Answer :
Answer:The possible dimensions for the triangular prism are:
- Base: 40 cm
- Other sides: (1, 40), (2, 39), (3, 38), (4, 37), (5, 36), (6, 35), (7, 34), (8, 33), (9, 32), (10, 31), (11, 30), (12, 29), (13, 28), (14, 27), (15, 26), (16, 25), (17, 24), (18, 23), (19, 22), (20, 21)
Step-by-step explanation:
To find the possible dimensions of the triangular prism, we can use the formula for the volume of a prism:
\[ \text{Volume} = \text{Base Area} \times \text{Height} \]
For a triangular prism, the base area is the area of the triangle. The formula for the area of a triangle is:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
So, if we denote the base of the triangle as \( b \), the formula for the volume becomes:
\[ 200 = \frac{1}{2} \times b \times h \times 10 \]
Given that the height is 10 cm, we can rearrange the equation to solve for the base:
\[ b = \frac{2 \times 200}{10} \]
\[ b = 40 \]
So, the base of the triangle must be 40 cm.
Now, let's find the possible dimensions for the sides of the triangle, keeping in mind that they must all be integers. Since we know one side is 40 cm, let's denote the other two sides as \( a \) and \( c \).
To satisfy the inequality \( a + c > b \) (Triangle Inequality Theorem), where \( b = 40 \), we have:
\[ a + c > 40 \]
Since all sides are integers, we can start with \( a = 1 \) and \( c = 40 \). Then we can iterate through all possible integer values for \( a \) and \( c \) such that \( a + c > 40 \).
Here are some possible sets of dimensions (a, c) for the triangle:
1. (1, 40)
2. (2, 39)
3. (3, 38)
4. (4, 37)
5. (5, 36)
6. (6, 35)
7. (7, 34)
8. (8, 33)
9. (9, 32)
10. (10, 31)
11. (11, 30)
12. (12, 29)
13. (13, 28)
14. (14, 27)
15. (15, 26)
16. (16, 25)
17. (17, 24)
18. (18, 23)
19. (19, 22)
20. (20, 21)
These are some possible combinations of dimensions for the triangular prism with a volume of 200 cm³ and a height of 10 cm, where all lengths are integers.
