Landon is standing in a hole that is 6.5 ft deep. He throws a rock, and it goes up into the air, out of the hole,
and then lands on the ground above. The path of the rock can be modeled by the equation y = 0.005x +
0.42x 6.5, where x is the horizontal distance of the rock, in feet, from Landon and y is the height, in feet, of
the rock above the ground. How far horizontally from Landon will the rock land?
a. 63.54 ft c. 10.23 ft
b. 20.46 ft d. 31.77 ft
To find where the rock will land, we need to find the point where y = 0 (i.e. the height of the rock is 0, indicating it has landed on the ground).
0 = 0.005x + 0.42x - 6.5
Combining like terms:
0.425x - 6.5 = 0
Adding 6.5 to both sides:
0.425x = 6.5
Dividing by 0.425:
x = 15.29
Therefore, the rock will land 15.29 feet horizontally from Landon.
The closest answer choice is (c) 10.23 ft, which is not correct.
To find the horizontal distance that the rock will land from Landon, we need to determine the value of x when y is equal to 0 (since the rock will land on the ground).
Given the equation y = 0.005x + 0.42x + 6.5, let's set y = 0:
0 = 0.005x + 0.42x + 6.5
Combining like terms, we have:
0.425x + 6.5 = 0
Subtracting 6.5 from both sides:
0.425x = -6.5
Now, divide both sides by 0.425:
x = -6.5 / 0.425
Simplifying this value gives approximately x = -15.294, but distance cannot be negative. Therefore, we can exclude this solution.
Let's consider the other possibility which is that the rock will land after it has traveled through the hole. In this case, the equation would be:
y = 0.005x + 0.42x - 6.5
Setting y = 0:
0 = 0.005x + 0.42x - 6.5
Combining the x terms:
0.425x - 6.5 = 0
Adding 6.5 to both sides:
0.425x = 6.5
Dividing both sides by 0.425:
x = 6.5 / 0.425
Simplifying this value, we find that x is approximately equal to 15.294.
Since the rock will land after traveling through the hole, the correct answer is 15.294 ft. Therefore, the closest option is:
b. 20.46 ft