Question

hw 20 - absolute extrema section 3.6: problem 10
(1 point)
an employees monthly productivity m, in number of units produced, is found to be a function of the number t of years of service. for a certain product, a productivity function is shown below. find the maximum productivity and the year in which it is achieved.
$m(t)=-8t^{2}+344t + 180$ for $0\leq t\leq43$
absolute maximum value:
(round to three decimal places as needed.)

Explanation:

Step1: Find the derivative of \(M(t)\)

The derivative of \(M(t)=-8t^{2}+344t + 180\) is \(M^{\prime}(t)=-16t + 344\) using the power rule \((x^{n})^\prime=nx^{n - 1}\).

Step2: Set the derivative equal to zero and solve for \(t\)

Set \(M^{\prime}(t)=0\), so \(-16t+344 = 0\).

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Step3: Check the endpoints and the critical point

• When \(t = 0\), \(M(0)=-8(0)^{2}+344(0)+180 = 180\)
• When \(t = 21.5\), \(M(21.5)=-8(21.5)^{2}+344(21.5)+180\)

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• When \(t = 43\), \(M(43)=-8(43)^{2}+344(43)+180\)

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Answer:

\(3978.000\)