Let's analyze the function given: [tex]\( n(t) = -2.82 t^2 + 25.74 t + 60.87 \)[/tex]. This is a quadratic function, which generally takes the form [tex]\( n(t) = at^2 + bt + c\)[/tex]. In this form, [tex]\(a = -2.82\)[/tex], [tex]\(b = 25.74\)[/tex], and [tex]\(c = 60.87\)[/tex].
Since the coefficient [tex]\(a\)[/tex] is negative ([tex]\(-2.82\)[/tex]), the parabola opens downwards. This indicates that the number of customers, [tex]\(n(t)\)[/tex], will reach a maximum value at some point and then start to decrease.
To find the vertex of the parabola, which gives us the time [tex]\( t \)[/tex] at which the maximum number of customers occurs, we can use the vertex formula for a parabola [tex]\( t = -\frac{b}{2a} \)[/tex].
Upon calculation:
[tex]\[ t = -\frac{25.74}{2(-2.82)} = 4.5638297872340425 \][/tex]
This value indicates that the maximum number of customers occurs at approximately [tex]\( t = 4.56 \)[/tex] hours.
Next, we substitute [tex]\( t = 4.56 \)[/tex] back into the function to find the maximum number of customers:
[tex]\[ n(4.56) = -2.82(4.56)^2 + 25.74(4.56) + 60.87 = 119.60648936170213 \][/tex]
Therefore, the peak number of customers is approximately 119.61.
Additionally, it's useful to know the values of [tex]\( n(t) \)[/tex] at some specific [tex]\( t \)[/tex] values:
- At [tex]\( t = 0 \)[/tex]:
[tex]\[ n(0) = 60.87 \][/tex]
- At [tex]\( t = 1 \)[/tex]:
[tex]\[ n(1) = -2.82(1)^2 + 25.74(1) + 60.87 = 83.79 \][/tex]
- At [tex]\( t = 2 \)[/tex]:
[tex]\[ n(2) = -2.82(2)^2 + 25.74(2) + 60.87 = 101.07 \][/tex]
By understanding these key points:
- The graph is a parabola opening downwards.
- The peak occurs at [tex]\( t \approx 4.56 \)[/tex] with [tex]\( n(t) \approx 119.61 \)[/tex].
- At [tex]\( t = 0 \)[/tex], [tex]\( n(t) = 60.87 \)[/tex].
You can now match these characteristics to the provided graphs to select the correct one. Be on the lookout for:
- A graph that peaks between [tex]\( t = 4 \)[/tex] and [tex]\( t = 5 \)[/tex].
- The peak height should be just under 120.
- The starting value at [tex]\( t = 0 \)[/tex] should be around 60.87.
Choose the graph that best aligns with these features.
