Because of the increase in traffic between Springfield and Orangeville, a new road was built to connect the two towns. The old road goes south x miles from Springfield to Freeport and then goes east x + 5 miles from Freeport to Orangeville. The new road is 9 miles long and goes straight from Springfield to Orangeville. Find the number of miles that a person drove using the old road. Please help ASAP!!!! :(
To find the number of miles a person drove using the old road, we need to determine the values of "x" and calculate the total distance.
Let's break down the information given.
1. The old road goes south x miles from Springfield to Freeport.
2. From Freeport, the old road goes east x + 5 miles to Orangeville.
3. The new road is 9 miles long and goes straight from Springfield to Orangeville.
Since the new road is a straight line from Springfield to Orangeville, we can use the Pythagorean Theorem to relate the distances involved.
According to the theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides in a right triangle.
In this scenario, the new road is the hypotenuse, connecting the southward and eastward sections of the old road.
Using the Pythagorean Theorem, we can set up the equation as follows:
x^2 + (x+5)^2 = 9^2
Expanding and simplifying the equation:
x^2 + x^2 + 10x + 25 = 81
2x^2 + 10x + 25 = 81
2x^2 + 10x - 56 = 0
Now, we can solve this quadratic equation to find the values of "x." There are a few ways to solve it, such as factoring or using the quadratic formula.
If we factor out a common 2 from the equation:
2(x^2 + 5x - 28) = 0
Next, we can factor the quadratic expression within the parentheses:
2(x + 7)(x - 4) = 0
Setting each factor equal to zero and solving for x, we have:
x + 7 = 0 -> x = -7
x - 4 = 0 -> x = 4
Since distance cannot be negative, we discard the negative value of x.
Therefore, x = 4.
Now that we have the value of x, we can calculate the total distance using the old road.
Distance = southward distance + eastward distance
Distance = x + (x + 5)
Distance = 4 + (4 + 5)
Distance = 4 + 9
Distance = 13 miles
Therefore, a person drove 13 miles using the old road.
The old road is 2x+5 miles long, where
x^2 + (x+5)^2 = 9^2