Question

find the length of side b.
a
b c
110° 30° 12
a = 40° c = 9.3 b = ?
round to the

Explanation:

Step1: Identify the Law of Sines

We can use the Law of Sines, which states that in any triangle, $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$. Here, we know side $BC = 12$ (let's call this side $a$), angle $A = 40^\circ$, angle $B = 110^\circ$, and we need to find side $b$.

Step2: Apply the Law of Sines

From the Law of Sines, we have $\frac{b}{\sin B}=\frac{a}{\sin A}$. We know $a = 12$, $\sin B=\sin 110^\circ$, $\sin A=\sin 40^\circ$. First, calculate $\sin 110^\circ\approx\sin(70^\circ)\approx0.9397$ and $\sin 40^\circ\approx0.6428$. Then, rearrange the formula to solve for $b$: $b=\frac{a\sin B}{\sin A}$.

Step3: Substitute the values

Substitute $a = 12$, $\sin B\approx0.9397$, and $\sin A\approx0.6428$ into the formula: $b=\frac{12\times0.9397}{0.6428}$.

Step4: Calculate the value

First, calculate the numerator: $12\times0.9397 = 11.2764$. Then divide by $0.6428$: $\frac{11.2764}{0.6428}\approx17.54$. Rounding to a reasonable decimal place (assuming to the nearest tenth), we get approximately $17.5$ or more accurately, let's check the calculation again. Wait, maybe we should use more precise values. $\sin 110^\circ=\sin(180 - 70)=\sin 70^\circ\approx0.9396926$, $\sin 40^\circ\approx0.6427876$. So $b=\frac{12\times0.9396926}{0.6427876}=\frac{11.2763112}{0.6427876}\approx17.54$, which rounds to $17.5$ or maybe the problem expects a different approach? Wait, also, we can check the angles: $A = 40^\circ$, $B = 110^\circ$, $C = 30^\circ$, so side $a$ is opposite angle $A$? Wait, no, in triangle notation, side $a$ is opposite angle $A$, side $b$ opposite angle $B$, side $c$ opposite angle $C$. Wait, in the diagram, side $BC$ is length 12, which is side $a$ (opposite angle $A$), side $AC$ is side $b$ (opposite angle $B$), side $AB$ is side $c$ (opposite angle $C$). So angle $A = 40^\circ$, angle $B = 110^\circ$, angle $C = 30^\circ$, side $a = 12$ (BC), side $b = AC$ (what we need to find), side $c = AB = 9.3$ (given). So using Law of Sines: $\frac{b}{\sin B}=\frac{a}{\sin A}$. So $b=\frac{a\sin B}{\sin A}=\frac{12\times\sin 110^\circ}{\sin 40^\circ}$. Let's compute this: $\sin 110^\circ\approx0.9397$, $\sin 40^\circ\approx0.6428$. So $12\times0.9397 = 11.2764$, divided by $0.6428$ is approximately $17.54$, so rounding to the nearest tenth, $17.5$ or maybe the problem wants to the nearest whole number, $18$? Wait, let's check with more precise calculation. $\sin 110^\circ = \sin(110) \approx 0.9396926208$, $\sin 40^\circ \approx 0.6427876097$. So $12\times0.9396926208 = 11.27631145$, divided by $0.6427876097$ is $11.27631145\div0.6427876097\approx17.54$, so approximately $17.5$ or $18$ depending on rounding.

Answer:

Approximately $\boxed{17.5}$ (or $\boxed{18}$ if rounded to the nearest whole number)