Question

142r = 47ten

Answer 1

To convert the number \( 142_r \) from base \( r \) to base 10, we need to evaluate it using the formula for base conversion. The number in base \( r \) can be expressed as:

\[
1 \cdot r^2 + 4 \cdot r^1 + 2 \cdot r^0
\]

This simplifies to:

\[
1 \cdot r^2 + 4 \cdot r + 2
\]

According to the problem, this is equal to \( 47_{10} \). Therefore, we have:

\[
r^2 + 4r + 2 = 47
\]

Now, subtract 47 from both sides:

\[
r^2 + 4r + 2 - 47 = 0
\]

This simplifies to:

\[
r^2 + 4r - 45 = 0
\]

Next, we can use the quadratic formula to solve for \( r \):

\[
r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \( a = 1 \), \( b = 4 \), and \( c = -45 \). Substituting these values into the formula gives:

\[
r = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 1 \cdot (-45)}}{2 \cdot 1}
\]

Calculating the discriminant:

\[
4^2 - 4 \cdot 1 \cdot (-45) = 16 + 180 = 196
\]

Taking the square root,

\[
\sqrt{196} = 14
\]

Now substituting back into the formula for \( r \):

\[
r = \frac{-4 \pm 14}{2}
\]

This results in two possible solutions:

1. \( r = \frac{10}{2} = 5 \)
2. \( r = \frac{-18}{2} = -9 \) (not valid for a base)

Thus, the only valid solution is:

\[
r = 5
\]

Hence, the base \( r \) is \( 5 \).