To find the average velocity of the object between $x = 1$ and $x = 1.2$, we will use the formula for average velocity:

$\text{Average Velocity} = \frac{h(x_2) - h(x_1)}{x_2 - x_1}$

where:

• $h(x_1)$ is the height at $x_1 = 1$
• $h(x_2)$ is the height at $x_2 = 1.2$
• $x_1 = 1$ and $x_2 = 1.2$

Step 1: Calculate $h(1)$

Using the given function $h(x) = -16x^2 + 147x + 83$:

= -16 + 147 + 83 = 214$$ ### Step 2: Calculate $$h(1.2)$$ Now, we calculate the height at $$x = 1.2$$: $$h(1.2) = -16(1.2)^2 + 147(1.2) + 83 = -16(1.44) + 176.4 + 83 = -23.04 + 176.4 + 83 = 236.36$$ ### Step 3: Calculate the average velocity Now, apply the average velocity formula: $$\text{Average Velocity} = \frac{h(1.2) - h(1)}{1.2 - 1} = \frac{236.36 - 214}{0.2} = \frac{22.36}{0.2} = 111.8 \, \text{feet per second}$$ So, the average velocity between $$x = 1$$ and $$x = 1.2$$ is approximately **111.8 feet per second**. --- Would you like more details or have any questions about this? Here are some related questions to consider: 1. How do you interpret the significance of average velocity in this scenario? 2. How would the average velocity change if we used different time intervals? 3. What does the quadratic nature of the function tell you about the motion of the object? 4. Can you calculate the instantaneous velocity at $$x = 1.1$$? 5. What is the object's velocity when it reaches its maximum height? **Tip:** Average velocity measures the overall rate of change in height over time, but it doesn’t account for variations in speed between the two points.