Construction Costs The town of Foxton lies 10 mi north of an abandoned east-west road that runs
through Grimley, as shown in the figure. The point on the abandoned road closest to Foxton is 40
mi from Grimley. County officials are about to build a new road connecting the two towns. They
have determined that restoring the old road would cost $100,000 per mile, whereas building a
new road would cost $200,000 per mile. How much of the abandoned road should be used (as
indicated in the figure) if the officials intend to spend exactly $6.8 million? Would it cost less
than this amount to build a new road connecting the towns directly?
No figure. Cannot copy and paste here.
the length of a direct road is √(40^2 + 10^2) = √1700 = 41.23, and its cost is thus about $.2M*41.23 = $8.246M, more than the budget allows
If x miles of the old road are used, the distance is
x + √((40-x)^2 + 10^2)
so to spend the budgeted amount, you want (suppressing all the extra 5 zeros)
x + 2√((40-x)^2 + 10^2) = 68
x = 16 mi
To determine how much of the abandoned road should be used, we need to find the distance from Grimley to the point on the abandoned road closest to Foxton.
Let's call this point P.
From the given information:
- Grimley lies on the abandoned road.
- Foxton is located 10 miles north of the abandoned road.
Since point P is closest to Foxton on the abandoned road, we can draw a right-angled triangle with the hypotenuse being the shortest distance from P to Foxton (10 miles) and the other leg being 40 miles (distance from P to Grimley).
Using the Pythagorean theorem, we can find the distance between P and Foxton:
c² = a² + b²
10² = 40² + b²
100 = 1600 + b²
b² = 100 - 1600
b² = -1500
Since we have a negative value for b², it means that point P does not exist on the abandoned road. Therefore, the entire abandoned road cannot be used to build a road connecting the towns.
Now let's determine the cost of building a new road directly from Grimley to Foxton.
The distance between Grimley and Foxton is the hypotenuse of the right-angled triangle we just considered, which is:
c = sqrt(a² + b²)
c = sqrt(40² + 10²)
c = sqrt(1600 + 100)
c = sqrt(1700)
c ≈ 41.23 miles
The cost of building a new road would be $200,000 per mile. Thus, the cost of building a new road connecting the towns directly would be:
Cost = Distance * Cost per mile
Cost = 41.23 * $200,000
Cost ≈ $8,246,000
Therefore, it would cost more than $6.8 million to build a new road connecting the towns directly, considering the information given.
To find out how much of the abandoned road should be used, we need to determine the distance between Foxton and Grimley.
We are given that the point on the abandoned road closest to Foxton is 40 miles from Grimley. Since the road is abandoned, we can assume that it runs directly east-west. Therefore, the distance between Foxton and Grimley is the hypotenuse of a right triangle with one leg measuring 10 miles (north-south distance) and the other leg measuring 40 miles (east-west distance). We can use the Pythagorean Theorem to find the distance between Foxton and Grimley.
The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Let's call the distance between Foxton and Grimley "d".
Applying the Pythagorean Theorem:
d^2 = 10^2 + 40^2
d^2 = 100 + 1600
d^2 = 1700
d ≈ 41.23 miles
So, the distance between Foxton and Grimley is approximately 41.23 miles.
Now, let's determine how much of the abandoned road should be used to connect the two towns.
The total length of the abandoned road is 40 miles. Since the road runs directly east-west, the section of the road to be used can be represented by a horizontal line segment. Let's call this distance "x".
Since the distance between Foxton and Grimley is approximately 41.23 miles, we can express x in terms of the distance between Foxton and Grimley as follows:
x = 41.23 - 10
x ≈ 31.23 miles
Therefore, approximately 31.23 miles of the abandoned road should be used to connect the two towns.
To determine if it would cost less than $6.8 million to build a new road directly connecting the towns, we need to compare the cost of using the abandoned road versus building a new road.
Using the abandoned road for approximately 31.23 miles would cost $100,000 per mile, so the cost of using the abandoned road would be:
Cost of using the abandoned road = $100,000 * 31.23 miles ≈ $3,123,000
Building a new road would cost $200,000 per mile, so the cost of building a new road directly connecting the towns would be:
Cost of building a new road = $200,000 * 41.23 miles ≈ $8,246,000
Therefore, it would cost less than $6.8 million to use the abandoned road, as the cost is approximately $3,123,000.