$$ y $$ is directly proportional to $$ \sqrt[3]{x} $$ $$ y=1 \frac{1}{5} \quad $$ when $$ x=27 $$ Find the value of $$ y $$ when $$ x=64 $$

Question

UpStudy AI · Accepted Answer

Since $$ y $$ is directly proportional to $$ \sqrt[3]{x} $$, we can express this relationship as:

$$
y = k \sqrt[3]{x}
$$

where $$ k $$ is the constant of proportionality.

We are given that $$ y = 1 \frac{1}{5} $$ when $$ x = 27 $$. First, we convert $$ 1 \frac{1}{5} $$ to an improper fraction:

$$
1 \frac{1}{5} = \frac{6}{5}
$$

Now we can substitute $$ y $$ and $$ x $$ into the equation to find $$ k $$:

$$
\frac{6}{5} = k \sqrt[3]{27}
$$

Since $$ \sqrt[3]{27} = 3 $$, we have:

$$
\frac{6}{5} = k \cdot 3
$$

To solve for $$ k $$, we divide both sides by 3:

$$
k = \frac{6}{5} \cdot \frac{1}{3} = \frac{6}{15} = \frac{2}{5}
$$

Now that we have $$ k $$, we can find $$ y $$ when $$ x = 64 $$:

$$
y = k \sqrt[3]{64}
$$

Calculating $$ \sqrt[3]{64} $$:

$$
\sqrt[3]{64} = 4
$$

Now substituting $$ k $$ and $$ \sqrt[3]{64} $$ into the equation for $$ y $$:

$$
y = \frac{2}{5} \cdot 4 = \frac{8}{5}
$$

Thus, the value of $$ y $$ when $$ x = 64 $$ is:

$$
\boxed{\frac{8}{5}}
$$
The value of $$ y $$ when $$ x = 64 $$ is $$ \frac{8}{5} $$.

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