question 6\nthe per capita consumption of breakfast cereal in the us has the following model that appears…

question 6\nthe per capita consumption of breakfast cereal in the us has the following model that appears above\n$c(t)=-0.0038t^{3}+0.12t^{2}-0.357t + 12.45$ pounds, where t is the number of years since 1990.\ndecide whether the rate of consumption was changing more rapidly in 2004 or 2012.\n$c(14)=$ lbs/year\n$c(22)=$ lbs/year\nthe rate was changing more rapidly in 2012 2004.\nquestion help: message instructor

question 6\nthe per capita consumption of breakfast cereal in the us has the following model that appears above\n$c(t)=-0.0038t^{3}+0.12t^{2}-0.357t + 12.45$ pounds, where t is the number of years since 1990.\ndecide whether the rate of consumption was changing more rapidly in 2004 or 2012.\n$c(14)=$ lbs/year\n$c(22)=$ lbs/year\nthe rate was changing more rapidly in 2012 2004.\nquestion help: message instructor

Answer

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)): [ \begin{align*} C^\prime(14)&=-0.0114\times14^{2}+0.24\times14 - 0.357\ &=-0.0114\times196+3.36-0.357\ &=-2.2344 + 3.36-0.357\ &=0.7686 \end{align*} ]

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

Substitute (t = 22) into (C^\prime(t)): [ \begin{align*} C^\prime(22)&=-0.0114\times22^{2}+0.24\times22-0.357\ &=-0.0114\times484+5.28-0.357\ &=-5.5176+5.28 - 0.357\ &=-0.5946 \end{align*} ] 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.