The height of a solid metal cylinder is 20cm. Its radius is 1.5cm. The cylinder is melted and cast into spheres of each radius 1.5cm. How many such spheres can be cast from the cylinder?
Answer
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Hint: First find the volume of the cylinder using formula $\pi {{r}^{2}}h$ using data given in the question, then find the volume of one sphere using the dimension given. Then divide the volume of the cylinder by the volume of the sphere you will get your answer.
Complete step-by-step answer:
At first we will find the volume of the cylinder by using $\pi {{r}^{2}}h$ formula.
In the formula $\pi =3.14$, ‘r’ depicts radius and ‘h’ describes height.
From the given information, the radius of the cylinder is 1.5 cm and height is 20 cm.
Therefore, the volume of cylinder after substituting the values, we get
\[{{\pi }^{2}}h=3.14\times 1.5\times 1.5\times 20=141.3c{{m}^{2}}\]
It is given that the cylinder was melted down to form spheres. We can say that ‘n’ numbers of spheres were made so the total volume of the cylinder is equal to the total sum of volume of all spheres.
We know that the volume of one sphere can be written as $\dfrac{4}{3}\pi {{r}^{3}}$.
Here ‘r’ is the radius of the sphere.
As per the given information, the radius of the sphere is 1.5 cm.
So the volume of one sphere will be,
$=\dfrac{4}{3}\times 3.14\times {{\left( 1.5 \right)}^{3}}$
So we can write as,
$\text{volume of cylinder = n }\!\!\times\!\!\text{ volume of 1 sphere}$
Here we know the volume of 1 cylinder is $\text{141}\text{.3c}{{\text{m}}^{3}}$.
So we can write it as,
$\text{141}\text{.3=n}\times \dfrac{4}{3}\times 3.14\times {{\left( 1.5 \right)}^{3}}$
$\therefore n=\dfrac{141.3}{\dfrac{4}{3}\times 3.14\times {{\left( 1.5 \right)}^{3}}}=10$
So, there are a total 10 spheres that can be cast out of the given cylinder.
Note: Students should know that the formulas of volume of sphere and cylinder by heart. They should be careful about the calculation mistake as it includes lots of calculations. Students often make mistakes in calculating the formula $\text{volume of cylinder = n }\!\!\times\!\!\text{ volume of 1sphere}$. In this way they get the wrong answer.
Complete step-by-step answer:
At first we will find the volume of the cylinder by using $\pi {{r}^{2}}h$ formula.
In the formula $\pi =3.14$, ‘r’ depicts radius and ‘h’ describes height.
From the given information, the radius of the cylinder is 1.5 cm and height is 20 cm.
Therefore, the volume of cylinder after substituting the values, we get
\[{{\pi }^{2}}h=3.14\times 1.5\times 1.5\times 20=141.3c{{m}^{2}}\]
It is given that the cylinder was melted down to form spheres. We can say that ‘n’ numbers of spheres were made so the total volume of the cylinder is equal to the total sum of volume of all spheres.
We know that the volume of one sphere can be written as $\dfrac{4}{3}\pi {{r}^{3}}$.
Here ‘r’ is the radius of the sphere.
As per the given information, the radius of the sphere is 1.5 cm.
So the volume of one sphere will be,
$=\dfrac{4}{3}\times 3.14\times {{\left( 1.5 \right)}^{3}}$
So we can write as,
$\text{volume of cylinder = n }\!\!\times\!\!\text{ volume of 1 sphere}$
Here we know the volume of 1 cylinder is $\text{141}\text{.3c}{{\text{m}}^{3}}$.
So we can write it as,
$\text{141}\text{.3=n}\times \dfrac{4}{3}\times 3.14\times {{\left( 1.5 \right)}^{3}}$
$\therefore n=\dfrac{141.3}{\dfrac{4}{3}\times 3.14\times {{\left( 1.5 \right)}^{3}}}=10$
So, there are a total 10 spheres that can be cast out of the given cylinder.
Note: Students should know that the formulas of volume of sphere and cylinder by heart. They should be careful about the calculation mistake as it includes lots of calculations. Students often make mistakes in calculating the formula $\text{volume of cylinder = n }\!\!\times\!\!\text{ volume of 1sphere}$. In this way they get the wrong answer.
Last updated date: 25th Sep 2023
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