https://file.aiquickdraw.com/mathbot/file/1727175513952po2rztvd.jpg

Let's break down and solve each of the problems shown in the image:

### 1. Luas dari segi enam dalam lingkaran berjari-jari 14

The problem describes a regular hexagon inscribed in a circle with a radius \( r = 14 \).

- The side length \( s \) of a regular hexagon inscribed in a circle is equal to the radius of the circle. Thus, \( s = 14 \).
- The area \( A \) of a regular hexagon can be calculated using the formula:
  \[
  A = \frac{3 \sqrt{3}}{2} s^2
  \]
  Substituting \( s = 14 \):
  \[
  A = \frac{3 \sqrt{3}}{2} \times 14^2 = \frac{3 \sqrt{3}}{2} \times 196 = 294 \sqrt{3}
  \]

So, the area of the hexagon is \( 294 \sqrt{3} \) square units.

### 2. Volume maksimal bola dalam kubus dengan sisi 10

The problem asks for the maximum volume of a sphere inscribed in a cube with side length \( 10 \).

- The diameter of the sphere will be equal to the side length of the cube, which is \( 10 \).
- Therefore, the radius \( r \) of the sphere is \( \frac{10}{2} = 5 \).
- The volume \( V \) of a sphere is given by:
  \[
  V = \frac{4}{3} \pi r^3
  \]
  Substituting \( r = 5 \):
  \[
  V = \frac{4}{3} \pi (5^3) = \frac{4}{3} \pi (125) = \frac{500}{3} \pi \approx 523.6 \text{ cubic units}
  \]

Thus, the maximum volume of the sphere is approximately \( 523.6 \) cubic units.

### 3. Hitung Nilai Sin(75°) dan Cos(75°)

To calculate \( \sin(75^\circ) \) and \( \cos(75^\circ) \), we can use trigonometric identities:
\[
\sin(75^\circ) = \sin(45^\circ + 30^\circ)
\]
\[
\sin(75^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ
\]
Substituting known values:
\[
\sin(75^\circ) = \left(\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right) \left(\frac{1}{2}\right)
\]
\[
\sin(75^\circ) = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}
\]

For \( \cos(75^\circ) \):
\[
\cos(75^\circ) = \cos(45^\circ + 30^\circ)
\]
\[
\cos(75^\circ) = \cos 45^\circ \cos 30^\circ - \sin 45^\circ \sin 30^\circ
\]
Substituting known values:
\[
\cos(75^\circ) = \left(\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right) \left(\frac{1}{2}\right)
\]
\[
\cos(75^\circ) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4}
\]

### 4. Gambar Ellips dengan persamaan \( 9x^2 + 16y^2 = 144 \)

The equation is in the form:
\[
\frac{x^2}{16} + \frac{y^2}{9} = 1
\]
This is the equation of an ellipse centered at the origin with:
- Semi-major axis \( a = 4 \) (in the \( x \)-direction),
- Semi-minor axis \( b = 3 \) (in the \( y \)-direction).

To sketch the ellipse:
- The ellipse stretches from \( x = -4 \) to \( x = 4 \) and from \( y = -3 \) to \( y = 3 \).

Would you like any further explanation or more detailed steps on any of these problems?

### Related Questions:
1. How do you derive the formula for the area of a regular hexagon?
2. What are the properties of a sphere inscribed in a cube?
3. Can you prove the trigonometric identities used for \( \sin(75^\circ) \) and \( \cos(75^\circ) \)?
4. How do you convert an ellipse equation into standard form?
5. What are the geometric properties of an ellipse?

### Tip:
For trigonometric functions of non-standard angles, using sum/difference identities is a powerful way to calculate exact values.

1. Apabila terdapat lingkaran dengan jari-jari r = 14 dan didalamnya diletakkan segienam sehingga tepat menyinggung sisi-sisi segitiga, maka hitung luas dari segienam tersebut.
2. Diberikan kubus dengan panjang sisi 10 dan didalamnya diletakan bola, berapa volume maksimal dari bola yang berada dalam kubus tersebut.
3. Hitung Nilai Sin(75°) dan Cos(75°)
4. Gambarkan ellips dengan persamaan 9x² + 16y² = 144.

This page provides solutions to various math problems involving the area of a hexagon inscribed in a circle with radius 14, the maximum volume of a sphere inside a cube with side length 10, calculations for Sin(75°) and Cos(75°) using trigonometric identities, and the sketching of an ellipse with the equation 9x² + 16y² = 144.

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