Noah and Brianna want to calculate the distance between their houses which are opposite sides of a water park. they mark a point, A, 150 m along the edge of the water park from brianna's house. the measure �ÚNBA as 80�‹and �ÚBAN as 75�‹. Determine the distance between their houses.
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To determine the distance between Noah and Brianna's houses, we can use the law of cosines. The law of cosines states that for any triangle, the square of one side is equal to the sum of squares of the other two sides minus twice the product of the lengths of those two sides multiplied by the cosine of the angle between them.
In this case, we want to find the length of side AB, which is the distance between Noah's house and the marked point A.
Let's assign the following values:
- Length of side AB (distance between houses) = x
- Length of side AN = 150m
- Angle NBA = 80 degrees
- Angle BAN = 75 degrees
Using the law of cosines, the formula for side AB is:
x^2 = (150)^2 + (x)^2 - 2(150)(x)cos(80)
Now, we can solve this equation to find the value of x (distance between houses).
Simplifying the equation, we get:
x^2 = 22500 + x^2 - 300x(cos(80))
Since the equation has an x^2 term on both sides, we can subtract x^2 from both sides of the equation:
0 = 22500 - 300x(cos(80))
Rearranging the equation:
300x(cos(80)) = 22500
Dividing both sides of the equation by 300(cos(80)):
x = 22500 / (300(cos(80)))
Using a calculator, we can determine that cos(80 degrees) is approximately 0.1736. Substituting this value back into the equation:
x = 22500 / (300(0.1736))
Calculating this expression, we get:
x ≈ 115.323
Therefore, the distance between Noah and Brianna's houses is approximately 115.323 meters.