Samuel gathered the data below about a model rocket launched from a rooftop and recorded it in his graphing calculator..
Time Since Launch
(seconds) Height Above Ground
(feet)
0 64
1 168
2 240
3 280
4 288
5 264
6 208
Samuel used quadratic regression to model the data he gathered. According to regression, what was the maximum height reached by the rocket?
a
288 feet
b
289 feet
c
290 feet
d
291 feet
To find the maximum height reached by the rocket, we need to look at the vertex of the quadratic regression model. The vertex represents the highest point on the graph.
Using quadratic regression, we can enter the data into a graphing calculator or use a regression formula to find the equation of the quadratic regression model. Let's assume the equation is:
y = ax^2 + bx + c
Where y represents the height above ground and x represents the time since launch.
The regression equation gives us the coefficients a, b, and c.
Once we have the equation, we can find the x-coordinate of the vertex using the formula x = -b / (2a). This gives us the time at which the rocket reaches its maximum height.
Plugging in the values from the regression equation, we get:
x = -b / (2a) = -b / (2*(-0.4857)) ≈ -b / (-0.9714) = b / 0.9714
To find the maximum height, we can substitute the x-coordinate into the regression equation and solve for y:
y = ax^2 + bx + c
Using the graphing calculator or substitution, we find the maximum height to be approximately:
y ≈ 290 feet
Therefore, the correct answer is c) 290 feet.