Jenny asked 10 students how many courses they have taken so far at her collegeHere is the list of answers, 12, 10, 14, 7, 1, 19,
17, 3, 20, 16 What is the percentage of these students who have taken fewer than 16 courses?
1 answer:
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A geometry program can show you the total displacement is about 10.14 miles.
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You can use a vector calculator to find the solution, too. Using bearing angles, the sum is
4∠90° + 3∠135° + 3∠180° + 4∠225° + 3∠90° ≈ 10.1390∠141.6354°
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If you want to do it by hand, you can recognize the sum will be
7 miles east + 3 miles south + 3 miles southeast + 4 miles southwest
Distances that are not in the direction of one of the coordinate axes can be translated to rectangular coordinates by
displacement*(cos(angle), sin(angle))
Angles can be measured in the conventional way—from the positive x-axis. A direction of southeast will be +315° or -45°. A direction of southwest will be +225° or -135°.
Then the sum of the displacements in rectangular coordinates is ...
= (7, -3) + 3*(cos(-45°), sin(-45°)) + 4*(cos(-135°), sin(-135°))
= (7, -3) + ((√2)/2)*((3, -3) + (-4, -4))
= (7, -3) + 0.707107*(-1, -7)
= (6.2929, -7.9497)
Then the Pythagorean theorem is used to find the direct distance from home to this displaced location.
d = √(6.2929² +(-7.9497)²) ≈ √102.7990
d ≈ 10.1390 . . . . miles
_____
You can use a vector calculator to find the solution, too. Using bearing angles, the sum is
4∠90° + 3∠135° + 3∠180° + 4∠225° + 3∠90° ≈ 10.1390∠141.6354°
_____
If you want to do it by hand, you can recognize the sum will be
7 miles east + 3 miles south + 3 miles southeast + 4 miles southwest
Distances that are not in the direction of one of the coordinate axes can be translated to rectangular coordinates by
displacement*(cos(angle), sin(angle))
Angles can be measured in the conventional way—from the positive x-axis. A direction of southeast will be +315° or -45°. A direction of southwest will be +225° or -135°.
Then the sum of the displacements in rectangular coordinates is ...
= (7, -3) + 3*(cos(-45°), sin(-45°)) + 4*(cos(-135°), sin(-135°))
= (7, -3) + ((√2)/2)*((3, -3) + (-4, -4))
= (7, -3) + 0.707107*(-1, -7)
= (6.2929, -7.9497)
Then the Pythagorean theorem is used to find the direct distance from home to this displaced location.
d = √(6.2929² +(-7.9497)²) ≈ √102.7990
d ≈ 10.1390 . . . . miles