5) use the standard hyperbola equation to find the y-values to the nearest integer from the x-values in the table.
$ \frac { ( x - 2 ) ^ { 2 } } { 4 ^ { 2 } } - \frac { ( y - 1 ) ^ { 2 } } { 2 ^ { 2 } } = 1 $
$ \begin{array} { c c } { x } & { y } \ { 11 } & { - } \ { 11 } & { - } end{array} $
5
-1
-3
-5
3

Question

Accepted Answer

# Explanation:
## Step1: Substitute $x = 11$ into the hyperbola equation
Given the hyperbola equation $\frac{(x - 2)^{2}}{4^{2}}-\frac{(y - 1)^{2}}{2^{2}}=1$. Substitute $x = 11$:
$$
\begin{align*}
\frac{(11 - 2)^{2}}{16}-\frac{(y - 1)^{2}}{4}&=1\
\frac{81}{16}-\frac{(y - 1)^{2}}{4}&=1
\end{align*}
$$
## Step2: Solve for $(y - 1)^{2}$
Multiply through by $16$ to clear the fractions:
$$
\begin{align*}
81-4(y - 1)^{2}&=16\
-4(y - 1)^{2}&=16 - 81\
-4(y - 1)^{2}&=- 65\
(y - 1)^{2}&=\frac{65}{4}=16.25
\end{align*}
$$
## Step3: Solve for $y$
Take the square root of both sides:
$$
\begin{align*}
y-1&=\pm\sqrt{16.25}\
y&=1\pm\sqrt{16.25}
\end{align*}
$$
$\sqrt{16.25}\approx4.03$. So $y_1=1 + 4.03\approx5$ and $y_2=1-4.03\approx - 3$

# Answer:
$5$, $-3$

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QUESTION IMAGE

Question

Explanation:

Step1: Substitute \(x = 11\) into the hyperbola equation

Step2: Solve for \((y - 1)^{2}\)

Step3: Solve for \(y\)

Answer: