Martina and Carl are part of a team that is studying weather patterns. The team is about to launch a weather balloon to collect data. Martina's rope is 7.8 m long and makes an angle of 36.0 deg with the ground. Carl's rope is 5.9 m long. Assuming that Martina and Carl form a triangle in a vertical plabne with the weather balloon, what is the distance between Martina and Carl, to the nearest tenth of a meter?
Draw a diagram, with triangle MCB where
M is Martina
C is Carl
B is Balloon
Let P be the base of the altitude from B to MC
MP/7.8 = cos 36°
BP/7.8 = sin 36°
BP^2 + PC^2 = 5.9^2
we want MC = MP+PC
Plug and chug
To find the distance between Martina and Carl, we can use the Law of Cosines. The formula for the Law of Cosines is:
c² = a² + b² - 2ab*cos(C)
Where:
c is the length of the side opposite angle C (the distance between Martina and Carl),
a and b are the lengths of the other two sides (Martina's and Carl's ropes),
C is the angle between the sides a and b (the angle between Martina's and Carl's ropes).
Let's calculate the distance between Martina and Carl:
c² = 7.8² + 5.9² - 2 * 7.8 * 5.9 * cos(36.0)
c² ≈ 60.84 + 34.81 - 92.232 * 0.809
c² ≈ 60.84 + 34.81 - 74.583088
c² ≈ 121.651912
c ≈ √(121.651912)
c ≈ 11.02 meters
Therefore, the distance between Martina and Carl is approximately 11.0 meters to the nearest tenth of a meter.
To find the distance between Martina and Carl, we can use the cosine rule. The cosine rule states that in a triangle with sides a, b, and c, and angle C opposite side c:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, let's call the distance between Martina and Carl d. We know the lengths of the two ropes and the angle between them.
Martina's rope length = 7.8 m
Carl's rope length = 5.9 m
Angle between the ropes = 36.0 degrees
Using the cosine rule, we can find the distance d:
d^2 = 7.8^2 + 5.9^2 - 2 * 7.8 * 5.9 * cos(36.0)
Plugging in the values, we get:
d^2 = 60.84 + 34.81 - 2 * 7.8 * 5.9 * cos(36.0)
Next, calculate the cosine of 36.0 degrees:
cos(36.0 degrees) ≈ 0.809
Substituting this value in:
d^2 = 60.84 + 34.81 - 2 * 7.8 * 5.9 * 0.809
d^2 ≈ 95.65
To find d, take the square root of both sides:
d ≈ √95.65
d ≈ 9.8 meters (rounded to the nearest tenth of a meter)
Therefore, the distance between Martina and Carl, to the nearest tenth of a meter, is approximately 9.8 meters.