A fuel tank has a length of 1 m and a cross-section that is a quarter of a circler having a radius of 40 cm. The tank is filled with gasoline to a height of 20 cm. What is the volume of the fuel? 

To find the volume of gasoline in the tank, let's break down the problem:

### Step-by-Step Explanation

1. **Understanding the tank geometry**:
   - The tank has a length of \( 1 \) meter.
   - The cross-section of the tank is a quarter of a circle with a radius \( R = 40 \) cm \( = 0.4 \) m.
   - Gasoline is filled to a height of \( 20 \) cm \( = 0.2 \) m.

2. **Finding the area of the cross-section filled with gasoline**:
   - The cross-section is a quarter-circle, and we need to find the area that is filled to a height of \( 0.2 \) m.
   
   The cross-section can be visualized as a quarter of a circle where gasoline forms a segment at a height of \( 0.2 \) m.

3. **Calculating the area of the filled segment**:
   - Let’s consider the quarter-circle centered at the origin in the coordinate system.
   - The formula to find the area \( A_{\text{segment}} \) of a circular segment with angle \( \theta \) in radians and height \( h \) is:
     \[
     A_{\text{segment}} = \frac{R^2}{2} \left( \theta - \sin(\theta) \right)
     \]
   
   - Since gasoline fills the tank to half of the quarter-circle’s height, the angle \( \theta \) corresponding to the gasoline level in a quarter-circle is \( 90^\circ \) or \( \frac{\pi}{2} \) radians.

   - Calculate the height ratio \( h/R = 0.2/0.4 = 0.5 \). For a quarter-circle, this corresponds to an area fraction.

4. **Finding the volume of gasoline**:
   - Once the filled area is found, multiply it by the length of the tank \( L = 1 \) meter to get the volume.
   
Let's perform the calculation:

### Solution
The volume of gasoline \( V \) is:
\[
V = A_{\text{filled\_segment}} \times L
\]To continue, let's calculate the filled area of the segment in detail.

The radius of the quarter-circle is \( R = 0.4 \) meters, and the gasoline level fills up to half of this radius (\( h = 0.2 \) meters). Here's the breakdown:

1. **Area of the quarter-circle:**
   \[
   A_{\text{quarter}} = \frac{1}{4} \times \pi R^2 = \frac{1}{4} \times \pi \times (0.4)^2
   \]
   \[
   A_{\text{quarter}} = \frac{1}{4} \times \pi \times 0.16 \approx 0.1256 \, \text{m}^2
   \]

2. Since the gasoline fills up exactly half of the quarter-circle’s height, it occupies half of the area of the quarter-circle:
   \[
   A_{\text{filled}} = \frac{1}{2} \times A_{\text{quarter}} \approx \frac{1}{2} \times 0.1256 \approx 0.0628 \, \text{m}^2
   \]

3. **Volume calculation**:
   \[
   V = A_{\text{filled}} \times L = 0.0628 \, \text{m}^2 \times 1 \, \text{m} = 0.0628 \, \text{m}^3
   \]

### Final Answer
The volume of gasoline in the tank is approximately:
\[
0.0628 \, \text{m}^3 \, \text{or} \, 62.8 \, \text{liters}
\]

Would you like a more detailed breakdown of the calculation steps or have other questions? Here are 5 related questions you might be interested in:

1. How do you find the volume if the tank is filled to a different height?
2. What if the tank had a semicircular cross-section instead of a quarter-circle?
3. How would you compute the volume if the tank’s cross-section was not a perfect shape?
4. How does the formula for a circular segment area change if the radius is different?
5. Can you derive the formula for the area of a segment for any angle \( \theta \)?

**Tip:** Remember that for problems involving circular segments, visualizing the problem using diagrams can make understanding areas and angles much clearer.

A fuel tank has a length of 1 m and a cross-section that is a quarter of a circle having a radius of 40 cm. The tank is filled with gasoline to a height of 20 cm. What is the volume of the fuel?

Find the volume of gasoline in a tank with a quarter-circle cross-section, radius 40 cm, filled to a height of 20 cm. This problem covers geometry concepts, including circular segments and volume calculations, suitable for Grades 10-12.

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