Question

suppose you are headed toward a plateau 53 m high. if the angle of elevation to the top of the plateau is 60°, how far are you from the base of the plateau? you are \square m from the base of the plateau. (do not round until the final answer. then round to the nearest hundredth as needed.)

Explanation:

Step1: Identify the trigonometric relationship

We have a right triangle where the height of the plateau (opposite side to the angle of elevation) is 53 m, the angle of elevation is \(60^\circ\), and we need to find the adjacent side (distance from the base, let's call it \(x\)). We use the tangent function, which is \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\). So \(\tan(60^\circ)=\frac{53}{x}\).

Step2: Solve for \(x\)

We know that \(\tan(60^\circ)=\sqrt{3}\), so the equation becomes \(\sqrt{3}=\frac{53}{x}\). To solve for \(x\), we can rearrange it as \(x = \frac{53}{\sqrt{3}}\). Rationalizing the denominator, we multiply numerator and denominator by \(\sqrt{3}\), getting \(x=\frac{53\sqrt{3}}{3}\).

Step3: Calculate the numerical value

Now we calculate the value of \(\frac{53\sqrt{3}}{3}\). \(\sqrt{3}\approx1.73205\), so \(53\times1.73205 = 91.79865\), and then divide by 3: \(\frac{91.79865}{3}\approx30.59955\). Rounding to the nearest hundredth, we get approximately 30.60.

Answer:

\(30.60\)