Firework portfolio first conduct some research to help you with later portions of this portfolio assessment. your local church is 50 feet tall. What are some of the initial velocity of different types of fireworks task to while setting up if I was just what you have a tour at the top of the building I need to drop it to a coworker below how long will it take for the tool to fall to the ground hint use the first equation you were given above h(t)=-16^2+h naught. For the buildings height, use the height of the building that you estimated and task one H naught represents the initial height of the tool. Let h(t) the zero to represent the height being zero solve for t (time) do you know how long it will take for the total reach the ground you must show your work. Draw a graph that represents the path of this to falling to the ground. Be sure to label your access with a title on a scale. Your graph should show the height of the tool h after t seconds have passed. Label this line tool. State whether the parabola represented by h(t)=-16t^2+250t opens up or down. Explain why your answer makes sense in the context of this problem. Consider with the coefficient of T squared is and how that affects the parabola. One of the fireworks has launched from the top of the building with an initial upward velocity of 150 ft./s. What is the equation for the situation? Use the height of your building from task one the initial velocity takes the place of the coefficient of t in the equation. When will the fireworks land? If it does not explode? What is the height of the firework when it lands on the ground plug that in for height and solve for t. You are looking for an X intercept you must show your work. Make a table for the situation so that it shows the height from time T equals zero until it hits the ground. Use random time values in between the time being zero in the time when it hits the ground to complete your table. Calculate the access of symmetry. Use the formula for finding the axis of symmetry. You must show your work. Calculate the coordinates of the vertex. Use the value of the axis of symmetry to find the Y coordinate of the vertex. You must show your work. Explain why negative values for t and h(t) represent in the context of this scenario, and why they could not be negative. On the same coordinate plane from number one draw a graph that represents the path of this firework. Make sure your graph is labeled appropriately. Label this graph firework number one choose an initial velocity for a firework based on your research from task one. Write an equation that represents the path for a firework that has launch from the ground with the initial velocity that you chose. Choose whatever initial velocity consider where that fit into the equation this time you are launching the firework from the ground, so consider how that changes the equation. Suppose this firework is set to explode three seconds after it is launched at what height will is firework be when it explodes. If the firework explodes after three seconds after it is launched, consider where that value of 3/5 into the equation to find the height of the firework at that time, you must show your work. On the same coordinate plane that you’ve been using dry grab the river to the path of this firework mark your graph to indicate the point at which the firework will explode. Label this craft firework number two. You launch a third firework decide whether you want to watch it from the ground or from the roof of the building. Also choose a height at which is firework will explode, and then initial velocity for this firework decide whether you are launching from the ground or the building and decide your own initial velocity choose at what height your firework will explode. Make sure you don’t use a hike that is greater than the maximum. How long after setting off the firework should the delay be set? At what time will the fireworks explode based on the height you wanted it to explode. consider where that height fits into the equation and solve for time you must show your work. On the same coordinate plane that you’ve been using drawl a graph that represents the path of this firework. Mark your graft indicate the point I watch the fireworks explode. Label this graph firework number three. What can you conclude about how the height of the building in the initial velocity of the item launched affect the maximum height and the time it takes to get there?
To calculate the initial velocity of different types of fireworks, you will need to conduct research as stated. However, I can help you with the other parts of your question.
Before we begin, let's assume that the standard unit of measurement is in feet.
1. Determine the time it takes for the tool to fall to the ground from the top of a 50-foot-tall building.
Given:
- h(t) = -16t^2 + h₀ (where h(t) represents the height at time t and h₀ is the initial height)
- h₀ = 50 ft
We need to find t, the time it takes for the tool to reach the ground (h(t) = 0).
Setting h(t) = 0:
-16t^2 + 50 = 0
Solving the quadratic equation:
16t^2 = 50
t^2 = 50/16
t^2 ≈ 3.125
t ≈ √3.125
t ≈ 1.766 seconds
Therefore, it will take approximately 1.766 seconds for the tool to fall to the ground.
2. Graph the path of the tool falling to the ground.
Draw a graph with the vertical axis representing the height (h) and the horizontal axis representing time (t). The graph should start at h₀ = 50 ft and decrease as time passes. Label the line "Tool."
3. The parabola represented by h(t) = -16t^2 + 250t opens downward.
The coefficient of t^2 is -16, which is negative. When the coefficient is negative, the parabola opens downward, which makes sense because the object is falling due to gravity.
Now let's move on to the next part of your question.
