Question

The elevation of the Fraser River at Hope is 41 meters. From there it flows approximately 147 kilometers to the sea. What is the average gradient of the river (meters per kilometre) over that distance?

Short answer

The average gradient of the Fraser River from Hope to the sea is approximately \(0.2786\) meters per kilometer.

Step-by-step solution

Understand the formula for gradient

The gradient represents the change in elevation over a specific distance, and it's calculated by dividing the change in elevation by the distance. The formula is: Gradient = \(\frac{\text{Change in Elevation}}{\text{Distance}}\)

Identify the given values in the problem

In this exercise, we are given: - The elevation at Hope: 41 meters - The distance from Hope to the sea: 147 kilometers

Calculate the average gradient of the river

Now, we can calculate the average gradient by substituting the given values into the gradient formula: Gradient = \(\frac{\text{Change in Elevation}}{\text{Distance}}\) = \(\frac{41\text{ meters}}{147\text{ kilometers}}\) It's important to note that the elevation is given in meters, and the distance is given in kilometers. As the problem asks for the gradient in meters per kilometer, we don't need to convert the units.

Perform the calculation

Now that the given values are inserted into the formula, we proceed to the calculation: Gradient = \(\frac{41\text{ meters}}{147\text{ kilometers}}\) ≈ \(0.2786\) So the average gradient of the Fraser River from Hope to the sea is approximately \(0.2786\) meters per kilometer.

Key concepts

Elevation Change

To understand the river gradient, the first thing we need to know is how much the river's elevation changes as it travels from one point to another. In this exercise, the Fraser River starts at an elevation of 41 meters in Hope and flows down to sea level, which is 0 meters.

The change in elevation is crucial because it is the difference between the starting and ending points of the river. Essentially, this is calculated by subtracting the ending elevation from the starting elevation.

• Starting elevation (Hope): 41 meters
• Ending elevation (sea level): 0 meters

The change in elevation is simply 41 meters - 0 meters = 41 meters. This figure tells us that the river drops a total of 41 meters as it travels from Hope to the sea. Understanding this concept lays the groundwork for calculating the river gradient.

Distance Calculation

Apart from knowing the elevation change, understanding the distance the river travels is equally important. In this instance, the distance from Hope to the sea is 147 kilometers. This distance will help us determine how rapidly or gradually the elevation changes over this linear path.

When measuring distance, make sure to use consistent units. Here, the distance is already given in kilometers, which matches the desired unit of measurement for the gradient calculation (meters per kilometer).

• Distance from Hope to sea: 147 kilometers

Knowing the distance is essential because it directly affects the gradient value we calculate. It represents how spread out the elevation change is, and influences whether the gradient is steep or gentle.

Gradient Formula

The gradient of a river is a measure of how steeply or gently the riverbed declines along a certain distance. It is calculated using a straightforward formula:

Gradient = \(\frac{\text{Change in Elevation}}{\text{Distance}}\).
This formula essentially reflects the elevation change per unit of distance. In simpler terms, it tells us how many meters the river drops for every kilometer it travels. By applying this formula, we can determine how steep or flat a given section of the river is.

• Change in Elevation: 41 meters
• Distance: 147 kilometers

Plugging these values into our formula, we get \(\frac{41 \text{ meters}}{147 \text{ kilometers}}\), resulting in a gradient of about \(0.2786\) meters per kilometer.

Average Gradient Calculation

After understanding the components, calculating the average gradient is quite simple. We use the gradient formula and input our known values to find the river's average slope.

By substituting the change in elevation and the distance into the formula, we get: Gradient = \(\frac{41 \text{ meters}}{147 \text{ kilometers}}\)
Simplifying this fraction gives us approximately \(0.2786\) meters per kilometer.

This number indicates the average gradient of the river from Hope to the sea. It tells us that for every kilometer the river travels, it drops about \(0.2786\) meters. Hence, this slope shows a gentle decline, which is typical of rivers as they approach their outflow points.