\]
B. \[20.02{\text{ liters ; 3867}}c{m^2}\]
C. \[25.02{\text{ liters ; 3067}}c{m^2}\]
D. \[10.02{\text{ liters ; 3067}}c{m^2}\]
Answer
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Hint: Here the volume of the bucket is equal to the capacity of the bucket in litres. We can find the amount of sheet required to make this bucket by calculating the total surface area of the bucket.
Complete step-by-step answer:
Given,
Height of the bucket \[h = 30cm\]
Radius of upper end of the bucket \[R = 21cm\]
Radius of lower end of the bucket \[r = 7cm\]
We know that volume of the bucket = \[\dfrac{1}{3}\pi h\left( {{R^2} + {r^2} + Rr} \right)c{m^3}\]
\[
= \dfrac{1}{3}\pi (30)\left( {{{\left( {21} \right)}^2} + {{\left( 7 \right)}^2} + \left( {21} \right)\left( 7 \right)} \right) \\
= \dfrac{1}{3}\pi \left( {30} \right)\left( {441 + 49 + 147} \right) \\
= \dfrac{1}{3}\pi \left( {30} \right)\left( {637} \right) \\
= \dfrac{1}{3} \times \dfrac{{22}}{7}\left( {19110} \right) \\
= \dfrac{{420420}}{{21}} = 20020c{m^3} \\
\]
Since, \[1c{m^3} = \dfrac{1}{{1000}}{\text{ litres}}\]
The capacity of the bucket is \[\dfrac{{20020}}{{1000}}{\text{ litres }} = {\text{ 20}}{\text{.02 litres}}\]
We know that slant height of the bucket \[l = \sqrt {{h^2} + {{\left( {R - r} \right)}^2}} \]
\[
l = \sqrt {{{30}^2} + {{\left( {21 - 7} \right)}^2}} \\
l = \sqrt {900 + {{\left( {14} \right)}^2}} \\
l = \sqrt {900 + 196} \\
l = \sqrt {1096} \\
\therefore l = 33.10cm \\
\]
Total area of the metal sheet required to make the bucket = \[\pi l\left( {R + r} \right) + \pi {r^2}\]
\[
= \pi \times 33.10\left( {21 + 7} \right) + \pi \times {7^2} \\
= \pi \times 33.10\left( {28} \right) + \pi \times 49 \\
= \pi \times 926.8 + \pi \times 49 \\
= \pi \left( {926.8 + 49} \right) \\
= \dfrac{{22}}{7} \times 975.8 \\
= \dfrac{{21467.6}}{7} \\
= 3066.8 \approx 3067c{m^2} \\
\]
Therefore, the amount of sheet required to make the bucket is \[{\text{3067}}c{m^2}\].
Thus, the capacity of the bucket is \[20.02{\text{ litres}}\] and the amount of sheet required to make the bucket is \[{\text{3067}}c{m^2}\].
So, the correct option is A. \[20.02{\text{ liters ; 3067}}c{m^2}\].
Note: The height given in the problem is the altitude of the bucket. To find the surface area we have to consider the slant height which is given by \[l = \sqrt {{h^2} + {{\left( {R - r} \right)}^2}} \]. We have converted the volume of the bucket in to capacity of the bucket in litres by using the conversion \[1c{m^3} = \dfrac{1}{{1000}}{\text{ litres}}\].
Complete step-by-step answer:
Given,
Height of the bucket \[h = 30cm\]
Radius of upper end of the bucket \[R = 21cm\]
Radius of lower end of the bucket \[r = 7cm\]
We know that volume of the bucket = \[\dfrac{1}{3}\pi h\left( {{R^2} + {r^2} + Rr} \right)c{m^3}\]
\[
= \dfrac{1}{3}\pi (30)\left( {{{\left( {21} \right)}^2} + {{\left( 7 \right)}^2} + \left( {21} \right)\left( 7 \right)} \right) \\
= \dfrac{1}{3}\pi \left( {30} \right)\left( {441 + 49 + 147} \right) \\
= \dfrac{1}{3}\pi \left( {30} \right)\left( {637} \right) \\
= \dfrac{1}{3} \times \dfrac{{22}}{7}\left( {19110} \right) \\
= \dfrac{{420420}}{{21}} = 20020c{m^3} \\
\]
Since, \[1c{m^3} = \dfrac{1}{{1000}}{\text{ litres}}\]
The capacity of the bucket is \[\dfrac{{20020}}{{1000}}{\text{ litres }} = {\text{ 20}}{\text{.02 litres}}\]
We know that slant height of the bucket \[l = \sqrt {{h^2} + {{\left( {R - r} \right)}^2}} \]
\[
l = \sqrt {{{30}^2} + {{\left( {21 - 7} \right)}^2}} \\
l = \sqrt {900 + {{\left( {14} \right)}^2}} \\
l = \sqrt {900 + 196} \\
l = \sqrt {1096} \\
\therefore l = 33.10cm \\
\]
Total area of the metal sheet required to make the bucket = \[\pi l\left( {R + r} \right) + \pi {r^2}\]
\[
= \pi \times 33.10\left( {21 + 7} \right) + \pi \times {7^2} \\
= \pi \times 33.10\left( {28} \right) + \pi \times 49 \\
= \pi \times 926.8 + \pi \times 49 \\
= \pi \left( {926.8 + 49} \right) \\
= \dfrac{{22}}{7} \times 975.8 \\
= \dfrac{{21467.6}}{7} \\
= 3066.8 \approx 3067c{m^2} \\
\]
Therefore, the amount of sheet required to make the bucket is \[{\text{3067}}c{m^2}\].
Thus, the capacity of the bucket is \[20.02{\text{ litres}}\] and the amount of sheet required to make the bucket is \[{\text{3067}}c{m^2}\].
So, the correct option is A. \[20.02{\text{ liters ; 3067}}c{m^2}\].
Note: The height given in the problem is the altitude of the bucket. To find the surface area we have to consider the slant height which is given by \[l = \sqrt {{h^2} + {{\left( {R - r} \right)}^2}} \]. We have converted the volume of the bucket in to capacity of the bucket in litres by using the conversion \[1c{m^3} = \dfrac{1}{{1000}}{\text{ litres}}\].
Last updated date: 16th Sep 2023
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