Town Hall is located 4.3 miles directly east of the middle school. The fire station is located 1.7 miles directly north of Town
Hall. Part B The hospital is 3.1 miles west of the fire station. What is the length of a straight line between the school and the hospital? Round to the nearest tenth. Enter your answer in the box.
1 answer:
Answer:
2.1 miles
Step-by-step explanation:
you can see a drawing of the situation in the attached picture (you need to draw all the places mentioned and the distance between them),
this way we can form a right triangle between the school and the hospital with measures of sides: 1.7 and 1.2 miles
and the green line is the straight line between the school and the hospital, so you need pythagoras to find the value:
rounding to the nearest tenth:
2.1 miles
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Answer:
distance: 320.624 miles
direction: 43.576°
Step-by-step explanation:
The speed and direction can be found by adding the given vectors.
... 320∠40° + 20∠130°
... = (320·cos(40°), 320·sin(40°)) + (20·cos(130°), 20·sin(130°))
... = (245.134, 205.692) +(-12.856, 15.321) = (232.278, 221.013)
The magnitude of the vector with these components is found using the Pythagorean theorem. The direction is found using the arctangent function.
... = √(232.278² +221.013²)∠arctan(221.013/232.278)
... = 320.624∠43.576°
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A suitable vector or graphing calculator can do this easily. In the screenshot of a TI-84 app below, the variable D has the value π/180. The display mode is set to degrees.
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<em>Comment on coordinate systems</em>
Navigation directions are generally measured clockwise from North. Angles in the usual x-y coordinate plane are measured counterclockwise from +x (effectively, East). You can consider the geometry of the navigation coordinate system to be a reflection across the line y=x of the geometry of the usual x-y coordinate system.
Reflection does not change lengths or angles within a given geometry. Hence, we can use all the usual tools of vector calculation using navigation coordinates, without bothering to translate them back and forth to x-y coordinates.
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Problems like this generally can be worked using the Law of Cosines and the Law of Sines, too. It generally helps to draw a diagram so you can find the values of the angles betwee the various vectors more easily.
