What is the product?
$\frac{5 k}{6} \cdot \frac{3}{2 k^3}$
We are tasked with finding the product of two rational expressions:
\[ \frac{5k}{6} \cdot \frac{3}{2k^3} \]
where \(k\) is a non-zero variable (since it appears in the denominator). To multiply two fractions, we multiply their numerators together to form the new numerator and their denominators together to form the new denominator. This can be expressed as:
\[ \frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d} \]
Applying this to our expression:
\[ \frac{5k \cdot 3}{6 \cdot 2k^3} \]
Now, we simplify both the numerator and the denominator of the resulting fraction:
Substituting these values, we get:
\[ \frac{15k}{12k^3} \]
Next, we simplify the fraction by reducing the numerical coefficient and the variable part separately.
\[ \frac{15}{12} = \frac{15 \div 3}{12 \div 3} = \frac{5}{4} \]
\[ \frac{k}{k^3} = \frac{1}{k^2} \]
Combining the simplified parts, we obtain:
\[ \frac{5}{4} \cdot \frac{1}{k^2} = \frac{5}{4k^2} \]
Comparing this result with the given choices, it matches the first option. Therefore, the simplified product is:
\[ \boxed{\frac{5}{4k^2}} \]
In various engineering and physics problems, you often encounter formulas involving rates or ratios that need to be combined. For example, if a machine's power output is modeled by \(\frac{5k}{6}\) units and its efficiency factor is \(\frac{3}{2k^3}\), the total net performance can be determined by multiplying these expressions. Simplifying such algebraic products is a fundamental skill in modeling systems, analyzing trends, or solving physics equations.