Let's break down the questions from the image you've provided.

Question Breakdown:

1. Part a): "What is the largest number of discs that can be stamped out of the rectangular sheet?"

   • You are given a rectangular metal sheet with dimensions 140 cm by 75 cm.
   • Discs have a radius of 3 cm.
   • The goal is to determine how many such discs can be stamped out of the rectangular sheet.

2. Part b): "What area of metal would be wasted? Give your answer to 1 decimal place."

   • This involves calculating the total area of the discs and comparing it to the area of the sheet to find the unused or wasted area.

Solution:

Part a) Finding the largest number of discs:

• Area of rectangular sheet: $\text{Area of sheet} = 140 \, \text{cm} \times 75 \, \text{cm} = 10,500 \, \text{cm}^2$

• Area of a single disc: The formula for the area of a circle is $A = \pi r^2$, where $r$ is the radius. $A_{\text{disc}} = \pi \times (3)^2 = \pi \times 9 \approx 28.27 \, \text{cm}^2$

• Number of discs: Now, to find out how many discs can fit within the rectangular area, we need to check the number of rows and columns of discs that can fit.

  • Diameter of each disc = $2 \times 3 = 6 \, \text{cm}$.
  • Number of discs along the length: $\frac{140}{6} = 23$ (approx).
  • Number of discs along the width: $\frac{75}{6} = 12$ (approx).

  Therefore, the maximum number of discs is approximately: $23 \times 12 = 276 \, \text{discs}$

Part b) Calculating wasted area:

• Total area of discs: $\text{Total area of 276 discs} = 276 \times 28.27 = 7,805.52 \, \text{cm}^2$

• Wasted area: $\text{Wasted area} = \text{Area of sheet} - \text{Total area of discs}$ $\text{Wasted area} = 10,500 - 7,805.52 = 2,694.48 \, \text{cm}^2$

Thus, the wasted area is approximately 2,694.5 cm² (to one decimal place).

Would you like further clarification on any part of this solution? Feel free to ask!

Here are some follow-up questions for practice:

1. How would the result change if the radius of the discs was 4 cm instead of 3 cm?
2. What if the dimensions of the sheet were 100 cm by 100 cm?
3. How much material would be wasted if 200 discs were stamped out?
4. Can you think of a way to arrange the discs to minimize wasted material?
5. What is the percentage of metal wasted compared to the total sheet area?

Tip: When calculating areas, always double-check units to ensure consistency (e.g., cm² for area).