The path of a firework is described by the function: h(1) = 4.97? + 491 + 1.5
where h(t) is the height of the firework, in meters, and t is the time in seconds since the launch.
a) Sketch the pathway of the firework,
b) Determine the maximum height of the firework.
a) To sketch the pathway of the firework, we need to plot the function h(t) on a coordinate plane.
First, we can notice that 4.97? + 491 + 1.5 is just a constant value of approximately 497.5. So we can rewrite the function as:
h(t) = 497.5
This means that the height of the firework is constant at 497.5 meters for the entire time it is in the air. Therefore, the pathway of the firework is a straight horizontal line at a height of 497.5 meters.
b) The maximum height of the firework is 497.5 meters, as shown in part a.
a) To sketch the pathway of the firework, we need to plot points on a graph representing the height of the firework at different times. Let's select a few values of t and calculate the corresponding values of h(t).
Let's take t = 0 seconds, h(0) = 4.97 * 0 + 491 + 1.5 = 492.5 meters.
For t = 1 second, h(1) = 4.97 * 1 + 491 + 1.5 = 497.47 meters.
Similarly, for t = 2, 3, and 4 seconds, we have:
h(2) = 4.97 * 2 + 491 + 1.5 = 502.44 meters.
h(3) = 4.97 * 3 + 491 + 1.5 = 507.41 meters.
h(4) = 4.97 * 4 + 491 + 1.5 = 512.38 meters.
Plotting these points on a graph, we have:
```
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| .
| .
| .
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+--------------------------------------------------------- t (time in seconds)
492 497.47 502.44 507.41 512.38 <- h (height in meters)
```
b) To determine the maximum height of the firework, we need to find the highest point on the graph. From our plotted points, we can see that the maximum height is reached at t = 4 seconds, where h(t) = 512.38 meters. Therefore, the maximum height of the firework is 512.38 meters.