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Type the correct answer in each box. Use numerals instead of words. If necessary, round your answer to the nearest hundredth.

Ana and Taylor are on the lacrosse team and practicing their passes. The vertical height, [tex]\(a(x)\)[/tex], of Ana's pass [tex]\(x\)[/tex] feet from where it was thrown is modeled in this table.

[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]

The vertical height, [tex]\(t(x)\)[/tex], of Taylor's pass [tex]\(x\)[/tex] feet from where it was thrown is modeled by this equation.
[tex]\[
t(x) = -0.05(x^2 - 50x)
\][/tex]

Complete the statements comparing their passes.

The difference of the maximum heights is [tex]\(\square\)[/tex] feet.

The difference of the total distances traveled is [tex]\(\square\)[/tex] feet.



Answer :

GinnyAnswer
To compare the maximum heights and the total distances traveled of Ana's and Taylor's passes, let's analyze the provided data:

For Ana:
- From the table, the vertical heights at various distances [tex]\(x\)[/tex] are given.
- The maximum height achieved by Ana's pass, [tex]\(a(x)\)[/tex], is 36 feet.
- The total distance traveled by Ana's pass is 60 feet.

For Taylor:
- The vertical height for Taylor's pass, [tex]\(t(x)\)[/tex], is given by the equation [tex]\( t(x) = -0.05(x^2 - 50x) \)[/tex].
- The maximum height Taylor's pass achieves is 31.25 feet.
- The total distance traveled by Taylor's pass, determined by solving the roots of the equation, is 50 feet.

From this data, we can compute the differences:
1. The difference in the maximum heights:
[tex]\[ 36 - 31.25 = 4.75 \text{ feet} \][/tex]

2. The difference in the total distances traveled:
[tex]\[ 60 - 50 = 10.0 \text{ feet} \][/tex]

Therefore:
- The difference of the maximum heights is [tex]\( 4.75 \)[/tex] feet.
- The difference of the total distances traveled is [tex]\( 10.0 \)[/tex] feet.

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