The table below shows the height, [tex]h(t)[/tex], in meters, of an object that is thrown off the top of a building, [tex]t[/tex] seconds after it is thrown.

\begin{tabular}{|c|r|r|r|r|r|r|}
\hline
[tex]t[/tex] & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 \\
\hline
[tex]h(t)[/tex] & 78.125 & 90.8 & 101.025 & 108.8 & 114.125 & 117 \\
\hline
\end{tabular}

A) Using your calculator to do a quadratic regression, express the height of the object as a function of the number of seconds that have passed since the object was thrown.
[tex]\[
h(t)= \boxed{}
\][/tex]
Round all numbers to 1 decimal place.

B) Using your regression equation, how high will the object be 2.8 seconds after it is thrown?
[tex]\[
\boxed{}
\][/tex]
Round to 3 decimal places.

C) Using your regression equation, how long will it take the object to reach 26 meters?
[tex]\[
\boxed{}
\][/tex]
Round to 3 decimal places.



Answer :

Let's solve the problem step by step:

### Part A: Finding the Quadratic Regression Equation
To express the height [tex]\( h(t) \)[/tex] of the object as a function of the number of seconds since it was thrown, we perform a quadratic regression using the given data points. The general form of the quadratic equation is:

[tex]\[ h(t) = at^2 + bt + c \][/tex]

After performing the quadratic regression with the given data points:

[tex]\[ (t, h(t)) = \{(0.5, 78.125), (1, 90.8), (1.5, 101.025), (2, 108.8), (2.5, 114.125), (3, 117)\} \][/tex]

We find the coefficients:
[tex]\[ a = 63.0 \][/tex]
[tex]\[ b = 32.7 \][/tex]
[tex]\[ c = -4.9 \][/tex]

Thus, the height of the object as a function of time [tex]\( t \)[/tex] is:

[tex]\[ h(t) = 63.0t^2 + 32.7t - 4.9 \][/tex]

### Part B: Calculating Height at [tex]\( t = 2.8 \)[/tex] seconds
Using the regression equation [tex]\( h(t) \)[/tex], we can find the height of the object at [tex]\( t = 2.8 \)[/tex] seconds:

[tex]\[ h(2.8) = 63.0(2.8)^2 + 32.7(2.8) - 4.9 \][/tex]
[tex]\[ h(2.8) = 580.580 \][/tex]

Therefore, the height of the object 2.8 seconds after it is thrown is:

[tex]\[ 580.580 \, \text{meters} \][/tex]

### Part C: Calculating the Time to Reach a Height of 26 Meters
Using the regression equation [tex]\( h(t) \)[/tex], we want to find the time [tex]\( t \)[/tex] when the height [tex]\( h(t) \)[/tex] is 26 meters. We set up the equation:

[tex]\[ 63.0t^2 + 32.7t - 4.9 = 26 \][/tex]

This simplifies to:

[tex]\[ 63.0t^2 + 32.7t - 30.9 = 0 \][/tex]

Solving this quadratic equation for [tex]\( t \)[/tex], we find:

[tex]\[ t = 0.487 \][/tex]

Therefore, it will take approximately:

[tex]\[ 0.487 \, \text{seconds} \][/tex]

for the object to reach a height of 26 meters.

### Summary of Results:
- Part A: The height function is [tex]\( h(t) = 63.0t^2 + 32.7t - 4.9 \)[/tex].
- Part B: The height of the object at [tex]\( t = 2.8 \)[/tex] seconds is [tex]\( 580.580 \)[/tex] meters.
- Part C: The time for the object to reach a height of 26 meters is [tex]\( 0.487 \)[/tex] seconds.

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