To evaluate [tex]\( Z = 4y^2 + x^2 \)[/tex] with the given conditions [tex]\( y = 2 \tan t \)[/tex] and [tex]\( x = \frac{4}{\sqrt{8 - q^2}} \)[/tex] at [tex]\( t = \frac{\pi}{3} \)[/tex] and [tex]\( q = 2.5 \)[/tex], we need to follow these steps:
1. Calculate [tex]\( y \)[/tex]:
Given [tex]\( y = 2 \tan t \)[/tex] and [tex]\( t = \frac{\pi}{3} \)[/tex],
[tex]\[
y = 2 \tan \left(\frac{\pi}{3}\right)
\][/tex]
We know that [tex]\( \tan \left(\frac{\pi}{3}\right) = \sqrt{3} \)[/tex], so:
[tex]\[
y = 2 \cdot \sqrt{3} = 2\sqrt{3}
\][/tex]
When rounded to 4 decimal places:
[tex]\[
y \approx 3.4641
\][/tex]
2. Calculate [tex]\( x \)[/tex]:
Given [tex]\( x = \frac{4}{\sqrt{8 - q^2}} \)[/tex] and [tex]\( q = 2.5 \)[/tex],
[tex]\[
x = \frac{4}{\sqrt{8 - (2.5)^2}}
\][/tex]
First, compute [tex]\( (2.5)^2 \)[/tex]:
[tex]\[
(2.5)^2 = 6.25
\][/tex]
Then,
[tex]\[
8 - 6.25 = 1.75
\][/tex]
Now,
[tex]\[
x = \frac{4}{\sqrt{1.75}}
\][/tex]
When rounded to 4 decimal places:
[tex]\[
x \approx 3.0237
\][/tex]
3. Compute [tex]\( Z \)[/tex]:
Given [tex]\( Z = 4y^2 + x^2 \)[/tex],
Substitute the computed values for [tex]\( y \)[/tex] and [tex]\( x \)[/tex]:
[tex]\[
y \approx 3.4641 \quad \text{and} \quad x \approx 3.0237
\][/tex]
Then,
[tex]\[
Z = 4y^2 + x^2
\][/tex]
Compute [tex]\( 4y^2 \)[/tex]:
[tex]\[
4 \times (3.4641)^2 \approx 48.0001
\][/tex]
Compute [tex]\( x^2 \)[/tex]:
[tex]\[
(3.0237)^2 \approx 9.1428
\][/tex]
Add these results together:
[tex]\[
Z \approx 48.0001 + 9.1428 = 57.1429
\][/tex]
Thus, the evaluated value of [tex]\( Z \)[/tex] correct to 4 decimal places is:
[tex]\[
Z \approx 57.1429
\][/tex]
Therefore, the final results are:
[tex]\[
y \approx 3.4641, \quad x \approx 3.0237, \quad \text{and} \quad Z \approx 57.1429
\][/tex]
