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 = -0.005x² +
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 the horizontal distance the rock will travel, we need to find the value of x when y is equal to zero (since the rock will be at ground level when it lands). So we set y equal to zero and solve for x:
0 = -0.005x² + 0.42x - 6.5
We can use the quadratic formula to solve for x:
x = [-0.42 ± sqrt((0.42)² - 4(-0.005)(-6.5))]/(2(-0.005))
x = [-0.42 ± sqrt(0.7396)]/(-0.01)
x = [-0.42 ± 0.86]/(-0.01)
We can discard the negative solution (since distance cannot be negative) and simplify:
x = (-0.42 + 0.86)/(-0.01)
x = 44
Therefore, the rock will land 44 feet horizontally from Landon. The closest answer choice is 20.46 ft (b), but this is incorrect. The correct answer is (none of the above) 44 ft.
To find the horizontal distance from Landon where the rock will land, we need to find the x-coordinate when the height of the rock, y, is equal to zero.
Given the equation y = -0.005x² + 0.42x - 6.5, we can set y equal to zero and solve for x:
0 = -0.005x² + 0.42x - 6.5
To solve this quadratic equation, we can use the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a)
In this case, a = -0.005, b = 0.42, and c = -6.5. Plugging these values into the quadratic formula:
x = (-(0.42) ± √((0.42)² - 4(-0.005)(-6.5))) / (2(-0.005))
Simplifying further:
x = (-0.42 ± √(0.1764 - 0.052)) / (-0.01)
x = (-0.42 ± √0.1244) / (-0.01)
Taking the positive square root and solving:
x = (-0.42 + 0.352) / (-0.01)
x = 0.032 / (-0.01)
x ≈ -3.2
Since distance cannot be negative, we can disregard the negative value. Therefore, the rock will land approximately 3.2 feet horizontally from Landon.
The closest option to 3.2 ft is 3.2 rounded to the nearest hundredth, which is 3.23 ft.
Therefore, the correct answer is c. 10.23 ft.