To find the distance from the house at A to the house at B, a surveyor measures the angle ACB, which is found to be 70°, and then walks off the distance to each house, 50 feet and 70 feet, respectively. How far apart are the houses?

feet

x^2 = 7400-((7000*(cos(70)) = 5005.858997

square root of 5005.858997 =

70.75209535 ft

Triangle ACB, C = 70 deg, a = 70, b = 50

find side c

Use Law of Cosines--Case III
(Given two sides and the included angle)

c^2 = a^2 + b^2 - 2ab cos C
c^2 = 70^2 + 50^2 - 2(70)(50) cos 70

Solve for c

Well, let's see here. According to the information you provided, the surveyor measured the angle ACB as 70°. Now, I'm no mathematician, but I'm pretty sure the Pythagorean theorem might come in handy here.

So, we have two sides of a triangle - 50 feet and 70 feet. We can call the distance between the two houses 'x'. Now, we can use a little trigonometry magic to figure out the third side.

Using the sine function, we can say that sin(70°) = 70 / x. Solving for x, we get x = 70 / sin(70°).

But let's not stop there! We don't want to end up with a boring decimal answer, do we? Let's call in the clown mathematicians and crunch some numbers.

Using their fancy math skills, they've determined that x is approximately 81.53 feet apart. So, according to my calculations, the distance between the houses is approximately 81.53 feet.

Remember, math problems don't always have to be serious - sometimes you just need a little clowning around!

To find the distance between the houses at A and B, we can use the Law of Cosines. According to the law, the square of the distance between two points is equal to the sum of the squares of the distances between each point and a third point, multiplied by twice the product of the distances.

Let's take point C as the third point, such that AC = 50 feet and BC = 70 feet. The angle ACB is given as 70°.

Using the Law of Cosines, we have:

AB^2 = AC^2 + BC^2 - 2 * AC * BC * cos(ACB)

Substituting the values we have:

AB^2 = 50^2 + 70^2 - 2 * 50 * 70 * cos(70°)

Now we'll calculate the value inside the square root:

AB^2 = 2500 + 4900 - 7000 * cos(70°)

AB^2 = 2400 - 7000 * cos(70°)

AB^2 = 2400 - 7000 * 0.3420

AB^2 = 2400 - 2394.0

AB^2 = 6.0

Taking the square root of both sides, we find:

AB ≈ √6 ≈ 2.449

Therefore, the houses are approximately 2.449 feet apart.

2966.76