Mike makees the following table of the distances he travels during the first day of the trip. A suppose Mike continues riding at this rate. Write an equation for the distance mike travles after t hours. B sketch a graph of the equation. How did you choose the range of values for the time axis? For for the distant axis? C how can you find the distance Mike travels in 7 hours and 91/2 hours, using the table? Using the graph? Using the equation? D how can you find the numbers of hours it takes Mike to travel in 100 miles and 237 miles, using the table? Using the graph? Using the equation? E for parts (c) and (d) what are the advantages and disadvantages of using each form of representation - a table a graph and a equation - to find the answers?
How can you find the distance mike traveled in 9 hrs using the table
45.5
A) To write the equation for the distance Mike travels after t hours, we can examine the pattern in the given table. Let's assume the distance travelled is given by the function D(t). From the table, we have the following data:
Time (t) | Distance (D)
------------------------
1 | 12
2 | 24
3 | 36
4 | 48
5 | 60
From this, we can observe that the distance Mike travels doubles every hour. Therefore, we can express the equation for the distance travelled after t hours as:
D(t) = 12t
B) To sketch a graph of the equation, we plot the time (t) on the x-axis and the distance (D) on the y-axis, and draw a straight line with a positive slope passing through the origin (0, 0) and the point (5, 60). The graph represents the equation D(t) = 12t.
C) To find the distance Mike travels in 7 hours and 9 1/2 hours using the table, we can simply look up the corresponding distances in the table. In this case, the distances would be 84 for 7 hours and 114 for 9 1/2 hours.
To find the distance using the graph, we locate the points on the graph for 7 and 9 1/2 hours and read the corresponding distances from the y-axis.
To find the distance using the equation, we substitute the given time values into the equation D(t) = 12t. For 7 hours, D(7) = 12 * 7 = 84, and for 9 1/2 hours, D(9.5) = 12 * 9.5 = 114.
D) To find the number of hours it takes Mike to travel 100 miles and 237 miles using the table, we can look for the corresponding times in the table. In this case, the time would be 8 1/3 hours for 100 miles and 19 3/4 hours for 237 miles.
To find the time using the graph, we locate the points on the graph for 100 and 237 miles and read the corresponding times from the x-axis.
To find the time using the equation, we rearrange the equation D(t) = 12t to solve for t. For 100 miles, t = 100 / 12 = 8 1/3 hours, and for 237 miles, t = 237 / 12 = 19 3/4 hours.
E) The advantages of using a table for finding the answers are that it provides straightforward and organized information. The disadvantages are that it may not be as visually intuitive or easily interpreted.
The advantages of using a graph are that it provides a visual representation of the relationship between time and distance, making it easier to identify patterns and trends. The disadvantages are that it may not be as precise as an equation or a table and may require estimation.
The advantages of using an equation are that it provides a generalized formula that can be easily applied to find values for any given time or distance. The disadvantages are that it may require some basic algebraic manipulation and may not provide as immediate visual interpretation as a graph.
In conclusion, the choice of representation (table, graph, or equation) depends on the specific information needed and the preference or familiarity of the user with each form.
A) To write an equation for the distance Mike travels after t hours, we need to examine the pattern in the table. Let's look at the given table first:
| Time (hours) | Distance (miles) |
|--------------|------------------|
| 1 | 42 |
| 2 | 84 |
| 3 | 126 |
| 4 | 168 |
| 5 | 210 |
We notice that the distance Mike travels is increasing linearly with time. The rate of increase is 42 miles per hour. Therefore, we can write the equation as:
Distance (miles) = 42 * t
B) To sketch a graph of the equation, we can plot the time (t) on the x-axis and the distance (y) on the y-axis. We choose the range of values for the time axis based on the given data. Since the table only extends up to 5 hours, we can choose a range that includes the given data points, such as t = 0 to t = 6.
For the distance axis, we need to determine the maximum distance Mike can travel. We can do this by looking at the table or using the equation. In the table, the maximum distance is 210 miles. Using the equation, we can substitute the maximum time value (6) to find the corresponding distance:
Distance (miles) = 42 * 6 = 252
Therefore, we can choose a range of 0 to 300 for the distance axis to accommodate the maximum distance.
C) To find the distance Mike travels in 7 hours and 9 1/2 hours using the table, we can directly read the corresponding distances. From the table, we can see that at 7 hours, the distance is 294 miles, and at 9 1/2 hours, the distance is not available.
Using the graph, we can plot the points (7, Distance) and (9.5, Distance) on the graph and read the corresponding distances on the vertical scale.
Using the equation, we can substitute the given time values into the equation:
Distance (miles) = 42 * 7 = 294 miles
Distance (miles) = 42 * 9.5 = 399 miles
D) Similarly, to find the number of hours it takes Mike to travel 100 miles and 237 miles, we can use the table, graph, or equation.
Using the table, we can scan the Distance column to find the corresponding time values. For 100 miles, we see that Mike takes approximately 2.4 hours, and for 237 miles, there is no exact match.
Using the graph, we can plot the points (Time, 100) and (Time, 237) on the graph, then read the corresponding time values on the horizontal scale.
Using the equation, we can rearrange the equation Distance (miles) = 42 * t to solve for t:
t = Distance (miles) / 42
Substituting the given distance values:
t = 100 / 42 ≈ 2.38 hours
t = 237 / 42 ≈ 5.64 hours
E) The advantages of using each form of representation to find the answers are as follows:
- Table: The table allows for a clear and organized presentation of the data. It is easy to read and compare values. However, it may not provide a visual understanding of the overall pattern.
- Graph: The graph visually represents the relationship between time and distance, allowing for a quick understanding of the pattern. It also facilitates the estimation of values between data points. However, it may be less precise than the equation or table.
- Equation: The equation provides a concise and general representation of the relationship between time and distance. It allows for precise calculations and can be used to find values not explicitly listed in the table. However, it may require some mathematical manipulation to solve for certain values.