A camper wants to know the width of a river. From point A, he walks downstream 100 feet to point B and sights a canoe across the river. It is determined that = 42°. About how wide is the river?
Let C be the point on the other side of the river so that AC is perpendicular to AB, so we want AC.
You have a right-angled triangle, but you don't state which angle is 42°
I will assume it is at B
so tan 42° = AC/100
AC = 100tan42° = ....
To find the width of the river, we can use trigonometry. Let's call the width of the river "w". We have the following information:
- The camper walks downstream from point A to point B, which is 100 feet.
- The angle between the line of sight of the canoe and the line connecting points A and B is 42°.
To solve for the width of the river, we can use the tangent ratio:
tan(angle) = opposite / adjacent
In this case, the opposite side is the width of the river (w), and the adjacent side is the distance traveled downstream (100 feet). Plugging in the values, we can solve for the width:
tan(42°) = w / 100
Rearranging the equation to solve for w:
w = tan(42°) * 100
Using a scientific calculator, we can find the value of tangent(42°) to be approximately 0.9004. Plugging this value into the equation:
w = 0.9004 * 100
w ≈ 90.04 feet
Therefore, the width of the river is approximately 90.04 feet.
To find the width of the river, we can use the trigonometric function tangent (tan).
The tangent of an angle is defined as the opposite side divided by the adjacent side. In this case, the opposite side is the width of the river and the adjacent side is the distance the camper walked downstream.
So, we have the equation: tan(θ) = opposite/adjacent
We know that θ = 42° and the adjacent side is 100 feet. We need to solve for the opposite side, which is the width of the river.
Rearranging the equation: opposite = tan(θ) * adjacent
Substituting the values: opposite = tan(42°) * 100 feet
Calculating the value of tan(42°), we find: tan(42°) ≈ 0.9004
Now we can plug in this value: opposite ≈ 0.9004 * 100 feet
Simplifying, we find that the approximate width of the river is: opposite ≈ 90.04 feet
Therefore, the width of the river is approximately 90.04 feet.