Question

galileo wanted to release a wooden ball and an iron ball from a height of 150 meters and measure the duration of their fall. he found a plane with an incline of 15° that he could climb until he gets to an altitude of 150 m. how far should galileo walk up the inclined plane? round your final answer to the nearest hundredth. meters

Explanation:

Step1: Use the sine function

In a right - triangle formed by the height (opposite side, \(h = 150\) m), the distance walked up the inclined plane (hypotenuse, \(d\)), and the horizontal distance. The sine of an angle \(\theta\) in a right - triangle is given by \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, \(\theta = 15^{\circ}\) and the opposite side \(h = 150\) m. So, \(\sin(15^{\circ})=\frac{150}{d}\).

Step2: Solve for \(d\)

We can rewrite the formula as \(d=\frac{150}{\sin(15^{\circ})}\). We know that \(\sin(15^{\circ})=\sin(45^{\circ}- 30^{\circ})=\sin45^{\circ}\cos30^{\circ}-\cos45^{\circ}\sin30^{\circ}=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}-\frac{\sqrt{2}}{2}\times\frac{1}{2}=\frac{\sqrt{6}-\sqrt{2}}{4}\approx0.2588\). Then \(d=\frac{150}{0.2588}\approx580.00\)

Answer:

\(580.00\)