The town of Mathville is triangular in shape and has a perimeter of 104 km. The interstate borders the southern part of the town, but town residents want to build two more roads, one on each side of the town. The length of one side of the town is ½ the length of the interstate. The length of the other side is 2/3 the length of the interstate. Determine the length of road necessary to complete the town’s task. That is, determine the length of each of the two sides of the town not bordered by the interstate.
Let the length of the interstate be x. Then, one side of the town is 1/2 x and the other side is 2/3 x.
We know that the perimeter of the town is 104 km, so we can write an equation:
x + 1/2 x + 2/3 x = 104
To solve for x, we can simplify the equation:
6/6 x + 3/6 x + 4/6 x = 104
13/6 x = 104
x = 48
Now we know that the length of the interstate is 48 km.
To find the length of the other sides of the town, we can use the ratios given to us:
One side is 1/2 x, so it is 1/2 (48) = 24 km.
The other side is 2/3 x, so it is 2/3 (48) = 32 km.
To find the total length of the two new roads, we simply add the lengths of the two sides:
24 km + 32 km = 56 km
Therefore, Mathville needs 56 km of new road to complete their task.
Let's start by assigning variables to the unknown lengths. We'll let "x" represent the length of the interstate.
According to the given information, one side of the town is ½ the length of the interstate. So, the length of this side would be (1/2)*x = x/2.
Another side of the town is 2/3 the length of the interstate. So, the length of this side would be (2/3)*x = 2x/3.
The perimeter of the town is the sum of all three sides. We are given that the perimeter is 104 km. Therefore, we can set up the following equation:
x + x/2 + 2x/3 = 104
To solve for x, we can first simplify the equation:
(6x + 3x + 4x)/6 = 104
13x/6 = 104
Multiplying both sides of the equation by 6/13:
x = (6/13)*104
x ≈ 48 km
Now that we have the length of the interstate (x), we can calculate the lengths of the other two sides:
One side is x/2 = 48/2 = 24 km.
The other side is 2x/3 = 2*(48/3) = 32 km.
Therefore, the lengths of the two sides of the town not bordered by the interstate are 24 km and 32 km.