This series of problems will build your fluency with using a graphing calculator to answer contextual questions about scenarios that can be modeled by quadratic functions. For each question below, graph the quadratic function on your graphing calculator and use the graph to answer the questions. ● Round all decimal values to three decimal places. ● Use the answer key to determine that you accurately answered the questions. ● If necessary, revise your work. 1. The height of a golf ball hit into the air is modeled by the equation ℎ(푡) = −16푡 2 +45푡, where h represents the height, in feet, and t represents the number of seconds that have passed since the ball was hit. a. After how many seconds does the golf ball reach the ground? b. What is the maximum height the golf ball reaches before beginning to fall to the ground? After how many seconds does the ball begin falling?

Question

This series of problems will build your fluency with using a graphing calculator to answer contextual questions about scenarios that can be modeled by quadratic functions. For each question below, graph the quadratic function on your graphing calculator and use the graph to answer the questions. ● Round all decimal values to three decimal places. ● Use the answer key to determine that you accurately answered the questions. ● If necessary, revise your work. 1. The height of a golf ball hit into the air is modeled by the equation ℎ(푡) = −16푡 2 +45푡, where h represents the height, in feet, and t represents the number of seconds that have passed since the ball was hit. a. After how many seconds does the golf ball reach the ground? b. What is the maximum height the golf ball reaches before beginning to fall to the ground? After how many seconds does the ball begin falling? 
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© Success Academy Charter Schools 4 In a contextual problem, how does understanding the context of the problem help determine where on the graph to find the answer to the question?

GPT-4o mini · Answer

To solve the problems given in your scenario, we will analyze the quadratic function:

\[ h(t) = -16t^2 + 45t \]

This equation models the height of a golf ball over time, where \( h \) is the height in feet and \( t \) is the time in seconds.

a. After how many seconds does the golf ball reach the ground?

To find out when the golf ball reaches the ground, we need to determine when the height \( h(t) = 0 \). This corresponds to the points where the graph of the function intersects the t-axis.

Set the equation to zero: \[-16t^2 + 45t = 0\]
Factor the equation: \[t(-16t + 45) = 0\]
Solve for \( t \): \[t = 0 \quad \text{or} \quad -16t + 45 = 0\]

From the second part: \[-16t + 45 = 0\] \[16t = 45\] \[t = \frac{45}{16} \approx 2.8125\]

Thus, the golf ball reaches the ground at \( t \approx 2.813 \) seconds.

b. What is the maximum height the golf ball reaches before beginning to fall to the ground? After how many seconds does the ball begin falling?

To find the maximum height, we need to determine the vertex of the parabola, as the vertex gives us the optimal (maximum) point for a quadratic function opening downwards.

The formula for the \( t \)-coordinate of the vertex for a quadratic equation in the form \( at^2 + bt + c \) is:

\[ t = -\frac{b}{2a} \]

For our function:

\( a = -16 \)
\( b = 45 \)

Substituting these values in:

\[ t = -\frac{45}{2 \times -16} = \frac{45}{32} \approx 1.40625 \]

At this time, we can find the maximum height \( h(t) \):

\[ h\left(\frac{45}{32}\right) = -16\left(\frac{45}{32}\right)^2 + 45\left(\frac{45}{32}\right) \]

Calculating the height:

Calculate \( \left(\frac{45}{32}\right)^2 \approx 1.953125 \)
Now substituting it back to find height: \[ h\left(\frac{45}{32}\right) \approx -16 \times 1.953125 + 45 \times \frac{45}{32} \] \[ = -31.25 + 63.28125 \approx 32.03125 \]

So, the maximum height is approximately \( 32.031 \) feet, and the ball reaches this height at approximately \( t \approx 1.406 \) seconds.

Summary of Answers:

a. The golf ball reaches the ground after \( t \approx 2.813 \) seconds.
b. The maximum height reached is approximately \( 32.031 \) feet, occurring at \( t \approx 1.406 \) seconds.

Understanding the Context:

Understanding the context of this problem helps in interpreting the results carefully. For example, we know that the height must be non-negative (as negative heights would not make sense physically in this situation), and we interpret the point where the graph intersects the x-axis as the moment the ball meets the ground. Moreover, knowing about projectile motion tells us that the graph will have a peak (the vertex) that indicates the maximum height of the ball before it descends, due to gravity acting on it. This context helps correctly orient oneself while analyzing the graph.