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Let’s go through the problems one by one.

### Problem 1:
You are given a circle with radius \( r = 14 \) and inside it, a regular hexagon is inscribed. We need to find the area of the hexagon.

- The radius of the circle is equal to the circumradius of the hexagon. The side length \( s \) of a regular hexagon inscribed in a circle is related to the radius by the formula:
  \[
  s = r
  \]
  Therefore, \( s = 14 \).

- The area \( A \) of a regular hexagon with side length \( s \) can be calculated using the formula:
  \[
  A = \frac{3\sqrt{3}}{2} s^2
  \]
  Substituting \( s = 14 \):
  \[
  A = \frac{3\sqrt{3}}{2} \times 14^2 = \frac{3\sqrt{3}}{2} \times 196
  \]

Let me compute the final value for the area.

### Problem 2:
You are given a cube with side length \( 10 \) and a sphere is inscribed in it. We need to calculate the volume of the sphere.

- The diameter of the sphere is equal to the side length of the cube, so the radius \( r \) of the sphere is:
  \[
  r = \frac{10}{2} = 5
  \]
- The volume \( V \) of a sphere is given by the formula:
  \[
  V = \frac{4}{3} \pi r^3
  \]
  Substituting \( r = 5 \):
  \[
  V = \frac{4}{3} \pi (5)^3 = \frac{4}{3} \pi \times 125
  \]

Let me compute the final value for the volume.

### Problem 3:
We need to compute \( \sin(75^\circ) \) and \( \cos(75^\circ) \).

- Using known trigonometric identities, we can use the angle sum formulas:
  \[
  \sin(75^\circ) = \sin(45^\circ + 30^\circ) = \sin(45^\circ)\cos(30^\circ) + \cos(45^\circ)\sin(30^\circ)
  \]
  \[
  \cos(75^\circ) = \cos(45^\circ + 30^\circ) = \cos(45^\circ)\cos(30^\circ) - \sin(45^\circ)\sin(30^\circ)
  \]
Let me compute these values.

### Problem 4:
We are asked to draw the ellipse with the equation:
\[
9x^2 + 16y^2 = 144
\]
We first simplify the equation by dividing through by 144:
\[
\frac{x^2}{16} + \frac{y^2}{9} = 1
\]
This is the standard form of an ellipse with semi-major axis \( a = 4 \) and semi-minor axis \( b = 3 \). You can sketch an ellipse centered at the origin with these dimensions.

Let me first compute the values for the previous steps before continuing.Here are the solutions to the problems:

### Problem 1:
The area of the hexagon inscribed in the circle with radius \( r = 14 \) is approximately:
\[
A = 509.22 \, \text{square units}
\]

### Problem 2:
The maximum volume of the sphere inscribed in the cube with side length 10 is approximately:
\[
V = 523.60 \, \text{cubic units}
\]

### Problem 3:
The values of the trigonometric functions are:
\[
\sin(75^\circ) \approx 0.966 \quad \text{and} \quad \cos(75^\circ) \approx 0.259
\]

### Problem 4:
The equation of the ellipse \( 9x^2 + 16y^2 = 144 \) simplifies to:
\[
\frac{x^2}{16} + \frac{y^2}{9} = 1
\]
This represents an ellipse with semi-major axis \( a = 4 \) and semi-minor axis \( b = 3 \), centered at the origin.

Do you want more details or have any further questions? Here are 5 follow-up questions:

1. How can you calculate the perimeter of the hexagon instead of its area?
2. What is the relationship between the side length of the hexagon and the radius of the circle?
3. How is the volume of a sphere affected if the cube’s side length doubles?
4. Can you explain how trigonometric identities like sum formulas help in computing exact values for non-standard angles?
5. How does the equation of the ellipse change if the coefficients of \( x^2 \) and \( y^2 \) are interchanged?

**Tip:** For large trigonometric computations, calculators can use radian mode, so it’s important to convert angles if needed.

Apabila terdapat lingkaran dengan jari-jari 14 dan didalamnya diletakkan segienam sehingga tepat menyinggung sisi-sisi segitiga, hitung luas segienam tersebut. Diberikan kubus dengan panjang sisi 10 dan didalamnya diletakkan bola, hitung volume maksimal bola. Hitung nilai Sin(75°) dan Cos(75°). Gambarkan ellips dengan persamaan 9x^2 + 16y^2 = 144.

Solve geometry and trigonometry problems: calculate the area of a hexagon inscribed in a circle, the volume of a sphere inside a cube, values of sin(75°) and cos(75°), and draw an ellipse from the equation 9x^2 + 16y^2 = 144. Suitable for Grades 9-11 students.

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