To find the length of the diagonal of the parallelogram, we will use the Law of Cosines. The Law of Cosines states that for any triangle with sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex], and an angle [tex]\(A\)[/tex] opposite side [tex]\(a\)[/tex]:
[tex]\[ a^2 = b^2 + c^2 - 2bc \cos(A) \][/tex]
Here, we are dealing with a parallelogram with two sides of lengths 13 and 17, and an angle of [tex]\(64^\circ\)[/tex] between them. We need to find the length of the diagonal, which we'll call [tex]\(x\)[/tex].
1. Identify the given information:
- Side [tex]\( a = 13 \)[/tex]
- Side [tex]\( b = 17 \)[/tex]
- Angle [tex]\( A = 64^\circ \)[/tex]
2. Set up the Law of Cosines formula:
[tex]\[ x^2 = 13^2 + 17^2 - 2 \cdot 13 \cdot 17 \cdot \cos(64^\circ) \][/tex]
3. Compute each term separately:
- [tex]\( 13^2 = 169 \)[/tex]
- [tex]\( 17^2 = 289 \)[/tex]
- [tex]\( 2 \cdot 13 \cdot 17 = 442 \)[/tex]
- [tex]\( \cos(64^\circ) \approx 0.43837 \)[/tex]
4. Substitute into the formula:
[tex]\[ x^2 = 169 + 289 - 442 \cdot 0.43837 \][/tex]
5. Calculate the product:
[tex]\[ 442 \cdot 0.43837 \approx 193.760047 \][/tex]
6. Complete the equation:
[tex]\[ x^2 = 169 + 289 - 193.760047 \][/tex]
[tex]\[ x^2 \approx 264.239953 \][/tex]
7. Take the square root to solve for [tex]\( x \)[/tex]:
[tex]\[ x \approx \sqrt{264.239953} \][/tex]
[tex]\[ x \approx 16.255459 \][/tex]
8. Round to the nearest whole number:
[tex]\[ x \approx 16 \][/tex]
Thus, the length of the diagonal to the nearest whole number is [tex]\( 16 \)[/tex].
Therefore, the correct answer is:
16
