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?
$\left\{\begin{aligned}(x-1)^2+(y-2)^2 & =5 \\ y & =\frac{14}{81}(x-2)^2-6\end{aligned}\right.$
$\left\{\begin{aligned}(x-1)^2+(y-2)^2 & =25 \\ y & =\frac{14}{81}(x-2)^2-6\end{aligned}\right.$
$\left\{\begin{aligned}(x-1)^2+(y-2)^2 & =5 \\ y & =-\frac{14}{81}(x+7)^2+8\end{aligned}\right.$
$\left\{\begin{aligned}(x-1)^2+(y-2)^2 & =25 \\ y & =-\frac{14}{81}(x+7)^2+8\end{aligned}\right.$
Answer (1)
Determine the equation of the circle representing the area within 5 miles of the lighthouse: ( x − 1 ) 2 + ( y − 2 ) 2 = 25 .
Find the equation of the sailboat's path as a quadratic function with vertex ( 2 , − 6 ) passing through ( − 7 , 8 ) : y = 81 14 ( x − 2 ) 2 − 6 .
Form the system of equations using the circle and the quadratic function.
The system of equations is { ( x − 1 ) 2 + ( y − 2 ) 2 = 25 y = 81 14 ( x − 2 ) 2 − 6 .
Explanation
Problem Analysis The problem asks for the system of equations that can be used to determine whether a sailboat comes within 5 miles of a lighthouse. The lighthouse is at ( 1 , 2 ) , and the sailboat's path is a quadratic function with vertex at ( 2 , − 6 ) and passes through ( − 7 , 8 ) .
Distance Equation The distance between the lighthouse at ( 1 , 2 ) and any point ( x , y ) on the sailboat's path must be less than or equal to 5 miles. The equation representing all points within 5 miles of the lighthouse is a circle centered at ( 1 , 2 ) with radius 5. Thus, the equation is ( x − 1 ) 2 + ( y − 2 ) 2 = 5 2 = 25 .
Quadratic Equation The path of the sailboat is a quadratic function with vertex ( 2 , − 6 ) . The general form of a quadratic with vertex ( h , k ) is y = a ( x − h ) 2 + k . In this case, y = a ( x − 2 ) 2 − 6 . Since the sailboat starts at ( − 7 , 8 ) , we can plug this point into the equation to find the value of a : 8 = a ( − 7 − 2 ) 2 − 6 , so 14 = a ( − 9 ) 2 = 81 a . Therefore, a = f r a c 14 81 .
Sailboat's Path The equation of the sailboat's path is y = f r a c 14 81 ( x − 2 ) 2 − 6 .
System of Equations The system of equations to determine if the boat comes within 5 miles of the lighthouse is ⎩ ⎨ ⎧ ( x − 1 ) 2 + ( y − 2 ) 2 = 25 y = f r a c 14 81 ( x − 2 ) 2 − 6
Final Answer Comparing this system of equations with the given options, we see that the correct system is: ⎩ ⎨ ⎧ ( x − 1 ) 2 + ( y − 2 ) 2 = 25 y = f r a c 14 81 ( x − 2 ) 2 − 6
Examples
Imagine you're tracking a ship at sea using radar. The lighthouse represents a fixed point, and the ship's path is described by a mathematical function. By setting up a system of equations like this, you can determine if the ship will come within a critical distance of the lighthouse, ensuring safe navigation and preventing potential collisions. This is crucial for maritime safety and efficient route planning.
