Question

Question

Answer 1

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)

Answer 2

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

$(x-a)^2+(y-b)^2=r^2$

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

Circle A

• center = (-2, 5)
• radius = 2 miles

$\implies (x+2)^2+(y-5)^2=4$

$\implies x^2+4x+4+y^2-10y+25=4$

$\implies x^2+4x+y^2-10y+25=0$

Circle B

• center = (2, 3)
• radius = 4 miles

$\implies (x-2)^2+(y-3)^2=16$

$\implies x^2-4x+4+y^2-6y+9=16$

$\implies x^2-4x+y^2-6y-3=0$

Circle C

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

$\implies (x+2)^2+(y+4)^2=49$

$\implies x^2+4x+4+y^2+8y+16=49$

$\implies x^2+4x+y^2+8y-29=0$

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

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

$\implies x^2-4x+y^2-6y-3=x^2+4x+y^2-10y+25$

$\implies -4x-6y-3=4x-10y+25$

Rearrange to isolate y:

$\implies -6y-3=8x-10y+25$

$\implies 4y-3=8x+25$

$\implies 4y=8x+28$

$\implies y=2x+7$

Substitute the expression for y into equation C and simplify:

$\implies x^2+4x+(2x+7)^2+8(2x+7)-29=0$

$\implies x^2+4x+4x^2+28x+49+16x+56-29=0$

$\implies 5x^2+48x+76=0$

Substitute the expression for y into equation B and simplify:

$\implies x^2-4x+(2x+7)^2-6(2x+7)-3=0$

$\implies x^2-4x+4x^2+28x+49-12x-42-3=0$

$\implies 5x^2+12x+4=0$

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

$\implies 5x^2+48x+76=5x^2+12x+4$

$\implies 48x+76=12x+4$

$\implies 36x+76=4$

$\implies 36x=-72$

$\implies x=-2$

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

$\implies y=2(-2)+7$

$\implies y=-4+7$

$\implies y=3$

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:

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