Answer: C. between 10 and 17 ft
Explanation: To find the possible length of \( AB \), we use the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Additionally, we can use the Law of Cosines to find the exact length of \( AB \).
Steps:
- Triangle Inequality Theorem:
- For \( \triangle ABC \):
\[
AB + BC > AC \quad \Rightarrow \quad AB + 7 > 10 \quad \Rightarrow \quad AB > 3
\]
\[
AB + AC > BC \quad \Rightarrow \quad AB + 10 > 7 \quad \Rightarrow \quad AB > -3 \quad (\text{not useful})
\]
\[
AC + BC > AB \quad \Rightarrow \quad 10 + 7 > AB \quad \Rightarrow \quad AB < 17
\]
- Law of Cosines:
- To find the exact length of \( AB \), use the Law of Cosines:
\[
AB^2 = AC^2 + BC^2 - 2 \cdot AC \cdot BC \cdot \cos(B)
\]
\[
AB^2 = 10^2 + 7^2 - 2 \cdot 10 \cdot 7 \cdot \cos(68^\circ)
\]
\[
AB^2 = 100 + 49 - 140 \cdot \cos(68^\circ)
\]
\[
AB^2 = 149 - 140 \cdot 0.3746 \quad (\text{using } \cos(68^\circ) \approx 0.3746)
\]
\[
AB^2 = 149 - 52.444
\]
\[
AB^2 = 96.556
\]
\[
AB \approx \sqrt{96.556} \approx 9.82
\]
Since \( AB \approx 9.82 \), it falls between 10 and 17 ft, making option C correct.