answer: \frac{8}{e^{8}}\#\frac{da}{dt}=t + 4;a(2)=11.\
particular solution: \
to advance in the circuit, solve a(t)=-5.

Question

Accepted Answer

# Explanation:
## Step1: Integrate the differential equation
We know that if $\frac{dA}{dt}=t + 4$, then by the power rule of integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n
eq- 1$), we have $A(t)=\int(t + 4)dt$.
$$
\begin{align*}
A(t)&=\int tdt+\int4dt\
&=\frac{t^{2}}{2}+4t + C
\end{align*}
$$
## Step2: Use the initial condition $A(2) = 11$ to find $C$
Substitute $t = 2$ and $A(2)=11$ into $A(t)=\frac{t^{2}}{2}+4t + C$.
$$
\begin{align*}
11&=\frac{2^{2}}{2}+4\times2+C\
11&=2 + 8+C\
11&=10 + C\
C&=1
\end{align*}
$$
So $A(t)=\frac{t^{2}}{2}+4t + 1$
## Step3: Solve $A(t)=-5$
Set $\frac{t^{2}}{2}+4t + 1=-5$.
Multiply through by $2$ to get $t^{2}+8t+2=-10$, or $t^{2}+8t + 12=0$.
Factor the quadratic equation: $t^{2}+8t + 12=(t + 2)(t+6)=0$

# Answer:
$t=-2$ or $t=-6$

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QUESTION IMAGE

Question

Explanation:

Step1: Integrate the differential equation

Step2: Use the initial condition \(A(2) = 11\) to find \(C\)

Step3: Solve \(A(t)=-5\)

Answer: