Landon is standing in a hole that is 6.7 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^2 + 0.48x - 6.7, where x is the horizontal distance of the rock, in meters, from Landon and y is the height, in meters, of the rock above the ground. How far horizontally from Landon will the rock land?
I got two answers: 17 and 79; however, I do not know which is x or y, nor horizontal. Please help!
y is vertical, x is horzontal. When the rock is at y=0, you need to find x.
0=-.005x^2+.48x-6.7
0=x^2-96x+1340
so x comes out 17, and 79 (rounded)
So there are two answers, and in the real world, he could have thrown it with two different velocities. Remember these problems are "contrived", and may not match the real world we live in.
There are two answers to chose from:
A: 16.95
B: 79.05
Which is horizontal?
To find the horizontal distance at which the rock will land, we need to determine the value of x when y is equal to 0. In other words, we need to find the x-intercepts of the equation.
Setting y to 0, we have:
0 = -0.005x^2 + 0.48x - 6.7
Now, let's solve this quadratic equation. There are multiple ways to solve it, but let me show you one method using the quadratic formula:
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
Here, the coefficients of the quadratic equation are as follows:
a = -0.005
b = 0.48
c = -6.7
Plugging these values into the quadratic formula, we get:
x = (-0.48 ± √(0.48^2 - 4(-0.005)(-6.7))) / (2(-0.005))
Simplifying further:
x = (-0.48 ± √(0.2304 - 0.134)) / (-0.01)
x = (-0.48 ± √0.0964) / (-0.01)
We have two potential solutions, one with the plus sign (+) and one with the minus sign (-). Let's calculate both of them:
x1 = (-0.48 + √0.0964) / (-0.01) ≈ 17.27563
x2 = (-0.48 - √0.0964) / (-0.01) ≈ 79.72437
Now, considering the context of the problem, we can determine that Landon is standing at the origin (0,0). Therefore, the horizontal distance at which the rock will land from Landon is given by the value of x.
From our calculations, we have two possible distances: approximately 17.28 meters and approximately 79.72 meters. However, since the rock was thrown upwards, it is more reasonable to consider the positive distance, 17.28 meters, as the answer.
Therefore, the rock will land approximately 17.28 meters horizontally from Landon's position.