Explore Westonci.ca, the premier Q&A site that helps you find precise answers to your questions, no matter the topic. Get detailed and precise answers to your questions from a dedicated community of experts on our Q&A platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
To solve this problem, we need to derive two equations: one for the quadratic boundary of the closed airspace and another for the linear path of the commuter jet. Follow these steps:
### Step 1: Find the Quadratic Equation
The quadratic boundary of the closed airspace has its vertex at [tex]\((10, 6)\)[/tex] and passes through the point [tex]\((12, 7)\)[/tex]. The general form of a quadratic equation with a vertex [tex]\((h, k)\)[/tex] is:
[tex]\[ y = a(x - h)^2 + k \][/tex]
Here, [tex]\( (h, k) = (10, 6) \)[/tex].
Substitute the vertex into the equation:
[tex]\[ y = a(x - 10)^2 + 6 \][/tex]
To find the value of [tex]\(a\)[/tex], use the point [tex]\((12, 7)\)[/tex]:
[tex]\[ 7 = a(12 - 10)^2 + 6 \][/tex]
Simplify:
[tex]\[ 7 = a(2)^2 + 6 \][/tex]
[tex]\[ 7 = 4a + 6 \][/tex]
[tex]\[ 1 = 4a \][/tex]
[tex]\[ a = \frac{1}{4} \][/tex]
Thus, the equation of the quadratic boundary is:
[tex]\[ y = \frac{1}{4}(x - 10)^2 + 6 \][/tex]
### Step 2: Find the Linear Equation
The linear path of the commuter jet goes from [tex]\((-18, 14)\)[/tex] to [tex]\((16, -13)\)[/tex]. Use the slope formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Here, [tex]\((x_1, y_1) = (-18, 14)\)[/tex] and [tex]\((x_2, y_2) = (16, -13)\)[/tex].
Calculate the slope [tex]\(m\)[/tex]:
[tex]\[ m = \frac{-13 - 14}{16 - (-18)} \][/tex]
[tex]\[ m = \frac{-27}{16 + 18} \][/tex]
[tex]\[ m = \frac{-27}{34} \][/tex]
Using the point-slope form [tex]\(y - y_1 = m(x - x_1)\)[/tex] with the point [tex]\((-18, 14)\)[/tex], the equation is:
[tex]\[ y - 14 = \frac{-27}{34}(x - (-18)) \][/tex]
[tex]\[ y - 14 = \frac{-27}{34}(x + 18) \][/tex]
Simplify:
[tex]\[ y = \frac{-27}{34}x + \frac{-27}{34} \cdot 18 + 14 \][/tex]
[tex]\[ y = \frac{-27}{34}x - \frac{486}{34} + 14 \][/tex]
[tex]\[ y = \frac{-27}{34}x - \frac{243}{17} + 14 \][/tex]
[tex]\[ y = \frac{-27}{34}x - 14.2941176470588235 \][/tex]
### Step 3: Form the System of Equations
The system of equations that represents the boundary of the closed airspace and the path of the commuter jet is:
[tex]\[ \left\{ \begin{array}{l} y = \frac{1}{4}(x - 10)^2 + 6 \\ y = \frac{-27}{34}x - \frac{5}{17} \end{array} \right. \][/tex]
None of the given options perfectly match the derived equations. Thus, the correct system of equations based on the given points should be:
[tex]\[ \left\{ \begin{array}{l} y = \frac{1}{4}(x - 10)^2 + 6 \\ y = \frac{-27}{34} x - \frac{5}{17} \end{array} \right. \][/tex]
It's important to match the form exactly to the correct problem parameters.
### Step 1: Find the Quadratic Equation
The quadratic boundary of the closed airspace has its vertex at [tex]\((10, 6)\)[/tex] and passes through the point [tex]\((12, 7)\)[/tex]. The general form of a quadratic equation with a vertex [tex]\((h, k)\)[/tex] is:
[tex]\[ y = a(x - h)^2 + k \][/tex]
Here, [tex]\( (h, k) = (10, 6) \)[/tex].
Substitute the vertex into the equation:
[tex]\[ y = a(x - 10)^2 + 6 \][/tex]
To find the value of [tex]\(a\)[/tex], use the point [tex]\((12, 7)\)[/tex]:
[tex]\[ 7 = a(12 - 10)^2 + 6 \][/tex]
Simplify:
[tex]\[ 7 = a(2)^2 + 6 \][/tex]
[tex]\[ 7 = 4a + 6 \][/tex]
[tex]\[ 1 = 4a \][/tex]
[tex]\[ a = \frac{1}{4} \][/tex]
Thus, the equation of the quadratic boundary is:
[tex]\[ y = \frac{1}{4}(x - 10)^2 + 6 \][/tex]
### Step 2: Find the Linear Equation
The linear path of the commuter jet goes from [tex]\((-18, 14)\)[/tex] to [tex]\((16, -13)\)[/tex]. Use the slope formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Here, [tex]\((x_1, y_1) = (-18, 14)\)[/tex] and [tex]\((x_2, y_2) = (16, -13)\)[/tex].
Calculate the slope [tex]\(m\)[/tex]:
[tex]\[ m = \frac{-13 - 14}{16 - (-18)} \][/tex]
[tex]\[ m = \frac{-27}{16 + 18} \][/tex]
[tex]\[ m = \frac{-27}{34} \][/tex]
Using the point-slope form [tex]\(y - y_1 = m(x - x_1)\)[/tex] with the point [tex]\((-18, 14)\)[/tex], the equation is:
[tex]\[ y - 14 = \frac{-27}{34}(x - (-18)) \][/tex]
[tex]\[ y - 14 = \frac{-27}{34}(x + 18) \][/tex]
Simplify:
[tex]\[ y = \frac{-27}{34}x + \frac{-27}{34} \cdot 18 + 14 \][/tex]
[tex]\[ y = \frac{-27}{34}x - \frac{486}{34} + 14 \][/tex]
[tex]\[ y = \frac{-27}{34}x - \frac{243}{17} + 14 \][/tex]
[tex]\[ y = \frac{-27}{34}x - 14.2941176470588235 \][/tex]
### Step 3: Form the System of Equations
The system of equations that represents the boundary of the closed airspace and the path of the commuter jet is:
[tex]\[ \left\{ \begin{array}{l} y = \frac{1}{4}(x - 10)^2 + 6 \\ y = \frac{-27}{34}x - \frac{5}{17} \end{array} \right. \][/tex]
None of the given options perfectly match the derived equations. Thus, the correct system of equations based on the given points should be:
[tex]\[ \left\{ \begin{array}{l} y = \frac{1}{4}(x - 10)^2 + 6 \\ y = \frac{-27}{34} x - \frac{5}{17} \end{array} \right. \][/tex]
It's important to match the form exactly to the correct problem parameters.
We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
