A plane is descending at an angle of $1^{\circ}$ from the horizontal. The plane's current altitude is $8.1 \mathrm{~km}$. See the figure below. (The figure is not drawn to scale.) Find the plane's new altitude, $h$, after 0.20 hours if it travels at a constant speed of $500 \mathrm{~km} / \mathrm{hr}$ along the direction of its flight. Carry your intermediate computations to at least four decimal places, and round your answer to the nearest tenth of a kilometer.
\[
h=\square \text { kilometers }
\]

Step 1 ：Given that the plane is descending at an angle of \(1^{\circ}\) from the horizontal, and the plane's current altitude is \(8.1 \mathrm{~km}\). The plane travels at a constant speed of \(500 \mathrm{~km} / \mathrm{hr}\) along the direction of its flight for 0.20 hours.

Step 2 ：First, we need to find the distance the plane travels in 0.20 hours at a speed of \(500 \mathrm{~km} / \mathrm{hr}\). The distance is calculated as \(distance = speed \times time\), which gives \(distance = 500 \times 0.2 = 100.0 \mathrm{~km}\).

Step 3 ：Next, we use the tangent function to find the change in altitude. The tangent of an angle in a right triangle is equal to the ratio of the opposite side to the adjacent side. In this case, the angle is \(1^{\circ}\), the opposite side is the change in altitude (which we're trying to find), and the adjacent side is the horizontal distance the plane travels. The change in altitude is calculated as \(delta\_altitude = tan(angle) \times distance\), which gives \(delta\_altitude = tan(1^{\circ}) \times 100.0 = 1.7455064928217585 \mathrm{~km}\).

Step 4 ：Finally, we subtract this change in altitude from the plane's current altitude to find its new altitude. The new altitude is calculated as \(new\_altitude = current\_altitude - delta\_altitude\), which gives \(new\_altitude = 8.1 - 1.7455064928217585 = 6.354493507178241 \mathrm{~km}\).

Step 5 ：Rounding to the nearest tenth of a kilometer, the plane's new altitude after 0.20 hours is approximately \(\boxed{6.4 \mathrm{~km}}\).