$$ m $$ varies inversely as the cube of $$ x $$ $$ m=200 $$ when $$ x=1 / 2 $$ Find $$ m $$ when $$ x=1 / 4 $$

Question

UpStudy AI · Accepted Answer

Given that $$ m $$ varies inversely as the cube of $$ x $$, we can express this relationship as:

$$
m = \frac{k}{x^3}
$$

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

**Step 1: Determine the constant $$ k $$ using the given values.**

When $$ m = 200 $$ and $$ x = \frac{1}{2} $$:

$$
200 = \frac{k}{\left(\frac{1}{2}\right)^3}
$$

$$
200 = \frac{k}{\frac{1}{8}} \implies 200 = 8k \implies k = \frac{200}{8} = 25
$$

**Step 2: Find $$ m $$ when $$ x = \frac{1}{4} $$.**

Using the constant $$ k = 25 $$:

$$
m = \frac{25}{\left(\frac{1}{4}\right)^3}
$$

$$
m = \frac{25}{\frac{1}{64}} = 25 \times 64 = 1600
$$

**Final Answer:**

When $$ x = \frac{1}{4} $$, the value of $$ m $$ is **1600**.
When $$ x = \frac{1}{4} $$, $$ m = 1600 $$.

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