Points A and B are separated by a lake. To find the distance between them, a surveyor locates a point C on land such that ∠CAB = 48.3°.

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 that ∠CAB = 48.3°. He also measures CA as 320 ft and CB as 527 ft. Find the distance between A and B. (Round your answer to one decimal place.)

??? ft

Answer 1

This is probably an unconventional way of solving it, but here goes:

Let x be the unknown distance. Note that the distance BC, 527 ft, is opposite the only known angle, 48.3^o. Call this angle A. Let a = 527 ft and b = 320 ft.

Use the Law of Cosines:

a² = b² + x² - 2bx*cos(A)

x² - 2*b*x*cos(A) + (b² - a² ) = 0

2*b*cos(A) = 2 * 320 * cos(48.3^o) = 425.75

b² - a² = 320² - 527² = -175329

x² - (425.75)x - 175329 = 0

Using a free online tool such as Octave-online, the roots are x = 682.60 and x = -256.85. Since x must be positive, our final answer is: 682.6 ft