- **Calculating the Dimensions of the Sheet Metal Required**
To create a cup with a height of 10 cm and a diameter of 5 cm, we need to consider the dimensions of the sheet metal required. The cup can be visualized as a cylinder, so we need to calculate the dimensions of the sheet metal that would form the side of the cylinder.
- **Calculating the Surface Area of the Cylinder**
The surface area of the side of the cylinder can be calculated using the formula for the lateral surface area of a cylinder: \(2\pi rh\), where \(r\) is the radius and \(h\) is the height. In this case, the radius would be half of the diameter (2.5 cm). So, the surface area would be \(2\pi \times 2.5 \times 10 = 50\pi\) cm².
- **Taking into Account the Thickness of the Sheet Metal**
Since the sheet metal has a thickness of 2 mm, we need to take this into account when calculating the dimensions. The sheet metal would wrap around the cylinder, so the length of the sheet metal needed would be the circumference of the cylinder: \(2\pi r\). Considering the thickness, the actual length of the sheet metal required would be \(2\pi r + 2t\), where \(t\) is the thickness (2 mm).
- **Calculating the Number of Deductions Necessary**
To create the cup, we would need to cut out the shape of the cylinder from a larger sheet of metal. The number of deductions necessary would be the total length of the sheet metal required divided by the width of the sheet. Since the width of the sheet is not given, we cannot provide an exact number of deductions needed without this information. However, we can calculate the total length of the sheet metal required and then divide it by the width of the sheet to determine the number of deductions necessary.