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 \(AB = 130\) yd, \(BC=125\) yd, and \(B = 88^{\circ}\).
Substitute the values: \(AC^{2}=130^{2}+125^{2}-2\times130\times125\times\cos88^{\circ}\)

Step2: Calculate each term

• \(130^{2}=16900\)
• \(125^{2}=15625\)
• \(\cos88^{\circ}\approx0.0349\)
• \(2\times130\times125\times\cos88^{\circ}=2\times130\times125\times0.0349 = 2\times16250\times0.0349=32500\times0.0349 = 1134.25\)
• Then \(AC^{2}=16900 + 15625-1134.25\)
• \(AC^{2}=32525 - 1134.25=31390.75\)

Step3: Find \(AC\)

Take the square - root: \(AC=\sqrt{31390.75}\approx177.2\)

Answer:

\(177\)