Understanding Direct Variation

The problem states that the variable S varies directly as the expression (R + 7). This means there's a constant relationship between S and (R + 7). We can write this relationship using a mathematical equation.

Direct variation can be expressed as:

$S \propto (R + 7)$

To turn this proportionality into an equation, we introduce a constant of variation, usually denoted by k:

$S = k(R + 7)$

Calculating the Constant of Variation (k)

We are given an initial condition: S = 42 when R = 17. We can use these values to find the value of the constant k.

Substitute the given values into the equation:

$42 = k(17 + 7)$

First, simplify the term in the parentheses:

$42 = k(24)$

Now, solve for k by dividing both sides by 24:

$k = \frac{42}{24}$

Simplify the fraction. Both 42 and 24 are divisible by 6:

$k = \frac{42 \div 6}{24 \div 6} = \frac{7}{4}$

So, the constant of variation is $k = \frac{7}{4}$.

Determining the New Value of S

Now we need to find the value of S when R = 29. We use the same equation $S = k(R + 7)$, but this time with the newly found value of k and the new value of R.

Substitute $k = \frac{7}{4}$ and $R = 29$ into the equation:

$S = \frac{7}{4}(29 + 7)$

First, calculate the sum inside the parentheses:

$S = \frac{7}{4}(36)$

Now, perform the multiplication. We can divide 36 by 4 first:

$S = 7 \times \frac{36}{4}$

$S = 7 \times 9$

Finally, calculate the result:

$S = 63$

Conclusion

Therefore, when R = 29, the value of S is 63.