4. Find the equation for a firework launched with an upward velocity of 150 ft/s from the top of the building.
Given:
- Initial upward velocity (v₀) = 150 ft/s
- h₀ (height) = 50 ft
The equation for the firework's height (h(t)) would be:
h(t) = -16t^2 + v₀t + h₀
Substituting the given values:
h(t) = -16t^2 + 150t + 50
5. Determine when the firework will land if it does not explode.
To find when the firework will land, we need to find the time when h(t) = 0.
Setting h(t) = 0:
-16t^2 + 150t + 50 = 0
Solving the quadratic equation:
16t^2 - 150t - 50 = 0
Using the quadratic formula, we find:
t ≈ 9.572 seconds or t ≈ 0.328 seconds
Since the firework cannot land before it is launched, we discard the negative value. Therefore, the firework will land after approximately 9.572 seconds.
6. Create a table showing the height of the firework at different times.
Choose random time values between t = 0 and t = 9.572, and calculate the corresponding height using the equation h(t) = -16t^2 + 150t + 50.
Time (t) | Height (h)
----------------------
t₁ | h(t₁)
t₂ | h(t₂)
t₃ | h(t₃)
... | ...
7. Calculate the axis of symmetry.
To calculate the axis of symmetry, use the formula:
t = -b / (2a)
Given the quadratic equation:
16t^2 - 150t - 50 = 0
a = 16, b = -150
t = -(-150) / (2 * 16)
t ≈ 4.688 seconds
The axis of symmetry is at t ≈ 4.688 seconds.
8. Calculate the coordinates of the vertex.
To find the coordinates of the vertex, substitute the axis of symmetry value (t ≈ 4.688) into the equation h(t) = -16t^2 + 150t + 50.
h(t) = -16(4.688)^2 + 150(4.688) + 50
≈ 118.687 ft
Thus, the coordinates of the vertex are (4.688, 118.687).
9. Negative values for t and h(t) represent invalid values in the context of this scenario.
In this scenario, negative values for t would represent a time before the object was launched or before t = 0. Similarly, negative values for h(t) would imply a negative height, which is nonsensical in the context of the problem.
Both time and height cannot be negative because they do not have a physical meaning in this context.
10. Plot the graph for the firework.
On the same coordinate plane, draw a graph representing the path of the firework. Label this graph "Firework Number One" and indicate the point of explosion.
11. Conclusion on the effects of building height and initial velocity.
From the calculations and graphs, we can conclude the following:
- The higher the building, the longer it takes for an object to fall to the ground.
- Increasing the initial velocity will result in the object reaching a greater maximum height.
- The time it takes for a firework to land increases with higher initial velocities.
- The maximal height of a firework increases with higher initial velocities.
These observations show that both the building height and initial velocity affect the maximum height achieved and the time it takes to reach that point.
Based on the given information, the height of the local church is 50 feet. In order to find the initial velocity of different types of fireworks, research would need to be conducted. Different fireworks can have varying initial velocities based on their design and purpose.
To determine the time it takes for an object to fall from the top of the building to the ground, the equation h(t) = -16t^2 + h0 can be used. In this equation, h(t) represents the height of the object at time t, h0 represents the initial height of the object, and -16 represents the acceleration due to gravity.
Using the height of the building as h0, which is 50 feet, the equation becomes h(t) = -16t^2 + 50. To find the time it takes for the tool to fall to the ground, we need to find the value of t when h(t) equals zero.
Setting h(t) = 0, we get the equation -16t^2 + 50 = 0. Solving this equation for t, we find:
16t^2 = 50
t^2 = 50/16
t = sqrt(50/16)
t ≈ 1.77 seconds
Therefore, it will take approximately 1.77 seconds for the tool to fall from the top of the building to the ground.
To represent this on a graph, the x-axis can represent time (t) in seconds and the y-axis can represent height (h) in feet. The line representing the tool's height would start at 50 feet and gradually decrease until it reaches zero at approximately 1.77 seconds. This graph would show a parabola opening downwards.
For the firework launched with an initial upward velocity of 150 ft/s, the equation representing its height can be determined using the same equation as before. In this case, however, the initial velocity replaces the coefficient of t. So the equation becomes h(t) = -16t^2 + 150t + h0, where h0 is the height of the building.
Using the height of the building as h0, the equation becomes h(t) = -16t^2 + 150t + 50. To find when the firework will land, we need to find the value of t when h(t) equals zero.
