A graphing calculator is needed for this section.

Source: Adjustable steel wrench, Bjorgvin Gudmundsson, Stockvault

The amount of force you must exert on a wrench handle to loosen a rusty bolt depends on how long the wrench handle is. Suppose that for a particular bolt, a wrench 6 inches long would require a force of 320 pounds, and a wrench 24 inches long would require 80 pounds.

1. How does force vary with length? Write the general equation.
   Check your answer As the length of the wrench increases, the force needed to loosen the bolt decreases. This means that force (F) is inversely proportional to the length (l) of the wrench. The general equation for this is k l , where k is a constant of proportionality. Close


2. Write the particular equation.
   Check your answerFrom part a, the general equation for this situation is F= k l , where k is a constant of proportionality. In the original problem, we were told that for a wrench 6 inches long, the force needed to loosen the bolt is 320 lbs. Substitute these values into our general equation and solve for k.

   

   In the original problem we were told a 24-inch wrench only needs 80 pounds of force. Substitute these values into our equation to check it.

   

   Notice that our equation F = 1920 l is a basic rational function.

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3. What are the units of the proportionality constant? (This quantity is called “torque,” and is a measure of the amount of “twisting” needed to loosen the bolt.)
   Check your answerIn Question 1 we found k, the proportionality constant, by multiplying the force by the length. The units of force were pounds, and the units of length were inches. Therefore, the units on the proportionality constant are inch-pounds. Close


4. Find the force needed for wrenches with handles 4, 12, 18, 32, and 64 inches long.
   Check your answerYou can quickly find the different forces needed by using a table on your calculator.
   • Enter the equation into Y1.
     
     
   • Press 2nd, WINDOW and change the table settings to ASK for the independent variable.
     
     
   • Press 2nd, GRAPH and enter the different lengths. Your calculator will compute the corresponding forces.
     
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5. How long a wrench would be needed by Jack, who can exert a force of 200 pounds?
   Check your answerSubstitute the 200-lb force that Jack can exert into the equation, solve for l and determine the length of the wrench he would need.

   

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6. How long a wrench would be needed by Jill, who can exert a force of 40 pounds?
   Check your answer

   Use the same process as in #5 or use a graph to solve this problem.

   • Y1 has our equation entered in it. Enter Y2=40.
     
     
   • Set an appropriate window for the graph. Since x stands for length, there can’t be a negative x-value. From Question 4 above, a 64-inch wrench produced a force of 30 pounds, so the x-values are not higher than 64. The y-values stand for force. The force has to be positive and not too much higher than the 40 pounds that Jill can exert.
     
     
   • Now we can have the calculator find the intersection of these two equations.
     

   A wrench 48 inches long will produce of a force of 40 pounds.

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