Question

find the distance across the lake from a to c, to the nearest yard, using the measurements shown in the figure.
the distance across the lake from a to c is □ yards.
(round to the nearest yard as needed.)

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines formula is \(AC^{2}=AB^{2}+BC^{2}-2(AB)(BC)\cos B\). Given \(AB = 155\) yd, \(BC=140\) yd, and \(B = 81^{\circ}\).
Substitute the values into the formula:
\(AC^{2}=155^{2}+140^{2}-2\times155\times140\times\cos(81^{\circ})\)
First, calculate \(155^{2}=24025\), \(140^{2}=19600\), and \(\cos(81^{\circ})\approx0.1564\).
Then \(2\times155\times140 = 43400\).
\(AC^{2}=24025 + 19600-43400\times0.1564\)

Step2: Calculate the value of \(AC^{2}\)

\(24025+19600=43625\)
\(43400\times0.1564 = 43400\times\frac{1564}{10000}=43400\times0.1564 = 6787.76\)
\(AC^{2}=43625-6787.76=36837.24\)

Step3: Find the value of \(AC\)

Take the square - root of \(AC^{2}\): \(AC=\sqrt{36837.24}\approx192\)

Answer:

\(192\)