Hey there! I'm happy to help!
We want to find how much paper we need to make the cup. So, we need to find the surface area of the cylinder, which is basically the "walls and floors" of the cup. So, this is the circle on the bottom and the surrounding stuff that goes up. There is no circular top to this cup, which will be important when finding the total area of paper.
So, first we need to find how much paper we need to make the bottom circle. The formula for the area of a circle is A=πr². This r represents the radius, or the distance from the center of the circle to the edge. We are given that the cup is 3 cm completely across, and the radius is be half of that, so the radius is 1.5 cm. Let's find the area. We will use 3.14 for π.
π×1.5²=7.065
Now, we need to find the area of the walls of the cup. The wall is a rectangle that is basically wrapped around the circle. The height of the rectangle is the height of the cup and the length of the rectangle is the circumference (perimeter) of the bottom circle because the length of the rectangle wraps completely around the circle.
First, let's find the circumference of the circle. The circumference is the diameter (distance across the circle, so 3 cm) multiplied by π. We will use 3.14 for π.
3π=9.42
So, this means that the long side of the rectangle is 9.42 cm, and now we multiply this by the height.
9.42×6.4=60.288
Now, we add the rectangle area and the bottom circle area.
60.288+7.065=67.353
But, they want us to round to the nearest centimetre, so we will just say 67 square centimetres.
An important thing to note is that this is not the area of a normal cylinder (one with a top and a bottom circle) because it is a cup so it does not have a top. If you have a cylinder with a top and a bottom, you would just multiply the area of the circle by two, so then you have the top and bottom. If that were the case, the area of our cylinder would be 74 cm with the numbers we have been given here.
Have a wonderful day!
