Let's go through each problem and solve them step by step.

1. Find the area and circumference of the circles:

• Circle 1 (radius = 10 m):

  • Area: $A = \pi r^2 = \pi \times 10^2 = 100\pi$ square meters.
  • Circumference: $C = 2\pi r = 2\pi \times 10 = 20\pi$ meters.

• Circle 2 (radius = 8 m):

  • Area: $A = \pi r^2 = \pi \times 8^2 = 64\pi$ square meters.
  • Circumference: $C = 2\pi r = 2\pi \times 8 = 16\pi$ meters.

2. Find the perimeter of the L-shaped figure:

The L-shaped figure has the following dimensions:

• Vertical segments: 5 cm and 4 cm.
• Horizontal segments: 4 cm and 2 cm.

We need to sum all the sides:

$\text{Perimeter} = 5 + 4 + 4 + 2 = 15 \, \text{cm}$

3. Find the surface area and volume of the rectangular prism:

• Dimensions: $7 \, \text{cm} \times 5 \, \text{cm} \times 3 \, \text{cm}$.

  • Surface area: The formula for surface area $A$ of a rectangular prism is: $A = 2(lw + lh + wh)$ where $l = 7 \, \text{cm}, w = 5 \, \text{cm}, h = 3 \, \text{cm}$.

    $A = 2(7 \times 5 + 7 \times 3 + 5 \times 3) = 2(35 + 21 + 15) = 2 \times 71 = 142 \, \text{cm}^2$

  • Volume: The volume $V$ of a rectangular prism is: $V = l \times w \times h = 7 \times 5 \times 3 = 105 \, \text{cm}^3$

4. Calculate the volume of the triangular prism:

The triangular prism has a triangular base with dimensions:

• Base = 6 cm,
• Height of triangle = 5 cm,
• Length of prism = 20 cm.

The formula for the volume $V$ of a triangular prism is: $V = \frac{1}{2} \times \text{base of triangle} \times \text{height of triangle} \times \text{length}$ $V = \frac{1}{2} \times 6 \times 5 \times 20 = 300 \, \text{cm}^3$

5. Find the area of the shape:

This problem involves a composite shape made up of two rectangles. The dimensions given are:

• Large rectangle: $11 \, \text{cm} \times 14 \, \text{cm}$,
• Small rectangle: $6 \, \text{cm} \times 4 \, \text{cm}$.

The total area is: $A = (11 \times 14) - (6 \times 4) = 154 - 24 = 130 \, \text{cm}^2$

6. Find $x$ in the triangle:

The given triangle has angles:

• $40^\circ$,
• $70^\circ$, and
• $x^\circ$.

Since the sum of angles in a triangle is $180^\circ$: $x = 180^\circ - (40^\circ + 70^\circ) = 180^\circ - 110^\circ = 70^\circ$

Would you like more details or have any questions about these solutions?

Here are 5 related questions to explore further:

1. How do you calculate the area of a sector of a circle?
2. Can you find the volume of a cone given its height and radius?
3. How would you calculate the surface area of a sphere?
4. What is the difference between perimeter and circumference?
5. How do you find the height of a triangle if only the base and area are known?

Tip: Always keep in mind the correct units when solving geometry problems. For areas, it's squared units, and for volume, it's cubed units!