Question

galileo wanted to release a wooden ball and an iron ball from a height of 150 meters to see how long it took them to fall to the ground. he found a ramp with an incline of (15^{\circ}) that he could climb until he gets to a height of 150 meters.

write the equation to solve for (x).

how far should galileo walk up the ramp? round your final answer to the nearest tenth.

galileo should walk meters up the ramp.

Explanation:

⚡ Using what you learned: right triangle trigonometry (soh cah toa)

Step 1: Identify the trigonometric ratio

Using the right triangle formed by the ramp:

• The angle of inclination is \( 15^\circ \).
• The side opposite to the angle is the height, which is \( 150 \) meters.
• The hypotenuse is the ramp length, represented by \( x \).

The sine function relates the opposite side to the hypotenuse:

$$ \sin(15^\circ) = \frac{150}{x} $$

Step 2: Solve for \( x \)

Rearrange the equation to isolate \( x \):

$$ x = \frac{150}{\sin(15^\circ)} $$

Step 3: Calculate the numerical value

Using a calculator to find \( \sin(15^\circ) \approx 0.2588 \):

$$ x \approx \frac{150}{0.2588} \approx 579.555 $$

Rounding to the nearest tenth gives:

$$ x \approx 579.6 $$

Answer:

Equation to solve for \( x \):

$$ \sin(15^\circ) = \frac{150}{x} $$

Distance Galileo should walk:
Galileo should walk 579.6 meters up the ramp.