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The area of the base is 60 m².  What is the longest piece of wood that can fit diagonally in the box?

metres

Give your answer to two decimal places.

To find the longest piece of wood that can fit diagonally in the box, we are looking for the space diagonal of the rectangular box, also known as the 3D diagonal. This diagonal stretches from one corner of the box to the opposite corner.

We can use the formula for the space diagonal \( d \) of a rectangular box:

\[
d = \sqrt{l^2 + w^2 + h^2}
\]

Where:
- \( l \) is the length of the box,
- \( w \) is the width of the box,
- \( h \) is the height of the box.

### Given dataTo find the longest piece of wood that can fit diagonally in the box, we are looking for the space diagonal of the rectangular box, also known as the 3D diagonal. This diagonal stretches from one corner of the box to the opposite corner.

We can use the formula for the space diagonal \( d \) of a rectangular box:

\[
d = \sqrt{l^2 + w^2 + h^2}
\]

Where:
- \( l \) is the length of the box,
- \( w \) is the width of the box,
- \( h \) is the height of the box.

### Given 
- The length \( l = 12 \, \text{m} \) (AB = 12 m),
- The area of the base of the box is \( 60 \, \text{m}^2 \), which gives us the relationship between length and width: 
\[
l \times w = 60 \implies 12 \times w = 60 \implies w = 5 \, \text{m}
\]
- To find the height, we use the angle of \( 22^\circ \). The height \( h \) is represented by the distance from point C to point E (vertical height). Since the angle \( 22^\circ \) is given at point A, the height is found using trigonometry in triangle ACD.

Using the tangent function:
\[
\tan(22^\circ) = \frac{h}{12}
\]
We can now calculate \( h \), and once we have all three dimensions, we can substitute them into the space diagonal formula.

Let me calculate that.The longest piece of wood that can fit diagonally in the box is approximately **13.87 meters**.

Would you like more details or have any questions about the solution?

Here are some related questions to explore:

1. How does changing the dimensions of the box affect the space diagonal?
2. What if the angle was different? How would that affect the height and diagonal?
3. How can we apply this method to find diagonals in other geometric shapes?
4. Can you derive a general formula for the diagonal of a rectangular prism in terms of the base area and angle?
5. How would the calculation change if the base was not rectangular?

**Tip:** Always visualize the geometry and use trigonometry when angles are involved, especially for determining unknown dimensions.

The area of the base is 60 m². What is the longest piece of wood that can fit diagonally in the box? Give your answer to two decimal places.

Learn how to calculate the longest diagonal that can fit inside a rectangular box with a base area of 60 m². This problem involves geometry, trigonometry, and applying the Pythagorean theorem to find the 3D diagonal. Suitable for high school students in Grades 10-12.

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