Final answer:
Without the value of angle A or assuming a right-angled triangle, it is not possible to calculate side b with the given information using the Law of Sines or any other trigonometric methods.
Step-by-step explanation:
To find side b in a triangle when given angle B, side c, and side a, you can use the Law of Sines, which states that the ratio of the length of a side to the sine of its opposite angle is constant for all sides and angles in a triangle. This gives us the equation:
\(\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\)
Since we are given angle B as 67°, side c as 13, and side a as 5, we need to find the sine of angle B and then rearrange the Law of Sines to solve for side b:
\(b = \frac{\sin(B) \cdot c}{\sin(C)}\)
We do not have angle C, but we can find it by using the fact that the sum of angles in a triangle is 180°. Since angle A is not given, angle C can be found as:
\(C = 180° - (A + B)\)
However, without angle A, we cannot proceed using this method. Instead, if angle A is a right angle (making it a right triangle), we can use the Pythagorean theorem or trigonometric identities directly. Assuming it's not provided, and this is not a right triangle, we are unable to calculate side b with the given information.
If additional information is provided that indicates this is a right triangle, or if angle A or C is known, the calculation could be completed. As it stands, with the current information, we cannot precisely calculate side b to one decimal place or select one of the provided options (a. 7.0, b. 9.6, c. 8.2, d. 6.5).