To find the length of the diagonal [tex]\( x \)[/tex] of a parallelogram with given side lengths [tex]\( b = 13 \)[/tex], [tex]\( c = 17 \)[/tex], and an angle [tex]\( A = 64^\circ \)[/tex] between them, we can use the law of cosines. Here's a detailed, step-by-step solution:
1. Convert the angle from degrees to radians:
Since trigonometric functions in standard mathematical contexts often use radians, we need to convert the given angle [tex]\( A = 64^\circ \)[/tex] to radians.
[tex]\[
\text{angle\_A\_rad} = \frac{64 \times \pi}{180}
\][/tex]
This gives us:
[tex]\[
\text{angle\_A\_rad} \approx 1.117 \text{ radians}
\][/tex]
2. Apply the law of cosines:
The law of cosines states:
[tex]\[
a^2 = b^2 + c^2 - 2bc \cos(A)
\][/tex]
Substituting the known values:
[tex]\[
a^2 = 13^2 + 17^2 - 2 \times 13 \times 17 \times \cos(64^\circ)
\][/tex]
3. Calculate the squared diagonal length [tex]\( a^2 \)[/tex]:
Let's substitute the values:
[tex]\[
a^2 = 169 + 289 - 2 \times 13 \times 17 \times \cos(1.117)
\][/tex]
We need to find [tex]\( \cos(1.117) \)[/tex]:
[tex]\[
\cos(1.117) \approx 0.438
\][/tex]
So:
[tex]\[
a^2 = 169 + 289 - 2 \times 13 \times 17 \times 0.438
\][/tex]
Multiplying these together:
[tex]\[
a^2 = 169 + 289 - 2 \times 13 \times 17 \times 0.438 \approx 169 + 289 - 193.76 = 264.24
\][/tex]
4. Find the length of the diagonal [tex]\( x \)[/tex]:
Taking the square root of both sides to get [tex]\( x \)[/tex]:
[tex]\[
x = \sqrt{264.24} \approx 16.26
\][/tex]
5. Round to the nearest whole number:
Finally, rounding [tex]\( 16.26 \)[/tex] to the nearest whole number, we get:
[tex]\[
x \approx 16
\][/tex]
Therefore, the length of the diagonal [tex]\( x \)[/tex], rounded to the nearest whole number, is [tex]\( \boxed{16} \)[/tex].
