To solve the question, we need to determine the quadratic regression equation that best fits the given data points, and then use that equation to estimate the height of the ball after 0.25 seconds.
Given data points:
- Time (seconds): [tex]\( 0, 0.5, 1, 1.5, 2, 2.5, 3 \)[/tex]
- Height (feet): [tex]\( 0, 35, 65, 85, 95, 100, 95 \)[/tex]
First, let's find the quadratic equation that models the data points. The general form of a quadratic equation is:
[tex]\[ f(x) = ax^2 + bx + c \][/tex]
By applying a quadratic regression to the data points, we can determine the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] for the equation. These coefficients, when rounded to the nearest whole number, are:
[tex]\[ a = -16 \][/tex]
[tex]\[ b = 81 \][/tex]
[tex]\[ c = 0 \][/tex]
So the quadratic regression equation is:
[tex]\[ f(x) = -16x^2 + 81x + 0 \][/tex]
or simply:
[tex]\[ f(x) = -16x^2 + 81x \][/tex]
Now, we need to estimate the height of the ball after 0.25 seconds. We substitute [tex]\( x = 0.25 \)[/tex] into the quadratic equation:
[tex]\[ f(0.25) = -16(0.25)^2 + 81(0.25) \][/tex]
Using the given coefficients and rounding to the nearest whole number:
[tex]\[ f(0.25) ≈ 19 \][/tex]
Thus, the quadratic regression equation that models the data, rounded to the nearest whole number, is:
[tex]\[ f(x) = -16x^2 + 81x + 0 \][/tex]
The estimated height of the ball after 0.25 seconds, rounded to the nearest whole number, is:
[tex]\[ 19 \, \text{feet} \][/tex]
