Question 128636
Since you are told that the distance (call it D) that an object falls is directly proportional
to the square of the time (let T represent the time), then you can write the proportional
relationship as:
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{{{D = K*T^2}}}
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in which K is the constant of proportionality that makes the equation balance.
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For this problem we first need to find K. The problem tells you that when T = 6 seconds,
then D = 1296 feet. Substituting these two values into the equation results in:
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{{{1296 = K*(6^2)}}}
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Square the 6 and the equation becomes:
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{{{1296 = K*(36)}}}
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Solve for K by dividing both sides of the equation by 36 and you get:
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{{{1296/36  = K}}}
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and after you do the division you have:
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{{{36 = K}}}
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Now return to the original equation for this falling body:
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{{{D = K*T^2}}}
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and substitute 36 for K to change the equation to:
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{{{D = 36*T^2}}}
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From this you can compute the total time it will take to fall a distance of 2304 feet by
substituting 2304 for D in the equation to get:
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{{{2304 = 36*T^2}}}
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Divide both sides of this equation by 36 and you then have:
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{{{64 = T^2}}}
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Solve for T by taking the square root of both sides to find that:
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{{{8 = T}}}
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This tells you that 8 seconds from the time that object is released it has fallen a
distance of 2304 feet.
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Hope this helps you to understand what is meant by "is directly proportional to" as well as the
details of solving this problem.
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