A freight company will move one truck load of material according to the following cost schedule: The base shipping fee is $400 for any distance up to and including 150 miles and an additional $50 for each 50 mile increment over 150 miles.
The cargo cost is 15 cents per pound of freight shipped
The total shipping cost is the sum of the base shipping fee and the cargo cost
In the context of the scenario above, suppose you need to ship 3000 lbs. discuss the continuity of the total cost as a function of the trip distance for total distances ranging from 50 to 350 miles. Then, for trips less than 150 miles, discuss the continuity of the total cost as a function of shipping weight for total freight weighing 1000 to 4000 lbs. What are the roles played by limits in analyzing the continuity of each function in your answers?
For the first part is the function 0<x < or = 1<0
For the second part is the function 1000 < or equal to x < or equal to 4000
What is the limit I can’t figure it out for both parts. Any help would be greatly appreciated.
To analyze the continuity of the total cost as a function of trip distance and shipping weight, we need to understand the roles played by limits.
For the first part, where the total distances range from 50 to 350 miles, we want to determine if there are any discontinuities in the total cost function. To do this, we should consider the behavior as the distance approaches the boundaries of the given range.
Let's calculate the total cost for various distances within the range:
For a distance of 50 miles:
Total cost = Base shipping fee + Cargo cost
Total cost = $400 + (15 cents/pound * 3000 pounds)
Total cost = $400 + $450 = $850
For a distance of 150 miles:
Total cost = $400 + (15 cents/pound * 3000 pounds)
Total cost = $400 + $450 = $850
For a distance of 200 miles:
Total cost = $400 + $50 (additional fee for 50-mile increment) + (15 cents/pound * 3000 pounds)
Total cost = $400 + $50 + $450 = $900
For a distance of 350 miles:
Total cost = $400 + $150 (additional fee for two 50-mile increments) + (15 cents/pound * 3000 pounds)
Total cost = $400 + $150 + $450 = $1000
From these calculations, we observe that the total cost is continuous within the given range of distances (50 to 350 miles). There are no abrupt changes or jumps in the cost. The function is continuous for 50 ≤ x ≤ 350.
Now, let's move on to the second part, where the total freight weight ranges from 1000 to 4000 pounds. Here, we need to determine if there are any discontinuities in the total cost as a function of shipping weight.
Again, let's calculate the total cost for various weights within the range:
For a weight of 1000 pounds:
Total cost = $400 + (15 cents/pound * 1000 pounds)
Total cost = $400 + $150 = $550
For a weight of 4000 pounds:
Total cost = $400 + (15 cents/pound * 4000 pounds)
Total cost = $400 + $600 = $1000
Similar to the previous case, we observe that the total cost is continuous within the given range of weights (1000 to 4000 pounds). There are no abrupt changes or jumps in the cost. The function is continuous for 1000 ≤ x ≤ 4000.
Limits play a crucial role in analyzing the continuity of each function. By taking the limits as the variables approach certain values (in this case, the boundaries of the given ranges), we can determine if there are any discontinuities or changes in behavior. If the limits exist and match the function's value at that point, we can conclude that the function is continuous. In these examples, the limits at the boundary points match the function's values, indicating continuity.