A pebble is tossed from the top of a cliff. The pebble's height in feet is given by y(t) = -16t2 + 6t + 200, where t is the time in seconds. Its horizontal distance in feet from the base of the cliff is given by d(t) = 5t.
How far will the pebble be from the base of the cliff when it hits the ground? Round to the nearest foot
17.9863 feet from the base of the tree
To find how far the pebble will be from the base of the cliff when it hits the ground, we need to determine the time when the pebble hits the ground and then substitute this time into the horizontal distance equation.
The equation for the height of the pebble is given by y(t) = -16t^2 + 6t + 200, where t is the time in seconds.
We can find the time when the pebble hits the ground by setting y(t) equal to 0 and solving for t.
-16t^2 + 6t + 200 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. In this case, factoring might not be straightforward, so let's use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = -16, b = 6, and c = 200. Plugging these values into the quadratic formula, we get:
t = (-6 ± √(6^2 - 4(-16)(200))) / (2(-16))
t = (-6 ± √(36 + 12800)) / (-32)
t = (-6 ± √12836) / (-32)
The square root of 12836 is approximately 113.24, so we have two possible solutions for t:
t = (-6 + 113.24) / (-32) ≈ -107.24 / -32 ≈ 3.35
t = (-6 - 113.24) / (-32) ≈ -119.24 / -32 ≈ 3.73
Since time cannot be negative, we discard the negative value for t. Hence, the pebble hits the ground at approximately t = 3.35 seconds.
Now that we have the time when the pebble hits the ground, we can substitute this time into the horizontal distance equation, d(t) = 5t, to find the distance from the base of the cliff.
d(3.35) = 5 * 3.35 ≈ 16.75
Therefore, the pebble will be approximately 16.75 feet from the base of the cliff when it hits the ground.
To find the horizontal distance the pebble will be from the base of the cliff when it hits the ground, we need to determine the time for which the pebble's height is zero.
The height of the pebble is given by the function y(t) = -16t^2 + 6t + 200, where t represents time in seconds.
To find the time at which the pebble hits the ground, we set y(t) = 0 and solve for t:
-16t^2 + 6t + 200 = 0
This is a quadratic equation. To solve it, you can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = -16, b = 6, and c = 200. Plugging these values into the formula, we get:
t = (-6 ± √(6^2 - 4(-16)(200))) / (2(-16))
Simplifying further:
t = (-6 ± √(36 + 12800)) / -32
t = (-6 ± √12836) / -32
Now, we will calculate the two possible values of t:
t1 = (-6 + √12836) / -32
t2 = (-6 - √12836) / -32
Calculating these values, t1 is approximately -0.5 and t2 is approximately 12.5.
Since time cannot be negative, we discard t1. Therefore, the pebble hits the ground after approximately 12.5 seconds.
To find the horizontal distance the pebble has traveled at this time, we can use the function d(t) = 5t, where d(t) represents the horizontal distance from the base of the cliff.
Substituting t = 12.5 into the equation, we have:
d(12.5) = 5 * 12.5
d(12.5) = 62.5
Therefore, the pebble will be approximately 62.5 feet from the base of the cliff when it hits the ground. Rounded to the nearest foot, it will be 63 feet from the base of the cliff.