Question

question 6
the per capita consumption of breakfast cereal in the us has the following model that appears above
$c(t)=-0.0038t^{3}+0.12t^{2}-0.357t + 12.45$ 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 2012 2004.
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Explanation:

Step1: Find the first derivative of \(C(t)\)

Given \(C(t)=- 0.0038t^{3}+0.12t^{2}-0.357t + 12.45\).
Using the power rule \((x^{n})^\prime=nx^{n - 1}\), we have \(C^\prime(t)=-0.0114t^{2}+0.24t-0.357\).

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

Substitute \(t = 14\) into \(C^\prime(t)\):

$$
LATEXBLOCK0
$$

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

Substitute \(t = 22\) into \(C^\prime(t)\):

$$
LATEXBLOCK1
$$

We compare the magnitudes of \(|C^\prime(14)| = 0.7686\) and \(|C^\prime(22)|=0.5946\).

Answer:

\(C^\prime(14)=0.7686\) lbs/year; \(C^\prime(22)=- 0.5946\) lbs/year; The rate was changing more rapidly in 2004.