What values of [tex]\( c \)[/tex] and [tex]\( d \)[/tex] make the equation true? Assume [tex]\( x \ \textgreater \ 0 \)[/tex] and [tex]\( y \geq 0 \)[/tex].

[tex]\[
\sqrt{\frac{50 x^6 y^3}{8 x^8}} - \frac{5 y^6 \sqrt{2 y}}{d x}
\][/tex]

A. [tex]\( c = 1, d = 3 \)[/tex]

B. [tex]\( c = 1, d = 9 \)[/tex]

C. [tex]\( c = 2, d = 8 \)[/tex]

D. [tex]\( c = 2, d = 9 \)[/tex]



Answer :

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.