Question

a blimp provides aerial television views of a soccer game. the television camera sights the stadium at a 14° angle of depression. the altitude of the blimp is 400m. what is the line - of - sight distance from the television camera to the base of the stadium? round to the nearest hundred meters.
the line - of - sight distance is approximately \boxed{5pi} m.

Explanation:

Step1: Identify the trigonometric relationship

We have a right triangle where the opposite side to the angle \(14^\circ\) is the altitude of the blimp (\(400\) m) and the hypotenuse is the line - of - sight distance (\(d\)). The sine function is defined as \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). So, \(\sin(14^\circ)=\frac{400}{d}\).

Step2: Solve for \(d\)

From \(\sin(14^\circ)=\frac{400}{d}\), we can re - arrange the formula to solve for \(d\). Cross - multiplying gives us \(d=\frac{400}{\sin(14^\circ)}\).

We know that \(\sin(14^\circ)\approx\sin(14)\approx0.2419\) (using a calculator to find the sine of \(14\) degrees).

Then \(d = \frac{400}{0.2419}\approx1653.6\) meters. Rounding to the nearest hundred meters, we look at the tens digit. The number is \(1653.6\), the tens digit is \(5\), so we round up the hundreds digit. \(1653.6\approx1700\) meters.

Answer:

The line - of - sight distance is approximately \(\boldsymbol{1700}\) meters.