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Question
The epicenter of an earthquake is the point on Earth's surface directly above the earthquake's origin. A seismograph can be used to determine the distance to the epicenter of an earthquake. Seismographs are needed in three different places to locate an earthquake's epicenter. Find the location of the earthquake's epicenter if it is 2 miles away from A(−2,5), 4 miles away from B(2,3), and 7 miles away from C(−2,−4).

Respuesta :

mhanifa

Answer:

  • The epicenter is ( - 2, 3)

Step-by-step explanation:

Let the epicenter be E(x, y).

Use the distance formula for each given distance:

  • AE² = (x + 2)² + (y - 5)² = 2²
  • BE² = (x - 2)² + (y - 3)² = 4²
  • CE² = (x + 2)² + (y + 4)² = 7²

Use the first and third equations, considering both have same term for x, and solve for y:

  • (y - 5)² - 4 = (y + 4)² - 49
  • y² - 10y + 25 - 4 = y² + 8y + 16 - 49
  • 18y = 54
  • y = 3

Substitute y into one of equations and solve for x:

  • (x + 2)² + (3 - 5)² = 4
  • (x + 2)² + 4 = 4
  • (x + 2)² = 0
  • x + 2 = 0
  • x = - 2

The epicenter is E( - 2, 3)

semsee45

Answer:

(-2, 3)

Step-by-step explanation:

Given:

  • A = (-2, 5)  →  2 miles away
  • B = (2, 3)  →  4 miles away
  • C = (-2, -4)  →  7 miles away

Model each given point as the center of a circle and the distance the epicenter is away from the point as the circle's radius.

The epicenter's location will be the point of intersection of the three circles.

Equation of a circle

[tex](x-a)^2+(y-b)^2=r^2[/tex]

where (a, b) is the center and r is the radius

Circle A

  • center = (-2, 5)
  • radius = 2 miles

[tex]\implies (x+2)^2+(y-5)^2=4[/tex]

[tex]\implies x^2+4x+4+y^2-10y+25=4[/tex]

[tex]\implies x^2+4x+y^2-10y+25=0[/tex]

Circle B

  • center = (2, 3)
  • radius = 4 miles

[tex]\implies (x-2)^2+(y-3)^2=16[/tex]

[tex]\implies x^2-4x+4+y^2-6y+9=16[/tex]

[tex]\implies x^2-4x+y^2-6y-3=0[/tex]

Circle C

  • center = (-2, -4)
  • radius = 7 miles

[tex]\implies (x+2)^2+(y+4)^2=49[/tex]

[tex]\implies x^2+4x+4+y^2+8y+16=49[/tex]

[tex]\implies x^2+4x+y^2+8y-29=0[/tex]

To find the point of intersection of the three circles, solve simultaneously.

Substitute equation B into equation A to eliminate x² and y²:

[tex]\implies x^2-4x+y^2-6y-3=x^2+4x+y^2-10y+25[/tex]

[tex]\implies -4x-6y-3=4x-10y+25[/tex]

Rearrange to isolate y:

[tex]\implies -6y-3=8x-10y+25[/tex]

[tex]\implies 4y-3=8x+25[/tex]

[tex]\implies 4y=8x+28[/tex]

[tex]\implies y=2x+7[/tex]

Substitute the expression for y into equation C and simplify:

[tex]\implies x^2+4x+(2x+7)^2+8(2x+7)-29=0[/tex]

[tex]\implies x^2+4x+4x^2+28x+49+16x+56-29=0[/tex]

[tex]\implies 5x^2+48x+76=0[/tex]

Substitute the expression for y into equation B and simplify:

[tex]\implies x^2-4x+(2x+7)^2-6(2x+7)-3=0[/tex]

[tex]\implies x^2-4x+4x^2+28x+49-12x-42-3=0[/tex]

[tex]\implies 5x^2+12x+4=0[/tex]

Equate the equations to eliminate 5x² and solve for x:

[tex]\implies 5x^2+48x+76=5x^2+12x+4[/tex]

[tex]\implies 48x+76=12x+4[/tex]

[tex]\implies 36x+76=4[/tex]

[tex]\implies 36x=-72[/tex]

[tex]\implies x=-2[/tex]

Substitute the found value of x into the found expression for y:

[tex]\implies y=2(-2)+7[/tex]

[tex]\implies y=-4+7[/tex]

[tex]\implies y=3[/tex]

Therefore, the point of intersection of the three circles if (-2, 3) and hence the location of the earthquake's epicenter is (-2, 3).

Learn more about circle equations here:

https://brainly.com/question/27953043

https://brainly.com/question/27979372

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