Mechanical and electrical engineers often study controls, which are used to manipulate how the inputs into a system affects

that system's output or response. These systems can be anything mechanical that involves motion or anything electrical that

involves a change in voltage or current. One such simple control system is a light switch. Though in practice light bulbs are

treated as turning on instantaneously, their brightness is actually a function of the voltage across the bulb, which is itself a

function of time. The brightness in a typical incandescent lightbulb is given by

$I=1\mathrm{.}5P*(1-1\mathrm{.}40e^{-t}sin(1\mathrm{.}43t+\frac{}{4}))$

Eq$.1$

where the intensity of the bulb's brightness I $[$Lumens$]$ can be found as a function of the power rating of the bulb $P[$Watts$]$

and the time since the switch was turned on $t[$microseconds$].$

As an electrical engineer, you are trying to better understand the effect of a power surge on a household lamp. The brightness

increase due to the power surge follows the equation

$I_s=10P{e}^{-1}sin(\frac{143}{2})$

Eq$.2$

where the surge brightness $I_s[$Lumens$]$ is a function of the power rating of the bulb $P[$Watts$].$ The power surge will increase

original brightness of the bulb by $I_s$ whenever the surge is taking place. You will use Equation $2$ to see how a power surge

would alter the nominal brightness I of the lamp.

Tasks:

TASK $1$: $(4-6$min$)$

Prompt the user to enter the lamp bulbs' power rating $[$Watts$]$ and a single time to analyze $[$microseconds$].$ Calculate and

output to the command window the brightness of a lightbulb with the given power rating at the time entered, not considering

the power surge.

TASK $2$: $(10-16$min$)$

Prompt the user to input the time at which the power surge starts $t_s($assume this time will be before the time entered in Task $1).$

Determine the brightness of the lamp at the time entered by the user. You will need to calculate the brightness of the lamp

considering the power surge considering the power surge increases the brightness of the lamp by the value of $I_s$ after the surge

begins $($after $t_s).$ Produce a formatted output to the command window that states the brightness at the time entered.

TASK $3$: $(6-13$min$)$

Create a vector of values for brightness from $0$ to the time entered by the user and create a plot of the theoretical result for

brightness $($ y $-$axis$)$ at each time $($x$-$axis$).$ Your plot should include the following:

X$-$axis label

Title

Y$-$axis label

Gridlines

Solid line $($theoretical data$)$

Red line color

Line thickness of $3$

Note: Use increments of $0\mathrm{.}01$ s $.$ The plot should go from $0$ to the total time by increments of $0\mathrm{.}01$ s $.$

Solve for matlab grader plz