171k views
4 votes
Composition of functions, formulas

User Kenglxn
by
8.4k points

1 Answer

3 votes

Step 1

State N(T) and T(t)


\begin{gathered} N(T)=24T^2-125T+73 \\ T(t)=9t+1.7 \end{gathered}

T is the temperature of the food

t is time in hours when the food is outside the refrigerator

Step 2

Find N(T(t))


\begin{gathered} N(T(t))=24(9t+1.7)^2-125(9t+1.7)\text{ + 73} \\ N(T(t))=24(9t+1.7)(9t+1.7)-125(9t+1.7)+73 \end{gathered}
\begin{gathered} N(T(t))=24(81t^2+15.3t+15.3t+2.89)-1125t-212.5+73 \\ N(T(t))=24(81t^2+30.6t+2.89)-1125t-139.5_{} \\ N(T(t))=1944t^2+734.4t+69.36-1125t-139.5 \end{gathered}
N(T(t))=1944t^2-390.6t-70.14

Step 3

Find the time when the bacteria count reaches 24722


\begin{gathered} N(T(t))=24722 \\ 24722=1944t^2-390.6t-70.14 \\ 1944t^2-390.6t-70.14-24722=0 \\ 1944t^2-390.6t-24792.14=0 \end{gathered}

Step 4

Find t using the quadratic formula.


\begin{gathered} t=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ \text{where} \\ a=1944 \\ b=-390.6 \\ c=-24792.14 \end{gathered}
\begin{gathered} t=\frac{-(-390.6)\pm\sqrt[]{(-390.6)^2-4*1944*-24792.14}}{2*1944} \\ t=\frac{390.6\pm\sqrt[]{152568.36+192783680.6}}{3888} \end{gathered}
\begin{gathered} t=\frac{390.6\pm\sqrt[]{192936249}}{3888} \\ t=(390.6\pm13890.14935)/(3888) \\ t=(390.6+13890.14935)/(3888)=3.67303224\text{ hrs} \\ or \\ t=(390.6-13890.14935)/(3888)=-3.472106314\text{ hrs} \end{gathered}

Since time cannot be negative t = 3.67303224 hrs

User JohnRudolfLewis
by
8.5k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories