According to a city's power authority, all streetlamps in the city can draw a percentage of power from a company's new solar panels. [tex]$E(n)$[/tex] is a function that represents the relationship between the number of solar panels installed and the amount of energy generated per day in megawatt hours (MWh), where E(n) [tex]$=0.3 n$[/tex]. [tex]$D ( e )$[/tex] is a function that represents the relationship between the number of days and the energy in (MWh) consumed by the street lamps in a city, where [tex]$D(e)=5 e$[/tex].
Which of the following functions could be used to determine the number of days the streetlamps stay on, based on the number of solar panels installed?
A. [tex]$E(D(e))=(0.3 n)(5 e)$[/tex]
B. [tex]$E(D(e))=0.3(5 e)$[/tex]
C. [tex]$D(E(n))=5(0.3 n)$[/tex]
D. [tex]$D(E(n))=5.3 n$[/tex]
Answer (2)
We have E ( n ) = 0.3 n and D ( e ) = 5 e .
We want to find D ( E ( n )) , which means substituting E ( n ) into D ( e ) .
D ( E ( n )) = 5 ( E ( n )) = 5 ( 0.3 n ) .
Therefore, the function is D ( E ( n )) = 5 ( 0.3 n ) .
Explanation
Understanding the Problem We are given two functions:
E ( n ) = 0.3 n , which represents the energy generated per day in MWh as a function of the number of solar panels n .
D ( e ) = 5 e , which represents the number of days the street lamps stay on as a function of the energy consumed e in MWh.
We want to find a function that determines the number of days the streetlamps stay on based on the number of solar panels installed. This means we want to find a function of the form d a ys = f ( n u mb ere wl in eo f e wl in eso l a re wl in e p an e l s ) .
Function Composition To find the number of days as a function of the number of solar panels, we need to substitute E ( n ) for e in the function D ( e ) . This is called function composition.
Substituting the Functions So we need to find D ( E ( n )) . We know that D ( e ) = 5 e , so we replace e with E ( n ) :
D ( E ( n )) = 5 ( E ( n ))
Since E ( n ) = 0.3 n , we substitute this into the equation:
D ( E ( n )) = 5 ( 0.3 n )
Simplifying the Expression Now we simplify the expression:
D ( E ( n )) = 5 ( 0.3 n ) = 1.5 n
Final Answer Therefore, the function we are looking for is D ( E ( n )) = 5 ( 0.3 n ) . This corresponds to the third option in the given choices.
Examples
Imagine you're planning a community project to power streetlights using solar panels. This problem helps you determine how many days the streetlights can stay on based on the number of solar panels you install. By understanding the relationship between solar panel quantity and streetlight duration, you can optimize your project for maximum efficiency and sustainability. This kind of problem is useful for planning and resource management in renewable energy projects.
The function that determines the number of days the streetlamps stay on based on the number of solar panels installed is D ( E ( n )) = 5 ( 0.3 n ) . This corresponds to option C in the provided choices. Therefore, the correct answer is C.
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