Question

attempt 1: 10 attempts remaining. the number of hats made by a manufacturer t days after the start of the month is modeled by n(t)=20t² + 6t, and the cost per hat, in dollars, is modeled by c(t)=ln(228t). the total cost of the hats made is v(t)=c(t)·n(t). how fast is the total cost of the hats changing on the 12th day? (round your answer to the nearest cent.)

Explanation:

Step1: Find the product - rule formula

The product - rule states that if \(v(t)=c(t)\cdot n(t)\), then \(v^\prime(t)=c^\prime(t)n(t)+c(t)n^\prime(t)\).
First, find \(n^\prime(t)\) and \(c^\prime(t)\).
For \(n(t) = 20t^{2}+6t\), using the power - rule \((x^n)^\prime=nx^{n - 1}\), we have \(n^\prime(t)=\frac{d}{dt}(20t^{2}+6t)=40t + 6\).
For \(c(t)=\ln(228t)\), using the chain - rule \((\ln(u))^\prime=\frac{u^\prime}{u}\), where \(u = 228t\) and \(u^\prime=228\), so \(c^\prime(t)=\frac{228}{228t}=\frac{1}{t}\).

Step2: Substitute \(t = 12\) into \(n(t)\), \(n^\prime(t)\), \(c(t)\) and \(c^\prime(t)\)

• Calculate \(n(12)\):

\(n(12)=20\times12^{2}+6\times12=20\times144 + 72=2880+72 = 2952\).

• Calculate \(n^\prime(12)\):

\(n^\prime(12)=40\times12 + 6=480+6=486\).

• Calculate \(c(12)\):

\(c(12)=\ln(228\times12)=\ln(2736)\approx7.91\).

• Calculate \(c^\prime(12)\):

\(c^\prime(12)=\frac{1}{12}\).

Step3: Calculate \(v^\prime(12)\)

Using the product - rule \(v^\prime(12)=c^\prime(12)n(12)+c(12)n^\prime(12)\)
\(v^\prime(12)=\frac{1}{12}\times2952+7.91\times486\)
\(v^\prime(12)=246+7.91\times486\)
\(v^\prime(12)=246 + 3844.26\)
\(v^\prime(12)=4090.26\)

Answer:

\(4090.26\)