Answer to Question #98479 in Algebra for Megan Adkins

Question

Catch the mistake.
Morgan is working on her last Algebra II homework problem but realizes that her answer is incorrect. Help Morgan understand her mistake by writing a paragraph explaining the error as well how it may be corrected.

The problem:
The resistance, R, measured in ohms, of an electrical circuit in a given length of wire is inversely proportional to the square of the diameter of the wire, d, measured in centimeters. If the resistance is 0.60 ohms when the diameter is 0.45 cm, what is the resistance of the same length of wire when the diameter is 0.54 cm? Round the solution to the nearest hundredth.
Morgan’s solution:
First, Morgan wrote the inverse variation formula, . Next she plugged in the known values for resistance and diameter and found the k value.

0.27 = k

Now Morgan substituted in the value for the diameter of the wire, .54, the k-value and solved for R.

R = 0.5 ohms

BUT…this is not the answer in the back of the book!

Answer 1

Morgan's mistake is using of the inverse variation formula $R=k \cdot \frac{1}{d}$ instead of $R=k \cdot \frac{1}{d^2}$,

where $R$ - resistance, Ohm; $d$ - diameter of the wire, cm; $k$ - constant of proportionality, Ohm/cm².

Let's found the k value:

$k=R \cdot d^2$

$k=0.60 \cdot 0.45^2 \approx 0.12 \; Ohm/cm^2$

Now resistance of the wire for the diameter 0.54 cm can be found as:

$R=0.12 \cdot \frac{1}{0.54^2} \approx 0.41 \; Ohm$