The problem that follows refers to triangle ABC . If and yd, find a
Apply the rules regarding the use of significant digits after all computations are complete.

A)49.1 yd
B)56.1 yd
C)53.1 yd
D)55.1 yd
E)54.1 yd

Question

The problem that follows refers to triangle ABC . ​ If   and   yd, find a
Apply the rules regarding the use of significant digits after all computations are complete.
​
A)49.1 yd 
B)56.1 yd 
C)53.1 yd 
D)55.1 yd 
E)54.1 yd

Quizplus · Accepted Answer

​Using the Law of Cosines, we have:
$ab^2 + bc^2 - 2abcos(\angle ABC) = ac^2$
$11^2 + 13^2 - 2\cdot 11\cdot 13cos(\angle ABC) = 12^2$
$cos(\angle ABC) = \frac{11^2 + 13^2 - 12^2}{2\cdot 11\cdot 13}$
$cos(\angle ABC) = 0.303$
Using the Law of Sines, we have:
$\frac{11}{sin(\angle ABC)} = \frac{yd}{sin(\angle A)}$
$\frac{13}{sin(\angle ABC)} = \frac{yd}{sin(\angle C)}$
$\frac{11}{sin^{-1}(0.303)} = \frac{13}{sin^{-1}(0.303)} = \frac{yd}{sin(\angle B)}$
$\angle A$ and $\angle C$ are acute angles, so we can use the approximation $sin(	heta) \approx 	heta$ when $	heta$ is small, in radians. Therefore:
$\frac{11}{0.309} \approx 35.6$ yd
$\frac{13}{0.309} \approx 42.1$ yd
$\frac{yd}{0.303} \approx 54.1$ yd
We take the value of $yd$ that has the least number of significant digits, which is the second calculation, giving $yd \approx 42.1$ yd. Thus, the answer is C, 53.1 yd.

The Problem That Follows Refers to Triangle ABC