A train hauling coal leaves the train yard traveling west at 32 mph. Five hours later a train carrying appliances leaves the train yard traveling north at 40 mph. 1.Construct a diagram of the situation and write a function to express the distance between the two trains x hours after the train carrying appliances has departed. 2. How long after the train carrying appliances departs will the distance between the two trains be 260 miles. Having a hard time with placing the values and x and the whole concept.
the western and northern tracks form a right angle
the distance between the trains is the hypotenuse of the triangle
western distance ... 32 (x + 5)
northern distance ... 40 x
Thanks!
1. To construct a diagram of the situation, we can use a coordinate system. Let's use the x-axis to represent the west-east direction (with positive values to the right) and the y-axis to represent the north-south direction (with positive values upwards). The train hauling coal starts at the origin (0,0) and travels west at a speed of 32 mph. The train carrying appliances starts 5 hours later at (0,0) and travels north at a speed of 40 mph.
To express the distance between the two trains x hours after the train carrying appliances has departed, we need to find the distance between the two trains at that time. Let's call this distance D(x).
For the train hauling coal, its westward distance can be expressed as -32x (negative sign because it's traveling west).
For the train carrying appliances, its northward distance can be expressed as 40(x - 5) (subtracting 5 hours from x because it starts 5 hours later).
The distance D(x) between the two trains can be calculated using the Pythagorean theorem, as the square root of the sum of the squares of the distances traveled in each direction:
D(x) = sqrt((-32x)^2 + (40(x - 5))^2)
2. To find out how long after the train carrying appliances departs will the distance between the two trains be 260 miles, we need to solve the equation D(x) = 260.
Let's set up the equation:
sqrt((-32x)^2 + (40(x - 5))^2) = 260
We can then square both sides of the equation and simplify:
(-32x)^2 + (40(x - 5))^2 = 260^2
Solve this equation to find the value of x.
To solve this problem, let's start by constructing a diagram:
```
Train carrying appliances (north)
|
|
|
+-------+-------+-------+
| |
| |
| |
Train hauling coal (west) x
```
In this diagram, the train carrying appliances is traveling north, and the train hauling coal is traveling west. The distance between the two trains will be represented by the variable 'x'.
Now, let's write a function to express the distance between the two trains x hours after the train carrying appliances departs.
```
Distance (x) = Distance traveled by the train carrying appliances + Distance traveled by the train hauling coal
```
Distance traveled by the train carrying appliances can be calculated using the formula: Distance = Speed x Time
Since the train carrying appliances is traveling at 40 mph, its distance traveled after x hours will be: 40x
Distance traveled by the train hauling coal can also be calculated using the formula: Distance = Speed x Time.
However, the train hauling coal departs 5 hours later than the train carrying appliances, so we need to consider this delay. The time traveled by the train hauling coal is (x - 5) hours, and its speed is 32 mph. Therefore, the distance traveled by the train hauling coal after x hours will be: 32(x - 5).
Combining these two distances, the function expressing the distance between the two trains x hours after the train carrying appliances departs is:
```
Distance (x) = 40x + 32(x - 5)
```
Now, let's move on to the second part of the question: How long after the train carrying appliances departs will the distance between the two trains be 260 miles?
To find the solution, we can set the Distance (x) function equal to 260 and solve for x:
```
40x + 32(x - 5) = 260
```
Simplifying the equation:
```
40x + 32x - 160 = 260
72x - 160 = 260
72x = 420
x = 420 / 72
x ≈ 5.83
```
Therefore, the distance between the two trains will be 260 miles approximately 5.83 hours after the train carrying appliances departs.