Problema Solution
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?
Answer provided by our tutors
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 radius 5 and center (1, 2) is:
(x - 1)^2 + (y - 2)^2 = 5^2
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