You work for the silver there is a home with a driveway 2 miles long extending to the house from a nearby highway. the nearest connection box is along the highway but 5 miles from the driveway. it costs the company $100 per mile to install cable along the highway and $140 per mile to install cable off the highway.
a) make a table of the possible integral values of x and the corresponding cost in each instance. does one choice appear to cost the least?
b) if you charge the owners of the house $800 for installation, would you be willing to let them choose which way the cable would go? explain
c) using a graphing calculator, graph the function and determine the value of X that would make the installation cost minimum?
d) check the local regulations for cable companies and find that there is pending state legislation that says that the cable that the cable cannot turn off the highway more than 0.5 mile from the house's driveway. if this legislation passes, what will be the ultimate cost of installing the house's cable?
e) if the cable companhy wishes to install cable in 5000 homes in this area, and assuming that the figures for the original house's installation are typical, how much will the new legislation cost the company overall if they cannot use the cheapest installation cost, but instead have to follow the new state regulation?
a) To make a table of the possible integral values of x and the corresponding cost in each instance, we have to consider the two installation options - along the highway and off the highway.
Let x represent the distance from the connection box to the driveway, measured in miles.
The cost of installing cable along the highway is $100 per mile, while the cost of installing cable off the highway is $140 per mile.
| x (miles) | Cost along highway | Cost off highway |
|-----------|--------------------|-----------------|
| 0 | $0 | $140 |
| 1 | $100 | $140 |
| 2 | $200 | $140 |
| 3 | $300 | $140 |
| 4 | $400 | $140 |
| 5 | $500 | $700 |
From the table, we can see that when the distance from the connection box to the driveway is 5 miles, it becomes more cost-effective to install the cable off the highway since it would only cost $700 compared to $500 if installed along the highway.
b) If the owners of the house are willing to pay $800 for installation, it would be feasible to let them choose which way the cable would go. In this case, they can choose the more expensive option of installing the cable off the highway.
c) To graph the situation and determine the value of x that would minimize the installation cost, we can create a graph of the cost versus the distance.
The installation cost along the highway can be expressed as: Cost_along = 100x
The installation cost off the highway can be expressed as: Cost_off = 140(5 - x)
By summing these costs together, we get the total cost: Cost_total = Cost_along + Cost_off = 100x + 140(5 - x)
Using a graphing calculator, graph this function and find the x-value that corresponds to the minimum point on the graph. This x-value represents the distance that minimizes the installation cost.
d) If the legislation states that the cable cannot turn off the highway more than 0.5 miles from the house's driveway, the ultimate cost of installing the house's cable would differ.
For instance, if the driveway is 2 miles away from the nearest connection box along the highway, the cable would have to turn off 1.5 miles earlier to comply with the regulation. As a result, the cost for installing cable off the highway would increase.
Using the same calculation as in part a) with x = 1.5, the cost off the highway would be:
Cost_off = 140(5 - 1.5) = $490
Therefore, the overall cost of installing the house's cable would increase compared to the initial scenario.
e) If the cable company wishes to install cable in 5000 homes in the area and they have to follow the new state regulation, it would impact the overall cost.
Assuming the figures for the original house's installation are typical and the new regulation is applied to all 5000 homes, the cost difference can be calculated as follows:
Cost_difference = (Cost_off with regulation - Cost_off without regulation) x Number of homes
Using the previous cost off the highway with regulation ($490) and without regulation ($140) and assuming all 5000 homes have the same installation scenario, the cost difference would be:
Cost_difference = ($490 - $140) x 5000 = $1750000
Therefore, the new legislation would cost the company an additional $1,750,000 for installing cable in 5000 homes in that area.