Sure! Let's walk through the process of converting the given parametric equations into a rectangular form step-by-step.
Given parametric equations:
[tex]\[ y = 2t^2 + 6 \][/tex]
[tex]\[ x = 5t - 8 \][/tex]
1. Express t in terms of x from the x equation:
From the equation [tex]\( x = 5t - 8 \)[/tex]:
[tex]\[ 5t = x + 8 \][/tex]
[tex]\[ t = \frac{x + 8}{5} \][/tex]
2. Substitute this expression for t into the y equation:
The y equation is [tex]\( y = 2t^2 + 6 \)[/tex]. Substitute [tex]\( t = \frac{x + 8}{5} \)[/tex] into this:
[tex]\[ y = 2\left(\frac{x + 8}{5}\right)^2 + 6 \][/tex]
3. Simplify the equation:
Start by squaring the fraction:
[tex]\[ \left(\frac{x + 8}{5}\right)^2 = \frac{(x + 8)^2}{25} \][/tex]
Substitute this back into the y equation:
[tex]\[ y = 2\left(\frac{(x + 8)^2}{25}\right) + 6 \][/tex]
Multiply through by 2:
[tex]\[ y = \frac{2(x + 8)^2}{25} + 6 \][/tex]
Now expand [tex]\( (x + 8)^2 \)[/tex]:
[tex]\[ (x + 8)^2 = x^2 + 16x + 64 \][/tex]
Plug this into the expression:
[tex]\[ y = \frac{2(x^2 + 16x + 64)}{25} + 6 \][/tex]
[tex]\[ y = \frac{2x^2 + 32x + 128}{25} + 6 \][/tex]
4. Combine the terms:
Separate the fraction:
[tex]\[ y = \frac{2x^2}{25} + \frac{32x}{25} + \frac{128}{25} + 6 \][/tex]
Convert 6 to a fraction with denominator 25:
[tex]\[ 6 = \frac{150}{25} \][/tex]
Combine all the terms:
[tex]\[ y = \frac{2x^2}{25} + \frac{32x}{25} + \frac{128}{25} + \frac{150}{25} \][/tex]
[tex]\[ y = \frac{2x^2}{25} + \frac{32x}{25} + \frac{278}{25} \][/tex]
Therefore, the rectangular form of the given parametric equations is:
[tex]\[ y = \frac{2}{25}x^2 + \frac{32}{25}x + \frac{278}{25} \][/tex]
So, the correct option is:
[tex]\[ \boxed{y = \frac{2}{25}x^2 + \frac{32}{25}x + \frac{278}{25}} \][/tex]
