Final answer:
To solve this problem, we need to use the equation of motion for an object in free fall. The height of the Grand Canyon is 9,148 feet. By plugging the values into the quadratic formula, we find that it takes approximately 24.4 seconds for the rock to hit the ground.
Step-by-step explanation:
To solve this problem, we need to use the equation of motion for an object in free fall:
d = vit + (1/2)gt^2
where d is the distance or height, vi is the initial velocity, g is the acceleration due to gravity, and t is the time.
In this case, the height of the Grand Canyon is 8,800 feet and the hiker is starting 348 feet below that. So, the total height is 8,800 + 348 = 9,148 feet.
Using the equation, we can plug in the values as follows:
9,148 = -16t^2 + 18t
This is a quadratic equation that we can solve using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
Comparing to the equation at^2 + bt + c = 0, we have a = -16, b = 18, and c = -9,148.
Plugging in these values into the quadratic formula, we get:
t = (-18 ± √((18)^2 - 4(-16)(-9,148))) / (2(-16))
Simplifying the equation further, we get:
t = (-18 ± √(324 + 580,768)) / (-32)
t = (-18 ± √581,092) / -32
t = (-18 ± 762.127) / -32
Solving for t, we get two possible solutions:
t ≈ (-18 + 762.127) / -32 ≈ -24.461
t ≈ (-18 - 762.127) / -32 ≈ 24.409
Since time cannot be negative, the rock takes approximately 24.409 seconds to hit the ground. Rounding to the nearest tenth of a second, the answer is approximately 24.4 seconds.