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A lighthouse is located at (1, 2) in a coordinate system measured in miles. a sailboat starts at (–7, 8) and sails in a positive x-direction along a path that can be modeled by a quadratic function with a vertex at (2, –6). which system of equations can be used to determine whether the boat comes within 5 miles of the lighthouse? startlayout enlarged left-brace 1st row (x minus 1) squared (y minus 2) squared = 5 2nd row y = startfraction 14 over 81 endfraction (x minus 2) squared minus 6 endlayout startlayout enlarged left-brace 1st row (x minus 1) squared (y minus 2) squared = 25 2nd row y = startfraction 14 over 81 endfraction (x minus 2) squared minus 6 endlayout startlayout enlarged left-brace 1st row (x minus 1) squared (y minus 2) squared = 5 2nd row y = negative startfraction 14 over 81 endfraction (x 7) squared 8 endlayout startlayout enlarged left-brace 1st row (x minus 1) squared (y minus 2) squared = 25 2nd row y = negative startfraction 14 over 81 endfraction (x 7) squared 8 endlayout

Sagot :

The system of equations that can be used to determine whether the boat comes within 5 miles of the lighthouse is:

  • y = (2/7)(x - 2)^2 - 6
  • (x - 1)^2 + (y - 2)^2 = 5^2

What are equations?

  • The equation is described as the state of being equal and is commonly represented as a math expression with equal values on either side, or it refers to an issue in which many factors must be considered.
  • 2+2 = 3+1 is an example of an equation.

To find the system of equations that can be used to determine whether the boat comes within 5 miles of the lighthouse:

The vertex form of a quadratic function is given by: f(x) = a(x - h)^2 + k

Where (h, k) is the vertex of the parabola, a is constant.

For the sailboat we have vertex: (h, k) = (2, -6) and one point: (-7, 8) that is f(-7) = 8.

  • f(x) = a(x - 2)^2 - 6

We will find a using f(-7) = 8:

  • f(-7) = a(-7-2)^2 - 6
  • f(-7) = 49a - 6
  • 49a - 6 = 8
  • 49a = 14
  • a = 14/49
  • a = 2/7

The quadratic function for the sailboat is given by:

  • f(x) = (2/7)(x - 2)^2 - 6 or y = (2/7)(x - 2)^2 - 6

The equation for a circle with a radius of 5 and center (1, 2) is:

  • (x - 1)^2 + (y - 2)^2 = 5^2

Therefore, the system of equations that can be used to determine whether the boat comes within 5 miles of the lighthouse is:

  • y = (2/7)(x - 2)^2 - 6
  • (x - 1)^2 + (y - 2)^2 = 5^2

Know more about equations here:
https://brainly.com/question/22688504

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The correct form of the question is given below:
A lighthouse is located at (1, 2) in a coordinate system measured in miles. a sailboat starts at (–7, 8) and sails in a positive x-direction along a path that can be modeled by a quadratic function with a vertex at (2, –6). which system of equations can be used to determine whether the boat comes within 5 miles of the lighthouse?

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