Question
A golf course features a driving range with a circular boundary represented by the equation (x−3)2+(y+2)2=81
. Golfers receive a score whenever their ball lands inside the boundary. Four balls are hit, landing at the coordinates listed below. Which ball lands inside the boundary?(1 point)
Responses
(11,5)
open paren 11 comma 5 close paren
(−2,4)
open paren negative 2 comma 4 close paren
(−1,−11)
open paren negative 1 comma negative 11 close paren
(1,7)
. Golfers receive a score whenever their ball lands inside the boundary. Four balls are hit, landing at the coordinates listed below. Which ball lands inside the boundary?(1 point)
Responses
(11,5)
open paren 11 comma 5 close paren
(−2,4)
open paren negative 2 comma 4 close paren
(−1,−11)
open paren negative 1 comma negative 11 close paren
(1,7)
Answers
GPT-4o mini
To determine which ball lands inside the circular boundary of the driving range described by the equation \((x - 3)^2 + (y + 2)^2 = 81\), we first identify the center and radius of the circle.
The center of the circle is at \((3, -2)\) and the radius is \(\sqrt{81} = 9\).
Next, we can check whether each of the given coordinates is inside the circle by evaluating the expression \((x - 3)^2 + (y + 2)^2\) for each point. If the result is less than \(81\), the point is inside the circle. If the result equals \(81\), the point is on the boundary. If it is greater than \(81\), the point is outside the circle.
1. For the point \((11, 5)\):
\[
(11 - 3)^2 + (5 + 2)^2 = 8^2 + 7^2 = 64 + 49 = 113 \quad (\text{outside})
\]
2. For the point \((-2, 4)\):
\[
(-2 - 3)^2 + (4 + 2)^2 = (-5)^2 + 6^2 = 25 + 36 = 61 \quad (\text{inside})
\]
3. For the point \((-1, -11)\):
\[
(-1 - 3)^2 + (-11 + 2)^2 = (-4)^2 + (-9)^2 = 16 + 81 = 97 \quad (\text{outside})
\]
4. For the point \((1, 7)\):
\[
(1 - 3)^2 + (7 + 2)^2 = (-2)^2 + 9^2 = 4 + 81 = 85 \quad (\text{outside})
\]
After checking all points, the only ball that lands inside the boundary is:
\((-2, 4)\).
The center of the circle is at \((3, -2)\) and the radius is \(\sqrt{81} = 9\).
Next, we can check whether each of the given coordinates is inside the circle by evaluating the expression \((x - 3)^2 + (y + 2)^2\) for each point. If the result is less than \(81\), the point is inside the circle. If the result equals \(81\), the point is on the boundary. If it is greater than \(81\), the point is outside the circle.
1. For the point \((11, 5)\):
\[
(11 - 3)^2 + (5 + 2)^2 = 8^2 + 7^2 = 64 + 49 = 113 \quad (\text{outside})
\]
2. For the point \((-2, 4)\):
\[
(-2 - 3)^2 + (4 + 2)^2 = (-5)^2 + 6^2 = 25 + 36 = 61 \quad (\text{inside})
\]
3. For the point \((-1, -11)\):
\[
(-1 - 3)^2 + (-11 + 2)^2 = (-4)^2 + (-9)^2 = 16 + 81 = 97 \quad (\text{outside})
\]
4. For the point \((1, 7)\):
\[
(1 - 3)^2 + (7 + 2)^2 = (-2)^2 + 9^2 = 4 + 81 = 85 \quad (\text{outside})
\]
After checking all points, the only ball that lands inside the boundary is:
\((-2, 4)\).