Comparing Algebraic Expressions: Equivalence of (36t/r - 18rt²)

To solve this problem, let's simplify and factor the given expression:

Original expression: $\frac{36t}{r} - 18rt^2$

Step 1: Factor out the greatest common divisor, which is 18:

$18 \left( \frac{2t}{r} - rt^2 \right)$

This matches the structure of choice 1.

Step 2: Substitute this back into the given options and simplify.

Option 1: $18\left( \frac{2t}{r} - rt^2 \right)$

This is identical to the factored form of the original.

Option 2: $18r\left( \frac{2t}{r^2} - t^2 \right)$

Expand and simplify:

$= 18r \cdot \frac{2t}{r^2} - 18r \cdot t^2$

$= \frac{36t}{r} - 18rt^2$

This matches the original expression.

Option 3: $-18(rt + \frac{2t}{r})$

Expand and simplify:

$= -18rt - \frac{36t}{r}$

This does not match the original expression.

Option 4: $6t\left( \frac{6t}{r} - 3rt \right)$

Expand and simplify:

$= 6t \cdot \frac{6t}{r} - 6t \cdot 3rt$

$= \frac{36t^2}{r} - 18rt^2$

This does not match the original expression.

Therefore, the correct options that are equivalent to the given expression are 1 and 2.