A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in the figure. If the height of the cylinder is 10 centimetres and its base radius is 3.5 centimetres, find the total surface area of the article.
So first, let’s just add the information we’ve been given in the question to the diagram. We have a height of 10 centimetres and a base radius of 3.5 centimetres. We need to find the total surface area of this article. So we need to consider what the different components of this surface area are going to be.
Firstly, there’s the curved surface area of the cylinder, which is the part that wraps around the outside of the cylinder. Then, there’s also the domed surface area at the top and the base of the cylinder, which is the curved surface area of each hemisphere. Now, in fact cause we have two hemispheres, the sum of their surface areas will be equal to the surface area of one single sphere.
So we have two surface areas that we need to find. Let’s think about the curved surface area of the cylinder first of all. Now, although this surface area is curved, if you were to unwrap it from around the cylinder and spread it out, it would actually be a rectangle. The width of the rectangle is the same as the height of the cylinder and the length of the rectangle is the circumference of the circular base of the cylinder.
To find the circumference of a circle, we use the formula 𝜋𝑑 or two 𝜋𝑟. So to find the area of this rectangle, we multiply two 𝜋𝑟 by ℎ. This gives us our formula for the curved surface area of the cylinder.
Next, we need to consider the surface area of the sphere. The formula for this is four 𝜋𝑟 squared. And remember as we have two hemispheres, then we have one sphere in total. So we have a formula for working out the total surface area. It’s two 𝜋𝑟ℎ plus four 𝜋𝑟 squared. And as we know the values of both 𝑟 and ℎ, we can substitute them into this formula.
This gives two multiplied by 𝜋 multiplied by 3.5 multiplied by 10 for the curved surface area of the cylinder and four multiplied by 𝜋 multiplied by 3.5 squared for the surface area of the two hemispheres.
Now, the first calculation is relatively straightforward to work out in terms of 𝜋: two multiplied by 3.5 is seven and seven multiplied by 10 is 70. So we have 70𝜋. For the second calculation, we need to work out 3.5 squared. And as we do not have a calculator, we can instead work out 35 multiplied by 35 using a column multiplication method and it gives 1225.
So if 35 multiplied by 35 is 1225, then 3.5 multiplied by 3.5 is 12.25. You can work this out in a couple of ways. Firstly, we could use a bit of estimation three squared is nine and four squared is 16. So 3.5 squared needs to be somewhere between these two values, which means the decimal point needs to go after the first two to give a value of 12.25.
The other way of working out where the decimal point goes is to realize that to get from 35 to 3.5, we have to divide by 10. So we’ve done that twice, which means we have to divide our overall answer by 100. So 3.5 squared is 12.25. But remember we’re also multiplying this by four. Four multiplied by 12 is 48 and four multiplied by 0.25 is one. So four multiplied by 12.25 is 48 plus one which is 49. So the second term in our calculation of the total surface area simplifies to 49𝜋.
Now, we have 70 lots of 𝜋 plus 49 lots of 𝜋. So overall, we have 119𝜋. Now, as we don’t have a calculator, this would be a perfectly acceptable format to leave our answer in or we could use an approximation for 𝜋. A close approximation for 𝜋 is 22 over seven. So an approximate answer would be 119 multiplied by 22 over seven. We can actually cancel a factor of seven from the seven in the denominator and the 119 in the numerator to give 17 multiplied by 22 over one.
We can see that 119 divided by seven is 17 if we go back to this earlier stage of working out, where we see that that 119 was the sum of 70 and 49. As 70 divided by seven is 10 and 49 divided by seven is seven, then the total number of sevens that go into 119 is 17. Now, we can work out 17 multiplied by 22 without a calculator by breaking 22 down into the sum of 20 and two. 17 multiplied by 20 is 340 and 17 multiplied by two is 34. And the sum of these two values is 374. So 17 multiplied by 22 over one is equal to 374.
Now, the units for the surface area will be centimetres squared as the measurements for the original cylinder were given in centimetres. So we can give an exact answer of 119𝜋 centimetres squared for the total surface area or an approximate answer of 374 centimetres squared.