Absolutely! Let's go through the step-by-step solution for the given expression.
We need to evaluate the following expression:
[tex]\[
(40 \, \text{km})^2 + (30 \, \text{km})^2 - 2 \cdot (40 \, \text{km}) \cdot (30 \, \text{km}) \cdot \cos (180^\circ - 45^\circ)
\][/tex]
### Step 1: Squaring the sides
First, we square the given sides:
[tex]\[
(40 \, \text{km})^2 = 1600 \, \text{km}^2
\][/tex]
[tex]\[
(30 \, \text{km})^2 = 900 \, \text{km}^2
\][/tex]
### Step 2: Angle conversion
The angle provided is [tex]\(180^\circ - 45^\circ\)[/tex]. Simplify this to:
[tex]\[
180^\circ - 45^\circ = 135^\circ
\][/tex]
### Step 3: Cosine of the angle
We need to find [tex]\(\cos 135^\circ\)[/tex].
The cosine of [tex]\(135^\circ\)[/tex] is [tex]\(\cos(180^\circ - 45^\circ)\)[/tex], which equals [tex]\(-\cos 45^\circ\)[/tex].
Since [tex]\(\cos 45^\circ = \frac{1}{\sqrt{2}}\)[/tex],
[tex]\[
\cos 135^\circ = -\cos 45^\circ = -\frac{1}{\sqrt{2}} \approx -0.7071
\][/tex]
### Step 4: Compute the product term
Now compute the term:
[tex]\[
2 \cdot (40 \, \text{km}) \cdot (30 \, \text{km}) \cdot \cos (135^\circ)
\][/tex]
Substitute the values:
[tex]\[
2 \cdot 40 \cdot 30 \cdot -\frac{1}{\sqrt{2}} \approx 2 \cdot 40 \cdot 30 \cdot -0.7071 = -1697.06 \, \text{km}^2
\][/tex]
### Step 5: Combine all terms
Finally, we sum and subtract all the terms:
[tex]\[
1600 \, \text{km}^2 + 900 \, \text{km}^2 - (-1697.06 \, \text{km}^2)
\][/tex]
Which simplifies to:
[tex]\[
1600 + 900 + 1697.06 = 4197.06 \, \text{km}^2
\][/tex]
Therefore, the result of the expression is:
[tex]\[
4197.06 \, \text{km}^2
\][/tex]
So, the detailed solution step-by-step for the given question is:
[tex]\[
(40 \, \text{km})^2 + (30 \, \text{km})^2 - 2 \cdot (40 \, \text{km}) \cdot (30 \, \text{km}) \cdot \cos (180^\circ - 45^\circ) = 4197.06 \, \text{km}^2
\][/tex]
