Question

in a certain factory, assume that the number of workers is constant. the number of minutes ( n ) that it takes to make a single unit of a product and the number of units ( u ) of the product that are made per day satisfy the relationship ( u=\frac{k}{n} ), where ( k ) is a constant. which of the following best describes the relationship between the rate of change, with respect to time ( t ), of ( u ) and the rate of change, with respect to time ( t ), of ( n )?
a ( \frac{du}{dt}=\frac{k}{(\frac{dn}{dt})} )
b ( \frac{du}{dt}=\frac{-k}{(\frac{dn}{dt})} )
c ( \frac{du}{dt}=\frac{k}{n^{2}}(\frac{dn}{dt}) )
d ( \frac{du}{dt}=\frac{-k}{n^{2}}(\frac{dn}{dt}) )

Explanation:

Step1: Differentiate \(U = \frac{k}{N}\) with respect to \(t\)

Use the quotient rule \((\frac{u}{v})^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}\). Here \(u = k\) (so \(u^\prime=0\)) and \(v = N\) (so \(v^\prime=\frac{dN}{dt}\)).

Step2: Apply the quotient rule formula

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Answer:

D. \(\frac{dU}{dt}=\frac{-k}{N^{2}}(\frac{dN}{dt})\)