Super fine 40-gauge copper wire has a diameter of only 0.080 mm and weighs only $44.5 g / km$. Suppose a spool of 40-gauge wire weighs $575 g$ less after some wire is pulled off to wind a magnet. How could you calculate how much wire was used?
Set the math up. But don't do any of it. Just leave your answer as a math expression.
Also, be sure your answer includes all the correct unit symbols.
length of wire = $\square$
Answer (2)
The weight of the wire used is 575 g.
The wire weighs 44.5 g/km.
Set up the equation: 44.5 km g × l = 575 g .
Solve for l : l = 44.5 575 km . The length of the wire is 44.5 575 km .
Explanation
Problem Analysis We are given that a spool of wire weighs 575 g less after some wire is pulled off. We also know that the wire weighs 44.5 g/km. We want to find the length of the wire that was pulled off the spool.
Setting up the Equation Let l be the length of the wire in kilometers. The weight of the wire is given by the product of its length and its weight per unit length. In this case, the weight of the wire is 44.5 km g × l . We know that the weight of the wire pulled off the spool is 575 g. Therefore, we have the equation
44.5 km g × l = 575 g
Solving for Length To find the length l , we divide both sides of the equation by 44.5 km g :
l = 44.5 km g 575 g
l = 44.5 575 km
Final Answer Therefore, the length of the wire is 44.5 575 km .
Examples
Imagine you're setting up a model train layout and need a specific length of wire for the electrical connections. Knowing the weight per unit length of the wire allows you to determine the exact length you've used by simply measuring how much lighter the spool has become. This is also useful in industries like aerospace or automotive, where minimizing weight is crucial, and engineers need to calculate wire lengths accurately based on weight changes.
To find the length of wire used, we set up the equation 44.5 km g × l = 575 g . Rearranging gives us l = 44.5 575 km . Therefore, the final expression for the length of wire is 44.5 575 km .
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