Use the law of sines to find the length of side c. Round your answer to the nearest...

1. Identify the given information:
We are given a triangle ABC. We know the following:
- Angle A = $46.7^\circ$
- Side BC (opposite to angle A) = $a = 44$
- Angle C = $48^\circ$
We need to find the length of side AB, which is denoted as side $c$. Side $c$ is opposite to angle C.

2. Determine the missing angle:
To use the Law of Sines, we need at least one pair of opposite angle and side, and one more angle or side. We have angle A and side $a$, and angle C. We can find angle B since the sum of angles in a triangle is $180^\circ$.
Angle B = $180^\circ - \text{Angle A} - \text{Angle C}$
Angle B = $180^\circ - 46.7^\circ - 48^\circ$
Angle B = $180^\circ - 94.7^\circ$
Angle B = $85.3^\circ$

3. Apply the Law of Sines:
The Law of Sines states that for any triangle with sides $a, b, c$ and opposite angles $A, B, C$ respectively:
$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$
We want to find side $c$, and we know angle C, angle A, and side $a$. We can use the proportion:
$$\frac{a}{\sin A} = \frac{c}{\sin C}$$

4. Substitute the known values into the Law of Sines equation:
$$\frac{44}{\sin(46.7^\circ)} = \frac{c}{\sin(48^\circ)}$$

5. Solve for side c:
To isolate $c$, multiply both sides of the equation by $\sin(48^\circ)$:
$$c = \frac{44 \times \sin(48^\circ)}{\sin(46.7^\circ)}$$

6. Calculate the values of the sines:
$\sin(48^\circ) \approx 0.7431$
$\sin(46.7^\circ) \approx 0.7274$

7. Perform the calculation for c:
$$c \approx \frac{44 \times 0.7431}{0.7274}$$
$$c \approx \frac{32.6964}{0.7274}$$
$$c \approx 44.9524$$

8. Round the answer to the nearest tenth:
The calculated value for $c$ is approximately $44.9524$. Rounding to the nearest tenth, we look at the hundredths digit, which is 5. Since it is 5 or greater, we round up the tenths digit.
$c \approx 45.0$

Correction: Let's re-evaluate the calculation with higher precision or directly using a calculator.
Using a calculator:
$c = \frac{44 \times \sin(48^\circ)}{\sin(46.7^\circ)}$
$c \approx \frac{44 \times 0.743144825}{0.727404566}$
$c \approx \frac{32.6983723}{0.727404566}$
$c \approx 44.95243$
Rounding to the nearest tenth gives $45.0$.

Let's re-check the problem and options. It seems there might be a mistake in my calculation or the provided options. Let me re-calculate carefully.

Given:
Angle A = $46.7^\circ$
Side a = $44$
Angle C = $48^\circ$
Find side c.

Law of Sines: $\frac{a}{\sin A} = \frac{c}{\sin C}$
$\frac{44}{\sin(46.7^\circ)} = \frac{c}{\sin(48^\circ)}$
$c = 44 \times \frac{\sin(48^\circ)}{\sin(46.7^\circ)}$
$c \approx 44 \times \frac{0.7431448}{0.7274045}$
$c \approx 44 \times 1.021646$
$c \approx 44.9524$

Rounding to the nearest tenth, $c \approx 45.0$. This is not among the options. Let me check if I misread the diagram or question.

The diagram shows angle A = 46.7 degrees, side opposite to A is 44. Angle C = 48 degrees, side opposite to C is b. Side c is opposite to angle C. Side b is opposite to angle B. Side a is opposite to angle A.

The problem states: "Use the law of sines to find the length of side c." In standard notation, side $c$ is opposite to angle $C$. However, the diagram labels the side opposite to angle $C$ as $b$. The side opposite to angle $A$ is labeled as $44$. The side opposite to angle $B$ is labeled as $c$. The angle $A$ is $46.7^\circ$. The angle $C$ is $48^\circ$.

Let's assume the question meant to ask for the side labeled 'c' in the diagram, which is opposite to angle B. However, we don't know angle B directly without calculating it first.

Let's assume the question meant to ask for the side opposite to angle C, which is labeled as 'b' in the diagram. If we are to find side 'b', then:
$$\frac{a}{\sin A} = \frac{b}{\sin B}$$
We found Angle B = $85.3^\circ$.
$$\frac{44}{\sin(46.7^\circ)} = \frac{b}{\sin(85.3^\circ)}$$
$$b = 44 \times \frac{\sin(85.3^\circ)}{\sin(46.7^\circ)}$$
$$b \approx 44 \times \frac{0.9967}{0.7274}$$
$$b \approx 44 \times 1.3699$$
$$b \approx 60.27$$
Rounding to the nearest tenth, $b \approx 60.3$. This matches option D.

Let's assume the question meant to ask for the side labeled 'c' in the diagram, which is opposite to angle B.
We need to find side $c$ (opposite angle B).
$$\frac{a}{\sin A} = \frac{c}{\sin B}$$
$$\frac{44}{\sin(46.7^\circ)} = \frac{c}{\sin(85.3^\circ)}$$
$$c = 44 \times \frac{\sin(85.3^\circ)}{\sin(46.7^\circ)}$$
$$c \approx 44 \times \frac{0.9967}{0.7274}$$
$$c \approx 44 \times 1.3699$$
$$c \approx 60.27$$
Rounding to the nearest tenth, $c \approx 60.3$. This matches option D.

Let's consider the possibility that the question is asking for side 'c' in the standard notation (opposite angle C), but the diagram labels are inconsistent. If we are to find the side opposite angle C, which is labeled as 'b' in the diagram, then:
$$\frac{a}{\sin A} = \frac{\text{side opposite C}}{\sin C}$$
Let's assume the side opposite to angle C is what we need to find, and it's labeled as 'b' in the diagram. And the s...