Question Video: Relating the Volumes and Surface Areas of Multiple Shapes | Nagwa Question Video: Relating the Volumes and Surface Areas of Multiple Shapes | Nagwa

Question Video: Relating the Volumes and Surface Areas of Multiple Shapes Mathematics • Second Year of Preparatory School

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Three spheres are inscribed in a cylinder, as shown in the figure. If the volume of a sphere is 36𝜋 cm³, find the total surface area of the cylinder.

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Video Transcript

Three spheres are inscribed in a cylinder, as shown in the figure. If the volume of a sphere is 36𝜋 cubic centimeters, find the total surface area of the cylinder.

In this question, we need to consider how a single sphere relates to the dimensions of the cylinder that contains the three spheres. To do this, we make use of the fact that the spheres are inscribed, which means they fit exactly into the cylinder with no gaps in the top, bottom, or sides. In particular, it means that the radii of the spheres in the cylinder must be the same. It also means that the height of the cylinder must be equal to three times the diameter of a single sphere, or six times the radius of one. If we let 𝑟 be the radius, then the height of the cylinder is six 𝑟.

In order to find the total surface area of the cylinder, we firstly need to find the value of 𝑟, as this will give us the radius and height of the cylinder. We recall that the volume of a sphere is equal to four-thirds 𝜋𝑟 cubed. And in this question, we are told that the volume is equal to 36𝜋. Setting these equal to one another, we have four-thirds 𝜋𝑟 cubed is equal to 36𝜋.

We can solve this equation for 𝑟 by firstly dividing through by 𝜋. We then divide both sides of this equation by four-thirds, which is the same as multiplying by three-quarters, giving us 𝑟 cubed is equal to 27. Cube rooting both sides of this equation, we have 𝑟 is equal to three. The radius of each of the spheres and, hence, the cylinder is equal to three centimeters. Multiplying this by six, we see that the height of the cylinder is 18 centimeters.

Next, we recall that the total surface area of a cylinder is equal to two 𝜋𝑟ℎ plus two 𝜋𝑟 squared. This is made up of the curved surface, which is a rectangle, and the two circles at either end. Letting this total surface area be 𝑆 and substituting in our values of 𝑟 and ℎ, we have 𝑆 is equal to two 𝜋 multiplied by three multiplied by 18 plus two 𝜋 multiplied by three squared. This is equal to 108𝜋 plus 18𝜋, which simplifies to 126𝜋. The total surface area of the cylinder in the figure is 126𝜋 square centimeters.

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