Answer :
Sure! Let's analyze the quadratic function [tex]\( r(t) = -3.22t^2 + 40.63t + 11.17 \)[/tex].
1. Identify the basic parameters:
- The term [tex]\(-3.22t^2\)[/tex] indicates that the parabola opens downwards because the coefficient of [tex]\( t^2 \)[/tex] is negative.
- The term [tex]\( 40.63t \)[/tex] affects the position and slope of the parabola's axis of symmetry.
2. Determine the vertex:
- The vertex form of a quadratic equation [tex]\( at^2 + bt + c \)[/tex] is obtained using [tex]\( t = -\frac{b}{2a} \)[/tex].
- Here, [tex]\( a = -3.22 \)[/tex], [tex]\( b = 40.63 \)[/tex], so [tex]\( t = -\frac{40.63}{2(-3.22)} = \frac{40.63}{6.44} \approx 6.31 \)[/tex].
3. Calculate the revenue at the vertex (maximum revenue):
- Substitute [tex]\( t = 6.31 \)[/tex] back into the original equation to find [tex]\( r(6.31) \)[/tex].
- This would provide the peak revenue value.
4. Key characteristics of the graph:
- Since the parabola opens downward and the vertex occurs at [tex]\( t = 6.31 \)[/tex], the revenue increases until [tex]\( t = 6.31 \)[/tex] years and then decreases.
- The graph should show a peak at [tex]\( t = 6.31 \)[/tex], reaching the maximum revenue.
Given these factors, the graph most likely associated with this model will display an upward trend in revenue initially, reaching a peak, and then a downward trend. Specific options (A, B, C, or D) were not described in detail, so we can say that the correct graph should have these features:
- It is a downward opening parabola.
- It peaks around [tex]\( t = 6.31 \)[/tex].
Based on these criteria, identify the graph that fits this description among the provided choices.
1. Identify the basic parameters:
- The term [tex]\(-3.22t^2\)[/tex] indicates that the parabola opens downwards because the coefficient of [tex]\( t^2 \)[/tex] is negative.
- The term [tex]\( 40.63t \)[/tex] affects the position and slope of the parabola's axis of symmetry.
2. Determine the vertex:
- The vertex form of a quadratic equation [tex]\( at^2 + bt + c \)[/tex] is obtained using [tex]\( t = -\frac{b}{2a} \)[/tex].
- Here, [tex]\( a = -3.22 \)[/tex], [tex]\( b = 40.63 \)[/tex], so [tex]\( t = -\frac{40.63}{2(-3.22)} = \frac{40.63}{6.44} \approx 6.31 \)[/tex].
3. Calculate the revenue at the vertex (maximum revenue):
- Substitute [tex]\( t = 6.31 \)[/tex] back into the original equation to find [tex]\( r(6.31) \)[/tex].
- This would provide the peak revenue value.
4. Key characteristics of the graph:
- Since the parabola opens downward and the vertex occurs at [tex]\( t = 6.31 \)[/tex], the revenue increases until [tex]\( t = 6.31 \)[/tex] years and then decreases.
- The graph should show a peak at [tex]\( t = 6.31 \)[/tex], reaching the maximum revenue.
Given these factors, the graph most likely associated with this model will display an upward trend in revenue initially, reaching a peak, and then a downward trend. Specific options (A, B, C, or D) were not described in detail, so we can say that the correct graph should have these features:
- It is a downward opening parabola.
- It peaks around [tex]\( t = 6.31 \)[/tex].
Based on these criteria, identify the graph that fits this description among the provided choices.
