# Finding the width of the gorge

• MHB
• daveyc3000
In summary: So the width of the gorge to the nearest metre is 55 m.In summary, the width of the gorge, given that Kristen is at a 65 degree angle of elevation and Greg is at a 35 degree angle of elevation, is 55 m.
daveyc3000
"Greg and Kristine are on opposite ends of a zip line that crosses a gorge. Greg went across the gorge first, and he's now on a ledge that's 15 m above the bottom of the gorge. Kristen is at the top of a cliff that is 72 m above the bottom of the gorge. Jon is on the ground at the bottom of the gorge, below the zip line. He sees Kristen at a 65 degree angle of elevation and Greg at a 35 degree angle of elevation,. What is the width of the gorge to the nearest metre?"

Re: need help solving this problem..ims tuck

As Dr, Peterson asked you on FMH: "What have you tried so far?" (Aside from posting the problem on just about any Math forum.)

-Dan

Re: need help solving this problem..ims tuck

... and if you're stuck doing that try making a diagram if you haven't done so already. :)

Re: need help solving this problem..ims tuck

Nothing but I have found the answer and now understand the problem

Thanks !

Re: need help solving this problem..ims tuck

daveyc3000 said:
Nothing but I have found the answer and now understand the problem

Thanks !

I've given this thread a useful title, and now, let's make the content useful to others by actually showing the work.

We are not told where along the bottom of the gorge Jon is, so let's let his distance from the taller side be $$x$$. All measures are in meters.

And then we may state:

$$\displaystyle \tan\left(65^{\circ}\right)=\frac{72}{x}$$

$$\displaystyle \tan\left(35^{\circ}\right)=\frac{15}{w-x}$$

The second equation implies:

$$\displaystyle w=15\cot\left(35^{\circ}\right)+x$$

The first equation implies:

$$\displaystyle x=72\cot\left(65^{\circ}\right)$$

Hence:

$$\displaystyle w=15\cot\left(35^{\circ}\right)+72\cot\left(65^{\circ}\right)\approx54.99637148829162$$

## 1. What is the width of the gorge?

The width of a gorge can vary greatly and is dependent on many factors such as the type of rock and the amount of water flow. It is best to measure the width at multiple points along the gorge to get an accurate average.

## 2. How do you measure the width of a gorge?

The most accurate way to measure the width of a gorge is to use a surveying tool such as a total station or a laser distance measurer. These tools use triangulation to calculate the distance between two points on either side of the gorge.

## 3. What is the significance of measuring the width of a gorge?

Measuring the width of a gorge can provide important information about the geology and hydrology of an area. It can also help with understanding the potential for erosion and identifying potential hazards for hikers and other visitors.

## 4. How does the width of a gorge affect the ecosystem?

The width of a gorge can greatly impact the ecosystem within and around it. A wider gorge may allow for more sunlight to reach the bottom, creating a different microclimate and potentially supporting different plant and animal species. It can also affect the flow of water and sediment, which can impact the entire ecosystem.

## 5. Can the width of a gorge change over time?

Yes, the width of a gorge can change over time due to natural processes such as erosion and sedimentation. Human activities, such as construction or deforestation, can also impact the width of a gorge. It is important to regularly monitor the width of a gorge to track any changes and their potential impacts.

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