Setting h(t) = 0, we get the equation -16t^2 + 150t + 50 = 0. Solving this equation for t will provide the time at which the firework lands.
To complete the table for this situation, random time values can be chosen between 0 (when t = 0) and the time when the firework hits the ground. For each time value, the height (h) can be calculated using the equation -16t^2 + 150t + 50.
The axis of symmetry can be calculated using the formula t = -b/2a, where a and b are the coefficients in the equation -16t^2 + 150t + 50 = 0. The value of t obtained through this calculation will represent the time at which the firework reaches its maximum height.
To find the coordinates of the vertex, the axis of symmetry can be substituted into the equation -16t^2 + 150t + 50 to find the corresponding height (h). The coordinates of the vertex will be represented as (t, h).
Negative values for t and h(t) represent values before the object or firework is launched. In the context of the scenario, it does not make sense for the values to be negative because they would indicate a time or height before the object or firework was in motion.
Based on the data, a conclusion can be drawn regarding how the height of the building and the initial velocity of the launched item affect the maximum height and the time it takes to reach it. As the height of the building increases, the maximum height the object or firework can reach also increases. Similarly, as the initial velocity increases, the maximum height the object or firework can reach also increases. The time it takes to reach the maximum height will vary depending on the specific values of the height and initial velocity.
To answer this question and complete the firework portfolio assessment, let's break it down step by step:
1. Finding the time it takes for an object to fall from the top of the building:
- Use the formula h(t) = -16t^2 + h0, where h(t) represents the height at time t, and h0 is the initial height of the tool.
- In this case, h0 is the height of the church, which is given as 50 feet.
- Set h(t) = 0 to find when the tool reaches the ground.
- Solve the equation -16t^2 + 50 = 0 for t. This can be done by factoring or using the quadratic formula.
2. Drawing a graph of the falling object:
- Create a graph with the x-axis representing time (t) and the y-axis representing height (h).
- Label the graph "Tool" and plot points to represent the height of the tool at different time intervals. Connect these points to form a parabolic line.
3. Determining the orientation of the parabola:
- The equation of the parabola is h(t) = -16t^2 + 250t. The coefficient of t^2 is -16.
- The parabola opens downward because the coefficient of t^2 is negative.
- This makes sense in context since the tool is falling under the force of gravity.
4. Finding the equation for a firework launched upward:
- Use the formula h(t) = -16t^2 + v0t + h0, where v0 is the initial upward velocity of the firework.
- Substitute the given values, such as the height of the building (h0) and initial velocity (v0), to determine the specific equation.
5. Finding when the firework will land:
- Set h(t) = 0 to find when the firework reaches the ground.
- Solve the equation -16t^2 + v0t + h0 = 0 for t.
- This can be done by factoring or using the quadratic formula.
6. Creating a table to represent the height of the firework over time:
- Choose several time values between 0 (launch) and when the firework hits the ground.
- Calculate the corresponding heights using the equation h(t) = -16t^2 + v0t + h0 for each time value.
- Create a table with columns for time (t) and height (h).
7. Calculating the axis of symmetry and vertex:
- The axis of symmetry formula is t = -b / (2a), where a and b are coefficients from the equation -16t^2 + v0t + h0 = 0.
- Calculate the axis of symmetry using the equation for the firework, and this will give you the time (t) when the firework reaches its maximum height.
- Plug this value into the equation to find the height (h) at that time.
- The coordinates of the vertex will be (t, h).
8. Explaining the negative values for t and h(t):
- In this scenario, negative values for time (t) indicate the time before the firework is launched.
- Negative values for height (h) are not physically meaningful in this context because height cannot be negative.
- The negative values are necessary to fully represent the mathematical equation but should be disregarded in the context of the problem.
9. Drawing a graph of the firework:
- Use the same graph from step 2 and label it "Firework Number One."
- Plot points representing the height of the firework at different time intervals on the graph.
- Connect the points to form a parabolic line.
10. Repeating steps 5-9 for the second and third fireworks:
- Choose different initial velocities, launch heights, and explosion times for each firework.
- Find the equation of the firework's path, when it lands, create a table, determine the axis of symmetry and vertex, and draw the graph.
11. Analyzing the relationship between building height, initial velocity, maximum height, and time:
- Compare the initial velocity and building height for the different fireworks and observe how they affect maximum height.
- Generally, a higher initial velocity or taller building will result in a higher maximum height for the firework.
- The time taken to reach the maximum height may vary based on the specific values chosen for the initial velocity and building height.