Find the radius of the following: A cone. A square pyramid. A square (both toward planet). Modified to reducing a based pyre chart to expressed data. One sector represents the arc in degrees and forms a 120 degree angle. The chord of the chart is 25 cm. Centre O, Radius R.

$$
A = \frac{1}{3} \times \pi (784 \text{ cm}^2)
$$
$$
A = \frac{784}{3} \pi \text{ cm}^2
$$
$$
A \approx 261.33 \pi \text{ cm}^2
$$
$$
A \approx 821.01 \text{ cm}^2
$$
The area of the sector is $\boxed{\frac{784}{3} \pi \text{ cm}^2 \approx 821.01 \text{ cm}^2}$.

• b) What fraction of the total circle does this sector represent? The fraction is given by $\frac{\theta}{360^\circ}$.

$$
\text{Fraction} = \frac{1}{3}
$$
The sector represents $\boxed{\frac{1}{3}}$ of the total circle.

3. The diagram below shows a frustum TORS of a cone. Calculate its volume.
Given: Larger radius $R = 8 \text{ cm}$, smaller radius $r = 3 \text{ cm}$, height $h = 12 \text{ cm}$.
The volume of a frustum of a cone is given by the formula $V = \frac{1}{3} \pi h (R^2 + Rr + r^2)$.

The volume of the frustum is $\boxed{388 \pi \text{ cm}^3 \approx 1218.94 \text{ cm}^3}$.

Symmetry

4. Define the following terms and give one example of each:

• a) Symmetry: A property of an object where it remains unchanged under certain transformations, such as reflection or rotation. Example: A square has symmetry.
• b) Axis of symmetry: A line through the center of a shape such that if the shape is folded along this line, both halves match exactly. Example: The line passing through the center of a circle is an axis of symmetry.
• c) Rotational symmetry: A property of a shape where it looks the same after being rotated by a certain angle less than $360^\circ$ around a central point. Example: A pinwheel has rotational symmetry.
• d) Angle of rotational symmetry: The smallest angle (greater than $0^\circ$) through which a shape can be rotated to coincide with its original position. Example: A square has an angle of rotational symmetry of $90^\circ$.
• e) Plane of symmetry: A plane that divides a three-dimensional object into two mirror-image halves. Example: The plane passing through the center of a cube parallel to one of its faces is a plane of symmetry.

5. Describe the symmetry of the given fan-like shape.
The fan-like shape has rotational symmetry. It appears to have an order of rotational symmetry of 4, meaning it looks the same after rotations of $90^\circ$, $180^\circ$, and $270^\circ$. It also has point symmetry (rotational symmetry of order 2, or $180^\circ$). It does not have reflectional symmetry.

6. How many planes of symmetry does a cuboid have?
A cuboid has $\boxed{3}$ planes of symmetry. These planes pass through the center of the cuboid, parallel to its faces.

7. The diagram below shows a hexagonal prism.

• a) State the number of planes of symmetry. A regular hexagonal prism has $\boxed{7}$ planes of symmetry.
• b) Describe them.
  • One plane of symmetry is horizontal, bisecting the prism through its height.
  • Six planes of symmetry are vertical. Three of these pass through the midpoints of opposite rectangular faces, and three pass through opposite vertices of the hexagonal bases.

Remainder and Factor Theorem

8. How can one predict the leftover items without actually running the machine? Is there a faster way than doing the whole experiment?
The number of leftover items can be predicted using the Remainder Theorem.
The Remainder Theorem states that if a polynomial $P(x)$ is divided by $(x-a)$, then the remainder is $P(a)$.
In this case, $P(x) = x^3 - 8x^2 + 17x - 10$ and the manager sets the machine to $x=5$. So, $a=5$.

The remainder is 0. This means there would be no leftover items if the machine is set to $x=5$. This method is much faster than running the machine.

9. A factory wants to link its daily production with the polynomial $P(x) = x^3 - 7x^2 + 14x - 8$. Engineers discover that when the production model is divided by $x-1$, the remainder is 0.

• a) Use the remainder theorem to determine the value of $k$. The given polynomial is $P(x) = x^3 - 7x^2 + 14x - 8$. The question asks for the value of $k$. Assuming $k$ refers to the coefficient of $x$ in the polynomial, and given that the remainder is 0 when divided by $x-1$, we use the Remainder Theorem: $P(1) = 0$.

$$
k = 14
$$
The value of $k$ is $\boxed{14}$.

• b) Using the value of $k$, divide $P(x)$ by $(x-2)$ using synthetic division. The polynomial is $P(x) = x^3 - 7x^2 + 14x - 8$. We are dividing by $(x-2)$, so we use 2 for synthetic division.

• Bring down the first coefficient (1).
• Multiply 1 by 2, write 2 under -7. Add -7 and 2 to get -5.
• Multiply -5 by 2, write -10 under 14. Add 14 and -10 to get 4.
• Multiply 4 by 2, write 8 under -8. Add -8 and 8 to get 0.

• c) State the quotient and remainder. The coefficients of the quotient are $1, -5, 4$. The remainder is 0. The quotient is $\boxed{x^2 - 5x + 4}$. The remainder is $\boxed{0}$.