The sides of a triangle are 40 cm, 70 cm and 90 cm. If the area of the triangle is `k sqrt5" cm"^2,` then the value of k is____

Answer

To find the value of \( k \) in the area of the triangle given by \( k \sqrt{5} \, \text{cm}^2 \), we will use Heron's formula.

### Step-by-Step Solution:

1. **Identify the sides of the triangle**:
The sides of the triangle are given as:
- \( a = 40 \, \text{cm} \)
- \( b = 70 \, \text{cm} \)
- \( c = 90 \, \text{cm} \)

2. **Calculate the semi-perimeter \( s \)**:
The semi-perimeter \( s \) is calculated using the formula:
\[ s = \frac{a + b + c}{2} \]
Substituting the values:
\[ s = \frac{40 + 70 + 90}{2} = \frac{200}{2} = 100 \, \text{cm} \]

3. **Apply Heron's formula to find the area \( A \)**:
Heron's formula states that the area \( A \) of the triangle can be calculated as:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
Substituting the values of \( s \), \( a \), \( b \), and \( c \):
\[ A = \sqrt{100 \times (100 - 40) \times (100 - 70) \times (100 - 90)} \]
Simplifying the terms inside the square root:
\[ A = \sqrt{100 \times 60 \times 30 \times 10} \]

4. **Calculate the product inside the square root**:
First, calculate \( 100 \times 60 \times 30 \times 10 \):
\[ 100 \times 60 = 6000 \]
\[ 6000 \times 30 = 180000 \]
\[ 180000 \times 10 = 1800000 \]
Now, we can find the square root:
\[ A = \sqrt{1800000} \]

5. **Simplify the square root**:
We can express \( 1800000 \) as:
\[ 1800000 = 1800 \times 1000 = 1800 \times 10^3 \]
Now, simplifying \( \sqrt{1800} \):
\[ 1800 = 36 \times 50 = 6^2 \times 50 \]
Thus,
\[ \sqrt{1800} = 6 \sqrt{50} \]
And since \( \sqrt{1000} = 10 \sqrt{100} = 10 \times 10 = 100 \),
\[ \sqrt{1800000} = 100 \times 6 \sqrt{50} = 600 \sqrt{5} \]

6. **Relate the area to the given form**:
We have found that:
\[ A = 600 \sqrt{5} \, \text{cm}^2 \]
According to the problem, the area is given as \( k \sqrt{5} \, \text{cm}^2 \). Therefore, we can equate:
\[ k \sqrt{5} = 600 \sqrt{5} \]
From this, we can conclude:
\[ k = 600 \]

### Final Answer:
The value of \( k \) is \( \boxed{600} \).