A mountain road makes an angle θ = 8.4° with the horizontal direction. If the road has a total length of 3.4 km, how much does it climb? That is, find h.
tan 8.4 = h/3.4
h =3.4 tan 8.4° = .......
Well, we've got ourselves a mountain road that's feeling a bit inclined, don't we? That's quite a steep little angle there at 8.4°!
To determine how much our mountain road climbs, we'll need to channel our inner mathematician. Let's use a bit of trigonometry magic, shall we?
Since we know the length of the road (3.4 km) and the angle (8.4°), we can use a little trigonometry to find the height (h).
We'll use the formula: h = length × sin(angle)
Substituting in the given values: h = 3.4 km × sin(8.4°)
Now, let me do the calculations for you:
h ≈ 0.485 km
So, our mountain road climbs approximately 0.485 kilometers. That's quite a decent ascent for a road, isn't it? I hope it enjoys the view up there!
To find the height h that the road climbs, we need to use the trigonometric relationship between the angle θ and the height.
We can consider the road as the hypotenuse of a right-angled triangle, with the horizontal direction as the base and the height h as the perpendicular side.
Using the trigonometric function tangent (tan), we have the equation:
tan(θ) = h / L
where θ is the angle, h is the height, and L is the length of the road.
In this case, θ = 8.4° and L = 3.4 km. However, we need to convert the length from kilometers to meters, since the trigonometric function operates on SI units (meters).
So, L = 3.4 km * 1000 m/km = 3400 m.
Rearranging the equation, we have:
h = L * tan(θ)
Substituting the values, we get:
h = 3400 m * tan(8.4°) ≈ 514.8 m
Therefore, the road climbs approximately 514.8 meters.
To find the height climbed by the mountain road, we can use trigonometry.
First, let's identify the given information:
- The angle between the mountain road and the horizontal direction, θ = 8.4°.
- The total length of the road, L = 3.4 km.
Now, let's find the height climbed (h).
We can use the trigonometric relationship between the angle θ, the opposite side (h), and the adjacent side (L) in a right triangle:
tan(θ) = h / L
Rearranging the equation to solve for h, we have:
h = L * tan(θ)
Substituting the given values:
h = 3.4 km * tan(8.4°)
Now, let's calculate h:
h ≈ 3.4 km * tan(8.4°)
≈ 3.4 km * 0.147
≈ 0.4998 km
Therefore, the road climbs approximately 0.4998 km or 499.8 meters.