A stone fell from the top of a cliff into the ocean. In the air, it had an average speed of 16m//s. In the water, it had an average speed of 3m//s before hitting the seabed. The total distance from the top of the cliff to the seabed is 127 meters, and the stone's entire fall took 12 seconds. How long did the stone fall in the air and how long did it fall in the water?

Question

A stone fell from the top of a cliff into the ocean. In the air, it had an average speed of  16m//s. In the water, it had an average speed of  3m//s before hitting the seabed. The total distance from the top of the cliff to the seabed is 127 meters, and the stone's entire fall took 12 seconds. How long did the stone fall in the air and how long did it fall in the water?

Bytelearn · Accepted Answer

Set up equations: Set up the equations based on the given information. Let t_air be the time the stone falls in the air and t_water be the time it falls in the water. We know that the total time for the fall is 12 seconds, so we can write the following equation: t_air + t_water = 12Express total distance: Express the total distance fallen in terms of the distances in air and water. Let d_air be the distance the stone falls in the air and d_water be the distance it falls in the water. The total distance is given as 127 meters, so we can write the following equation: d_air + d_water = 127Use average speeds: Use the average speeds to relate time and distance for each part of the fall. The average speed in the air is 16 m/s, and the average speed in the water is 3 m/s. We can write two equations based on the definition of average speed (distance = speed × time): d_air = 16 * t_air d_water = 3 * t_waterSubstitute expressions: Substitute the expressions for d_air and d_water into the total distance equation. Using the equations from Step 3, we substitute the expressions for d_air and d_water into the total distance equation: 16 * t_air + 3 * t_water = 127Solve system of equations: Solve the system of equations to find t_air and t_water. We now have a system of two equations with two unknowns: 1) t_air + t_water = 12 2) 16 * t_air + 3 * t_water = 127 We can solve this system using substitution or elimination. Let's use substitution by expressing t_water in terms of t_air from the first equation: t_water = 12 - t_airFind t_air: Substitute t_water into the second equation and solve for t_air. Substituting t_water into the second equation gives us: 16 * t_air + 3 * (12 - t_air) = 127 Expanding this, we get: 16 * t_air + 36 - 3 * t_air = 127 Combining like terms, we have: 13 * t_air = 127 - 36 13 * t_air = 91 Dividing both sides by 13 gives us: t_air = 91 / 13 t_air = 7Find t_water: Use the value of t_air to find t_water. Now that we have t_air, we can find t_water using the equation from Step 5: t_water = 12 - t_air t_water = 12 - 7 t_water = 5

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