Question

solve the triangle.
b = 72° 15, c = 27 m, a = 71 m
what is the length of side b?
b = □ m
(round to the nearest whole number as needed.)
what is the measure of angle a?
a = □° □
(round to the nearest whole number as needed.)
what is the measure of angle c?
c = □° □
(round to the nearest whole number as needed.)

Explanation:

Step1: Convert angle B to decimal degrees

First, we convert \( 72^\circ 15' \) to decimal degrees. Since \( 1'=\frac{1}{60}^\circ \), so \( 15' = \frac{15}{60}=0.25^\circ \). Thus, \( B = 72.25^\circ \).

Step2: Use the Law of Cosines to find side b

The Law of Cosines states that \( b^{2}=a^{2}+c^{2}-2ac\cos B \). We know that \( a = 71 \), \( c = 27 \), and \( B=72.25^\circ \).

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We know that \( \cos(72.25^\circ)\approx\cos(72^\circ15')\approx0.3057 \)

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Then \( b=\sqrt{4597.9462}\approx68 \)

Step3: Use the Law of Sines to find angle A

The Law of Sines states that \( \frac{\sin A}{a}=\frac{\sin B}{b} \), so \( \sin A=\frac{a\sin B}{b} \)
We know \( a = 71 \), \( b\approx68 \), \( B = 72.25^\circ \)

$$ \sin A=\frac{71\times\sin(72.25^\circ)}{68} $$

\( \sin(72.25^\circ)\approx0.9526 \)

$$ \sin A=\frac{71\times0.9526}{68}=\frac{67.6346}{68}\approx0.9946 $$

So \( A=\arcsin(0.9946)\approx84^\circ \) (since \( \sin(84^\circ)\approx0.9945 \))

Step4: Find angle C

We know that the sum of angles in a triangle is \( 180^\circ \). So \( C=180^\circ - A - B \)
\( A\approx84^\circ \), \( B = 72.25^\circ=72^\circ15' \)
\( C=180^\circ-(84^\circ + 72^\circ15')=180^\circ - 156^\circ15'=23^\circ45'\approx24^\circ \) (rounded to nearest whole number for minutes, but we can also calculate using Law of Sines. Let's verify using Law of Sines: \( \frac{\sin C}{c}=\frac{\sin B}{b} \), \( \sin C=\frac{c\sin B}{b}=\frac{27\times0.9526}{68}=\frac{25.7202}{68}\approx0.3782 \), \( C=\arcsin(0.3782)\approx22.2^\circ\approx22^\circ12' \), but due to rounding errors in previous steps, we can also use the angle sum. The more accurate way: Since \( A + B + C=180^\circ \), \( C = 180^\circ - A - B \). If \( A\approx84^\circ \) and \( B = 72.25^\circ \), then \( C=180 - 84 - 72.25=23.75^\circ = 23^\circ45' \approx24^\circ \) (if we take \( A = 84^\circ \)) or if we calculate more accurately:

From Law of Sines: \( \sin A=\frac{71\sin(72.25^\circ)}{68}\approx\frac{71\times0.9526}{68}\approx0.9946 \), so \( A = 84^\circ \) (since \( \sin(84^\circ)\approx0.9945 \)) or \( A = 180 - 84=96^\circ \), but since \( a > b \), \( A > B \), but \( b\approx68 \), \( a = 71 \), so \( A \) is acute? Wait, \( \sin A = 0.9946 \), so \( A = 84^\circ \) or \( A=96^\circ \). But if \( A = 96^\circ \), then \( C=180 - 96 - 72.25 = 11.75^\circ \), and \( \sin C=\frac{27\sin(72.25^\circ)}{68}\approx\frac{27\times0.9526}{68}\approx0.378 \), \( \sin(11.75^\circ)\approx0.203 \), which is not equal. So \( A = 84^\circ \) is correct. Then \( C=180 - 84 - 72.25 = 23.75^\circ=23^\circ45'\approx24^\circ \) (or \( 23^\circ45' \))

Answer:

Length of side \( b \): \( \boxed{68} \) m

Measure of angle \( A \): \( \boxed{84^\circ 0'} \) (or \( 84^\circ \))

Measure of angle \( C \): \( \boxed{24^\circ 0'} \) (or \( 23^\circ45'\approx24^\circ \))

(Note: The values of angles may have slight variations due to rounding during the calculation process. If we calculate more precisely:

For angle \( A \):

\( \sin A=\frac{71\sin(72.25^\circ)}{68}=\frac{71\times0.952579}{68}=\frac{67.6331}{68}\approx0.9946 \)

\( A=\arcsin(0.9946)=84.0^\circ \) (since \( \sin(84^\circ)=0.994522 \))

For angle \( C \):

\( C = 180 - 84 - 72.25=23.75^\circ=23^\circ45' \))