The function [tex]$h(t)=-16 t^2+28 t+500$[/tex] represents the height of a rock [tex]$t$[/tex] seconds after it is propelled by a slingshot.
What does [tex]$h(3.2)$[/tex] represent?
A. the height of the rock 3.2 seconds before it reaches the ground
B. the time it takes the rock to reach the ground, or 3.2 seconds
C. the time it takes the rock to reach a height of 3.2 meters
D. the height of the rock 3.2 seconds after it is propelled
Answer (2)
The problem provides a function h ( t ) that models the height of a rock after t seconds.
We substitute t = 3.2 into the function: h ( 3.2 ) = − 16 ( 3.2 ) 2 + 28 ( 3.2 ) + 500 .
Calculate the result: h ( 3.2 ) = 425.76 .
h ( 3.2 ) represents the height of the rock 3.2 seconds after it is propelled. the height of the rock 3.2 seconds after it is propelled
Explanation
Understanding the Problem The problem states that the function h ( t ) = − 16 t 2 + 28 t + 500 represents the height of a rock t seconds after it is propelled by a slingshot. We need to determine what h ( 3.2 ) represents.
Substituting the Value of t To find out what h ( 3.2 ) represents, we need to evaluate the function h ( t ) at t = 3.2 . This means we substitute 3.2 for t in the equation: h ( 3.2 ) = − 16 ( 3.2 ) 2 + 28 ( 3.2 ) + 500
Calculating h(3.2) Now, let's calculate the value: h ( 3.2 ) = − 16 ( 10.24 ) + 89.6 + 500 h ( 3.2 ) = − 163.84 + 89.6 + 500 h ( 3.2 ) = − 163.84 + 589.6 h ( 3.2 ) = 425.76 So, h ( 3.2 ) = 425.76 .
Interpreting the Result Since h ( t ) represents the height of the rock t seconds after it is propelled, h ( 3.2 ) represents the height of the rock 3.2 seconds after it is propelled.
Final Answer Therefore, h ( 3.2 ) represents the height of the rock 3.2 seconds after it is propelled.
Examples
Understanding functions like h ( t ) is crucial in physics and engineering. For example, if you're designing a bridge or launching a rocket, you need to know how height, distance, and time are related. By plugging in different values for time t , engineers can predict the height of a projectile at any given moment, ensuring safety and accuracy in their designs. This helps in calculating trajectories, avoiding obstacles, and landing precisely where intended.
The function h ( t ) = − 16 t 2 + 28 t + 500 describes the height of a rock after t seconds. Evaluating h ( 3.2 ) yields a height of 425.76, indicating that it represents the height of the rock 3.2 seconds after launch. Therefore, the correct answer is option D.
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