Question

question 6
the per capita consumption of breakfast cereal in the us has the following model that appears above
$c(t)=-0.0036t^{3}+0.086t^{2}-0.348t + 12.6$ pounds, where t is the number of years since 1990.
decide whether the rate of consumption was changing more rapidly in 2004 or 2012.
$c(14)=$ lbs/year
$c(22)=$ lbs/year
the rate was changing more rapidly in 2004 2012.
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Explanation:

Step1: Differentiate the function

Differentiate \(C(t)=- 0.0036t^{3}+0.086t^{2}-0.348t + 12.6\) using the power rule \((x^{n})^\prime=nx^{n - 1}\).
\(C^\prime(t)=-0.0036\times3t^{2}+0.086\times2t-0.348\)
\(C^\prime(t)=-0.0108t^{2}+0.172t - 0.348\)

Step2: Calculate \(C^\prime(14)\)

Substitute \(t = 14\) into \(C^\prime(t)\):
\(C^\prime(14)=-0.0108\times14^{2}+0.172\times14-0.348\)
\(C^\prime(14)=-0.0108\times196 + 2.408-0.348\)
\(C^\prime(14)=-2.1168+2.408 - 0.348\)
\(C^\prime(14)=-0.0568\)

Step3: Calculate \(C^\prime(22)\)

Substitute \(t = 22\) into \(C^\prime(t)\):
\(C^\prime(22)=-0.0108\times22^{2}+0.172\times22-0.348\)
\(C^\prime(22)=-0.0108\times484+3.784-0.348\)
\(C^\prime(22)=-5.2272+3.784-0.348\)
\(C^\prime(22)=-1.7912\)

Step4: Compare the magnitudes

The magnitude of \(C^\prime(14)\) is \(|C^\prime(14)| = 0.0568\), and the magnitude of \(C^\prime(22)\) is \(|C^\prime(22)|=1.7912\)

Answer:

\(C^\prime(14)=-0.0568\) lbs/year
\(C^\prime(22)=-1.7912\) lbs/year
The rate was changing more rapidly in \(2012\)