Question

1. ( r ) varies directly as ( s ) and inversely as ( t ) cubed. if ( r = 607.5 ) when ( s = 12 ) and ( t = 2 ), find ( r ) when ( s = 42 ) and ( t = 6 ).

Explanation:

Step1: Define combined variation formula

Since \( r \) varies directly as \( s \) and inversely as \( t^3 \), the relationship is:
$$r = k \cdot \frac{s}{t^3}$$
where \( k \) is the constant of variation.

Step2: Solve for constant \( k \)

Substitute \( r = 607.5 \), \( s = 12 \), \( t = 2 \):
$$607.5 = k \cdot \frac{12}{2^3}$$
$$607.5 = k \cdot \frac{12}{8}$$
$$607.5 = k \cdot 1.5$$
$$k = \frac{607.5}{1.5} = 405$$

Step3: Calculate \( r \) for new values

Substitute \( k = 405 \), \( s = 42 \), \( t = 6 \):
$$r = 405 \cdot \frac{42}{6^3}$$
$$r = 405 \cdot \frac{42}{216}$$
$$r = 405 \cdot \frac{7}{36}$$
$$r = \frac{2835}{36} = 78.75$$

Answer:

78.75