Mathematics : asked on small77
 09.03.2022

How can you use the coordinates of Island A and Island B to find the distance between the islands?

. 5

Faq

Mathematics
Step-by-step answer
P Answered by PhD

9514 1404 393

  (13.06, 16.41)

Step-by-step explanation:

Any point not on a perpendicular bisector of the line between towns will be closer to one town than the other. So, the intersection of perpendicular bisectors will be the equidistant from all towns. That point is the circumcenter of the circle circumscribing the triangle with the towns as its vertices. This can be found as the intersection of two perpendicular bisectors of the segments between the towns.

The midpoint of AB is ...

  (A +B)/2 = ((15, 3) +(25, 10))/2 = (40, 13)/2 = (20, 6.5)

The midpoint of AC is ...

  (A +C)/2 = ((15, 3) +(0, 20))/2 = (15, 23)/2 = (7.5, 11.5)

__

The differences between points are ...

  B -A = (25, 10) -(15, 3) = (10, 7)

  C -A = (0, 20) -(15, 3) = (-15, 17)

Then the perpendicular bisector equations can be written as ...

  bisector of AB  ⇒  10(x -20) +7(y -6.5) = 0

     10x +7x -245.5 = 0

  bisector of AC  ⇒  -15(x -7.5) +17(y -11.5) = 0

     -15x +17y -83 = 0

These have coefficients that aren't particularly nice, so a method similar to Cramer's Rule is suitable for solving this pair of equations. Using the "cross multiplication method", we have ...

  x = (7(-83) -17(-245.5))/(10(17) -(-15)(7)) = 3592.5/275 = 13 7/110 ≈ 13.06

  y = (-245.5(-15) -(-83)(10))/275 = 4512.5/275 = 16 9/22 ≈ 16.41

The point as far as possible from each town within the boundary of the island has coordinates (13.06, 16.41).

_____

Further detail regarding the solution

Here, we have written the equation for the perpendicular bisector this way. The difference in coordinates between points R and S can be called ...

  R -S = (∆x, ∆y)

The midpoint of segment RS can be called ...

  (R +S)/2 = (h, k)

Then the line that is a perpendicular bisector of RS can be written as ...

  ∆x(x -h) +∆y(y -k) = 0

This is the form that is used above.

__

The "cross multiplication method" for solving a pair of general-form linear equations ...

  ax +by +c = 0

  dx +ey +g = 0

will give the solution as ...

  1/(ae -db) = x/(bg -ec) = y/(cd -ga)

If you write the coefficients in two rows of four, the pattern of products and differences is easy to see.

  \left[\begin{array}{cccc}a&b&c&a\\d&e&g&d\end{array}\right]


an island has three towns located at points with the coordinates a (15,3) b (25,10) and c (0,20) whe
StudenGPT
Step-by-step answer
P Answered by Studen AI
The section from paragraph 22 that best supports the answer to Part A is C. "anybody that says they weren't scared, well they're just not quite telling the truth." This section explains that the speaker, J.C. Alton, admits that he and others were scared during the attack on Pearl Harbor. It shows the emotional impact and fear experienced by the soldiers during the attack. This supports the idea that the soldiers were not just focused on their duty but also experienced fear and emotions during the attack.
Mathematics
Step-by-step answer
P Answered by PhD

9514 1404 393

  (13.06, 16.41)

Step-by-step explanation:

Any point not on a perpendicular bisector of the line between towns will be closer to one town than the other. So, the intersection of perpendicular bisectors will be the equidistant from all towns. That point is the circumcenter of the circle circumscribing the triangle with the towns as its vertices. This can be found as the intersection of two perpendicular bisectors of the segments between the towns.

The midpoint of AB is ...

  (A +B)/2 = ((15, 3) +(25, 10))/2 = (40, 13)/2 = (20, 6.5)

The midpoint of AC is ...

  (A +C)/2 = ((15, 3) +(0, 20))/2 = (15, 23)/2 = (7.5, 11.5)

__

The differences between points are ...

