Questions: 3. Ida hit a golf ball from the top of a hill. The height of the ball above the green can be modeled by the regression equation h(t)=-9.7 t^2+48.4 t+11.5 where h represent the height in meters and y represents the time in seconds. a) Use your knowledge of polynomial functions to describe the curve of this function. b) Determine the y-intercept. What does it represent in this context? Show your work. c) The roots of this equation are near t=-0.2 and t=5.2. What do these points represent, if anything?

Solution Steps

The given height function \( h(t) = -9.7t^2 + 48.4t + 11.5 \) is a quadratic equation of the form \( y = ax^2 + bx + c \). Since the coefficient \( a = -9.7 \) is negative, the parabola opens downwards. This indicates that the height of the ball will increase to a maximum point and then decrease as time progresses.

To find the \( y \)-intercept of the function, we evaluate \( h(0) \): \[ h(0) = -9.7(0)^2 + 48.4(0) + 11.5 = 11.5 \] Thus, the \( y \)-intercept is \( 11.5 \) meters. In this context, it represents the initial height of the ball above the green at time \( t = 0 \).

The roots of the equation are given as \( t = -0.2 \) and \( t = 5.2 \). These roots represent the times when the height of the ball is zero, i.e., when the ball hits the ground.

• The root \( t = -0.2 \) is not physically meaningful, as time cannot be negative.
• The root \( t = 5.2 \) indicates the time at which the ball reaches the ground (height \( h(t) = 0 \)).

Final Answer