we are a team of students studying for a test - asking again - thanks - The distance between Sue and a dude ranch was 30 mi. straight accross a rattlsnake-infested canyon. Instead of crossing the canyon, she hiked due north for a long time and then hiked due east for a shorter leg of the journey to the ranch. If she averaged 4 mph and it took her 9 hrs. to get to the ranch, then how far did she hike in a northernly direction? Round your answers to the nearest hundredth.
To determine the distance Sue hiked in a northernly direction, we can use the formula: Distance = Speed × Time.
We know that Sue averaged 4 mph and it took her 9 hours to reach the ranch. Therefore, the total distance she hiked would be:
Distance = 4 mph × 9 hours
Calculating the distance, we have:
Distance = 36 miles
However, this is the total distance Sue hiked, which includes both the northerly and easterly direction. To find out the distance she hiked in a northernly direction, we need to use the concept of a right-angled triangle.
As given in the problem, the total distance between Sue and the dude ranch is 30 miles, and she hiked due north (the longer leg of the journey) for a certain distance before hiking east (the shorter leg) to reach her destination.
Let's assume the distance Sue hiked in the northernly direction is x miles. Then, the distance she hiked in the easternly direction would be (30 - x) miles.
Using the Pythagorean theorem, we have:
x^2 + (30 - x)^2 = 36^2
Now, let's solve this equation to find the distance Sue hiked in the northernly direction:
x^2 + (30 - x)^2 = 1296
Expanding and simplifying the equation:
x^2 + 900 - 60x + x^2 = 1296
Combining like terms:
2x^2 - 60x + 900 - 1296 = 0
2x^2 - 60x - 396 = 0
Dividing the equation by 2 to simplify:
x^2 - 30x - 198 = 0
Now, we can solve this quadratic equation. Using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values:
x = (-(-30) ± √((-30)^2 - 4(1)(-198))) / (2(1))
Simplifying:
x = (30 ± √(900 + 792)) / 2
x = (30 ± √(1692)) / 2
x = (30 ± 41.15) / 2
Now, we can calculate the two possible values for x:
x1 = (30 + 41.15) / 2 = 71.15 / 2 = 35.58
x2 = (30 - 41.15) / 2 = -11.15 / 2 = -5.58
Since distance cannot be negative, we disregard x2 = -5.58.
Therefore, Sue hiked approximately 35.58 miles in a northernly direction, rounded to the nearest hundredth.