  B -A = (25, 10) -(15, 3) = (10, 7)

  C -A = (0, 20) -(15, 3) = (-15, 17)

Then the perpendicular bisector equations can be written as ...

  bisector of AB  ⇒  10(x -20) +7(y -6.5) = 0

     10x +7x -245.5 = 0

  bisector of AC  ⇒  -15(x -7.5) +17(y -11.5) = 0

     -15x +17y -83 = 0

These have coefficients that aren't particularly nice, so a method similar to Cramer's Rule is suitable for solving this pair of equations. Using the "cross multiplication method", we have ...

  x = (7(-83) -17(-245.5))/(10(17) -(-15)(7)) = 3592.5/275 = 13 7/110 ≈ 13.06

  y = (-245.5(-15) -(-83)(10))/275 = 4512.5/275 = 16 9/22 ≈ 16.41

The point as far as possible from each town within the boundary of the island has coordinates (13.06, 16.41).

_____

Further detail regarding the solution

Here, we have written the equation for the perpendicular bisector this way. The difference in coordinates between points R and S can be called ...

  R -S = (∆x, ∆y)

The midpoint of segment RS can be called ...

  (R +S)/2 = (h, k)

Then the line that is a perpendicular bisector of RS can be written as ...

  ∆x(x -h) +∆y(y -k) = 0

This is the form that is used above.

__

The "cross multiplication method" for solving a pair of general-form linear equations ...

  ax +by +c = 0

  dx +ey +g = 0

will give the solution as ...

  1/(ae -db) = x/(bg -ec) = y/(cd -ga)

If you write the coefficients in two rows of four, the pattern of products and differences is easy to see.

  \left[\begin{array}{cccc}a&b&c&a\\d&e&g&d\end{array}\right]


an island has three towns located at points with the coordinates a (15,3) b (25,10) and c (0,20) whe
StudenGPT
Step-by-step answer
P Answered by Studen AI
Based on the interview, we can infer that J.C. Alton has a high regard for his fellow soldiers and their experiences. Here are some pieces of evidence from the text to support this inference:

1. In response to Daniel Rowe's question about the emotions experienced during the attack, J.C. Alton states that nobody expected anything, and anyone who says they weren't scared is not telling the truth. This suggests that he understands and acknowledges the fear and vulnerability felt by his fellow soldiers during the attack on Pearl Harbor.

2. J.C. Alton mentions that the captain of the West Virginia was killed by shrapnel from a bomb that hit the nearby Maryland. He recalls that the West Virginia had come by to check on their ship, which shows concern and camaraderie among the soldiers.

3. When discussing the casualties of the attack, J.C. Alton mentions that different numbers are often shared, but he estimates around 2,600 lives were lost. By acknowledging the loss of lives, he demonstrates empathy and recognition of the sacrifices made by his fellow soldiers.

These evidences indicate that J.C. Alton has a deep understanding and respect for the experiences and hardships endured by his fellow soldiers during the attack on Pearl Harbor.
English
Step-by-step answer
P Answered by Specialist

Part A:

Answer

One of Roosevelt's purposes was to inform the American people of Japan's attack on the US and other places in the Pacific.

The other possible answer is the one that mentions Japan being an ally of the Nazis, however the word "Nazi" is never mentioned on the speech.


Part B:

Answer

He uses repetition, employing the word ¨attack¨ over and over, to inspire Americans to join the armed forces and aggressively take on the enemy.

This is definitely the rhetoric that applies to advance his purpose from part A. The other options wouldn't apply.

English
Step-by-step answer
P Answered by Specialist

Part A:

Answer

One of Roosevelt's purposes was to inform the American people of Japan's attack on the US and other places in the Pacific.

The other possible answer is the one that mentions Japan being an ally of the Nazis, however the word "Nazi" is never mentioned on the speech.


Part B:

Answer

He uses repetition, employing the word ¨attack¨ over and over, to inspire Americans to join the armed forces and aggressively take on the enemy.

This is definitely the rhetoric that applies to advance his purpose from part A. The other options wouldn't apply.

Try asking the Studen AI a question.

It will provide an instant answer!

FREE