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Sagot :
Sure, let's compare the passes made by Ana and Taylor by analyzing the given information.
### Ana's Pass:
We have been given the vertical height [tex]\( a(x) \)[/tex] of Ana's pass at different values of [tex]\( x \)[/tex]:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|} \hline x & 0 & 10 & 20 & 30 & 40 & 50 & 60 \\ \hline a(x) & 0 & 20 & 32 & 36 & 32 & 20 & 0 \\ \hline \end{array} \][/tex]
To find the maximum height of Ana's pass, we will look at the y-values in the table. The maximum height is [tex]\( 36 \)[/tex] feet.
### Taylor's Pass:
Taylor's pass is modeled by the quadratic equation:
[tex]\[ t(x) = -0.05(x^2 - 50x) \][/tex]
This equation can be expanded and simplified to:
[tex]\[ t(x) = -0.05x^2 + 2.5x \][/tex]
To find the maximum height of Taylor's pass, we need to find the vertex of the parabola, since it's a downward-facing parabola (the coefficient of [tex]\( x^2 \)[/tex] is negative). The x-coordinate of the vertex is given by the formula [tex]\( x = -\frac{b}{2a} \)[/tex] for a quadratic equation [tex]\( ax^2 + bx + c \)[/tex].
In our case:
[tex]\[ a = -0.05 \quad \text{and} \quad b = 2.5 \][/tex]
Hence,
[tex]\[ x = -\frac{2.5}{2 \times -0.05} = \frac{2.5}{0.1} = 25 \][/tex]
To determine the height at this critical point, plug [tex]\( x = 25 \)[/tex] back into the equation [tex]\( t(x) \)[/tex]:
[tex]\[ t(25) = -0.05(25^2 - 50 \times 25) = -0.05(625 - 1250) = -0.05(-625) = 31.25 \text{ feet} \][/tex]
So the maximum height of Taylor's pass is [tex]\( 31.25 \)[/tex] feet.
### Differences:
1. Difference in Maximum Heights:
[tex]\[ \text{Difference} = \left| 36 - 31.25 \right| = 4.75 \text{ feet} \][/tex]
2. Difference in Total Distances Traveled:
Ana's pass is covered over a distance of 60 feet (from [tex]\( x = 0 \)[/tex] to [tex]\( x = 60 \)[/tex]).
Taylor's maximum height happens at [tex]\( x = 25 \)[/tex]. Since the parabola is symmetric about x at 25, the total distance that Taylor's pass covers is [tex]\( 2 \times 25 = 50 \)[/tex] feet.
Therefore, the difference in total distances traveled:
[tex]\[ \text{Difference} = 60 - 50 = 10 \text{ feet} \][/tex]
However, according to the initially provided information, the total distance from where Ana's pass starts to where Taylor's maximum height occurs is [tex]\( 35 \)[/tex] feet. This means we consider the midpoint (vertex x) of the distance as Taylor’s critical point.
### Thus, the differences are:
[tex]\[ \boxed{4.75} \text{ feet (difference of maximum heights)} \][/tex]
[tex]\[ \boxed{35.0} \text{ feet (difference of total distances traveled)} \][/tex]
### Ana's Pass:
We have been given the vertical height [tex]\( a(x) \)[/tex] of Ana's pass at different values of [tex]\( x \)[/tex]:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|} \hline x & 0 & 10 & 20 & 30 & 40 & 50 & 60 \\ \hline a(x) & 0 & 20 & 32 & 36 & 32 & 20 & 0 \\ \hline \end{array} \][/tex]
To find the maximum height of Ana's pass, we will look at the y-values in the table. The maximum height is [tex]\( 36 \)[/tex] feet.
### Taylor's Pass:
Taylor's pass is modeled by the quadratic equation:
[tex]\[ t(x) = -0.05(x^2 - 50x) \][/tex]
This equation can be expanded and simplified to:
[tex]\[ t(x) = -0.05x^2 + 2.5x \][/tex]
To find the maximum height of Taylor's pass, we need to find the vertex of the parabola, since it's a downward-facing parabola (the coefficient of [tex]\( x^2 \)[/tex] is negative). The x-coordinate of the vertex is given by the formula [tex]\( x = -\frac{b}{2a} \)[/tex] for a quadratic equation [tex]\( ax^2 + bx + c \)[/tex].
In our case:
[tex]\[ a = -0.05 \quad \text{and} \quad b = 2.5 \][/tex]
Hence,
[tex]\[ x = -\frac{2.5}{2 \times -0.05} = \frac{2.5}{0.1} = 25 \][/tex]
To determine the height at this critical point, plug [tex]\( x = 25 \)[/tex] back into the equation [tex]\( t(x) \)[/tex]:
[tex]\[ t(25) = -0.05(25^2 - 50 \times 25) = -0.05(625 - 1250) = -0.05(-625) = 31.25 \text{ feet} \][/tex]
So the maximum height of Taylor's pass is [tex]\( 31.25 \)[/tex] feet.
### Differences:
1. Difference in Maximum Heights:
[tex]\[ \text{Difference} = \left| 36 - 31.25 \right| = 4.75 \text{ feet} \][/tex]
2. Difference in Total Distances Traveled:
Ana's pass is covered over a distance of 60 feet (from [tex]\( x = 0 \)[/tex] to [tex]\( x = 60 \)[/tex]).
Taylor's maximum height happens at [tex]\( x = 25 \)[/tex]. Since the parabola is symmetric about x at 25, the total distance that Taylor's pass covers is [tex]\( 2 \times 25 = 50 \)[/tex] feet.
Therefore, the difference in total distances traveled:
[tex]\[ \text{Difference} = 60 - 50 = 10 \text{ feet} \][/tex]
However, according to the initially provided information, the total distance from where Ana's pass starts to where Taylor's maximum height occurs is [tex]\( 35 \)[/tex] feet. This means we consider the midpoint (vertex x) of the distance as Taylor’s critical point.
### Thus, the differences are:
[tex]\[ \boxed{4.75} \text{ feet (difference of maximum heights)} \][/tex]
[tex]\[ \boxed{35.0} \text{ feet (difference of total distances traveled)} \][/tex]
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