Question

a train track through a mountain is 1500 feet long and makes an angle of 1.7° with the horizontal. what is the change in elevation from one end of the tunnel to the other?
the change in elevation from one end of the tunnel to the other is (square) ft.
(round to one - decimal place as needed.)

Explanation:

Step1: Identify the trigonometric relationship

We can use the sine function. In a right - triangle formed by the tunnel (hypotenuse), the horizontal, and the elevation change, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. Here, the length of the tunnel is the hypotenuse $c = 1500$ feet and the angle with the horizontal is $\theta=1.7^{\circ}$, and the change in elevation is the side opposite the given angle.

Step2: Calculate the change in elevation

We know that $\sin\theta=\frac{h}{c}$, where $h$ is the change in elevation. Rearranging for $h$, we get $h = c\times\sin\theta$. Substituting $c = 1500$ and $\theta = 1.7^{\circ}$ (and using $\sin(1.7^{\circ})\approx0.02967$), we have $h=1500\times\sin(1.7^{\circ})$.
$h = 1500\times0.02967 = 44.505$.

Answer:

$44.5$