Question

points a and b are separated by a lake. to find the distance between them, a surveyor locates a point c on land such than ∠cab = 45.3°. find the distance across the lake from a to b. b 458 ft 45.3° c 355 ft a note: the triangle is not drawn to scale. distance = ft enter your answer as a number; your answer should be accurate to 2 decimal places. question help: worked example 1 submit question

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines formula is \( c^{2}=a^{2}+b^{2}-2ab\cos C\). In \(\triangle ABC\), let \(a = 458\), \(b = 355\), and \(C=45.3^{\circ}\). We want to find the length of \(AB\) (let's call it \(c\)).

$$c^{2}=458^{2}+355^{2}-2\times458\times355\times\cos(45.3^{\circ})$$

Step2: Calculate each term

• Calculate \(458^{2}=458\times458 = 209764\)
• Calculate \(355^{2}=355\times355=126025\)
• Calculate \(2\times458\times355 = 2\times162590=325180\)
• Calculate \(\cos(45.3^{\circ})\approx0.70356\)
• Then \(325180\times0.70356\approx325180\times0.70356 = 228829.5368\)
• Now \(c^{2}=209764 + 126025-228829.5368\)
• \(c^{2}=335789 - 228829.5368=106959.4632\)

Step3: Find \(c\)

Take the square - root of \(c^{2}\). \(c=\sqrt{106959.4632}\approx327.05\)

Answer:

\(327.05\)