To determine whether the road intersects the boundary of the tower's signal, we need to formulate both the quadratic function representing the signal boundary and the linear equation representing the road. Let's proceed step by step:
### Step 1: Determine the quadratic function for the signal boundary
The quadratic function is given in vertex form:
[tex]\[ y = a(x - h)^2 + k \][/tex]
Given that the vertex is at [tex]\((4, 2)\)[/tex], we have:
[tex]\[ h = 4 \][/tex]
[tex]\[ k = 2 \][/tex]
The quadratic function passes through the point [tex]\((5, 4)\)[/tex]. We can use this point to determine the value of [tex]\(a\)[/tex]:
[tex]\[ y = a(x - h)^2 + k \][/tex]
Substituting the point [tex]\((5, 4)\)[/tex]:
[tex]\[ 4 = a(5 - 4)^2 + 2 \][/tex]
[tex]\[ 4 = a(1)^2 + 2 \][/tex]
[tex]\[ 4 = a + 2 \][/tex]
[tex]\[ a = 2 \][/tex]
Therefore, the quadratic function for the signal boundary is:
[tex]\[ y = 2(x - 4)^2 + 2 \][/tex]
To match the form given in the question, we rewrite it as:
[tex]\[ y - 2 = 2(x - 4)^2 \][/tex]
[tex]\[ y - 2(x - 4)^2 = 2 \][/tex]
### Step 2: Determine the linear function for the road
The road passes through the points [tex]\((-3, 7)\)[/tex] and [tex]\((8, 2)\)[/tex]. We first calculate the slope [tex]\(m\)[/tex] of the line:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the points [tex]\((-3, 7)\)[/tex] and [tex]\((8, 2)\)[/tex]:
[tex]\[ m = \frac{2 - 7}{8 - (-3)} \][/tex]
[tex]\[ m = \frac{-5}{11} \][/tex]
[tex]\[ m = -\frac{5}{11} \][/tex]
Next, we use the point-slope formula to write the equation of the line. Taking the point [tex]\((-3, 7)\)[/tex]:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
[tex]\[ y - 7 = -\frac{5}{11}(x + 3) \][/tex]
[tex]\[ y - 7 = -\frac{5}{11}x - \frac{15}{11} \][/tex]
[tex]\[ y = -\frac{5}{11}x + \frac{77}{11} - \frac{15}{11} \][/tex]
[tex]\[ y = -\frac{5}{11}x + \frac{62}{11} \][/tex]
Multiplying through by 11 to clear the fraction:
[tex]\[ 11y = -5x + 62 \][/tex]
[tex]\[ 5x + 11y = 62 \][/tex]
### Step 3: Form the system of equations
We now have two equations to form a system:
1. The quadratic boundary of the signal: [tex]\(y - 2(x - 4)^2 = 2\)[/tex]
2. The linear road equation: [tex]\(5x + 11y = 62\)[/tex]
Thus, the correct system of equations is:
[tex]\[
\begin{cases}
y - 2(x - 4)^2 = 2 \\
5x + 11y = 62 \\
\end{cases}
\][/tex]
The system of equations that can be used to determine whether the road intersects the boundary of the tower's signal is:
[tex]\[
\left\{\begin{aligned} y-2(x-4)^2 & =2 \\ 5 x+11 y & =62\end{aligned}\right.
\][/tex]
