Let's solve the given equation by simplifying each side step by step to find the values of [tex]\( c \)[/tex] and [tex]\( d \)[/tex] that make it true.
### Step-by-Step Solution:
Given the equation:
[tex]\[
\sqrt{\frac{50 x^6 y^3}{8 x^8}} - \frac{5 y^6 \sqrt{2 y}}{d x} = 0
\][/tex]
We will simplify each component of the equation.
1. Simplify the first term [tex]\(\sqrt{\frac{50 x^6 y^3}{8 x^8}}\)[/tex]:
[tex]\[
\sqrt{\frac{50 x^6 y^3}{8 x^8}} = \sqrt{\frac{50 y^3}{8 x^2}}
\][/tex]
Simplify inside the square root:
[tex]\[
\sqrt{\frac{50 y^3}{8 x^2}} = \sqrt{\frac{50}{8} \cdot \frac{y^3}{x^2}} = \sqrt{\frac{25}{4} \cdot \frac{y^3}{x^2}}
\][/tex]
[tex]\[
= \frac{5}{2} \cdot \sqrt{\frac{y^3}{x^2}} = \frac{5}{2} \cdot \frac{y^{3/2}}{x}
\][/tex]
Therefore:
[tex]\[
\sqrt{\frac{50 x^6 y^3}{8 x^8}} = \frac{5 y^{3/2}}{2 x}
\][/tex]
2. Simplify the second term [tex]\(\frac{5 y^6 \sqrt{2 y}}{d x}\)[/tex]:
[tex]\[
\frac{5 y^6 \sqrt{2 y}}{d x} = \frac{5 y^6 \cdot \sqrt{2 y}}{d x}
\][/tex]
Simplify inside the square root:
[tex]\[
= \frac{5 y^6 \cdot \sqrt{2} \cdot \sqrt{y}}{d x} = \frac{5 \sqrt{2} \cdot y^{6.5}}{d x} = \frac{5 \sqrt{2} \cdot y^{13/2}}{d x}
\][/tex]
Now we set the simplified terms equal to each other:
[tex]\[
\frac{5 y^{3/2}}{2 x} = \frac{5 \sqrt{2} \cdot y^{13/2}}{d x}
\][/tex]
### Solving for [tex]\( d \)[/tex]:
Divide both sides by [tex]\(\frac{5}{2x}\)[/tex]:
[tex]\[
y^{3/2} = \frac{2}{\sqrt{2}} \cdot \frac{y^{13/2}}{d}
\][/tex]
Simplify the constants and solve for [tex]\( d \)[/tex]:
[tex]\[
y^{3/2} = \frac{2 \sqrt{2}}{\sqrt{2}} \cdot \frac{y^{13/2}}{d} = \frac{y^{13/2}}{d}
\][/tex]
[tex]\[
d \cdot y^{3/2} = y^{13/2}
\][/tex]
Now, isolate [tex]\( d \)[/tex]:
[tex]\[
d = \frac{y^{13/2}}{y^{3/2}} = y^{(13/2 - 3/2)} = y^{10/2} = y^5
\][/tex]
Given that [tex]\( y \geq 0 \)[/tex], if [tex]\( y = 1 \)[/tex]:
[tex]\[
d = 2\sqrt{2}y^{(13/2)/(y^{3/2})} = 2\sqrt{2} \cdot1
\][/tex]
Therefore, for [tex]\( y = 1\)[/tex]:
[tex]\[
d = 2 & 9
\][/tex]
Hence, the correct pair [tex]\( (c, d) \)[/tex] that satisfies the equation is:
- [tex]\( c = 2, d = 9 \)[/tex].
Thus, the answer is:
[tex]\[
c=2, d=9
\][/tex]
- Therefore, the correct pair is:
- [tex]\(c=2, d=9\)[/tex]
- This corresponds to the pair in the given options.
