Question

an airplane climbs at an angle of 12° with the ground. find the altitude it has reached once it has traveled 5000 feet horizontally. round answer to the nearest tenth.
1039.6 feet
4890.7 feet
106.3 feet
1062.8 feet

Explanation:

Step1: Identify trigonometric ratio

We have a right triangle where the horizontal side (adjacent) is 5000 feet, the angle with the ground is \(12^\circ\), and we need to find the altitude (opposite side). The tangent function relates opposite and adjacent sides: \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\).

Step2: Set up the equation

Let \(h\) be the altitude. Then \(\tan(12^\circ)=\frac{h}{5000}\).

Step3: Solve for \(h\)

Multiply both sides by 5000: \(h = 5000\times\tan(12^\circ)\). Calculate \(\tan(12^\circ)\approx0.2126\), so \(h = 5000\times0.2126 = 1063\) (approx). Rounding to the nearest tenth, \(h\approx1062.8\) (wait, let's recalculate with more precise \(\tan(12^\circ)\)). Using a calculator, \(\tan(12^\circ)\approx0.2125565617\). Then \(h = 5000\times0.2125565617 = 1062.7828085\), which rounds to 1062.8. Wait, but let's check again. Wait, maybe I made a mistake. Wait, no, let's see the options. Wait, 1062.8 is an option. Wait, but let's re-express:

Wait, \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\), so opposite = adjacent \(\times\tan(\theta)\). So adjacent is 5000, \(\theta = 12^\circ\), so \(h = 5000\times\tan(12^\circ)\). Let's compute \(\tan(12^\circ)\):

Using calculator: \(\tan(12^\circ) \approx 0.2125565617\)

Then \(5000 \times 0.2125565617 = 1062.7828085\), which rounds to 1062.8.

Answer:

1062.8 feet (corresponding to the option with 1062.8 feet)