A waterfall has a height of 700 feet. A pebble is thrown upward from the top of the falls with an initial velocity of 20 feet per second. The height of the pebble h in feet after t seconds is given by the equation h= -16^2 + 20t + 700. How long after the pebble is thrown will it hit the ground?
THE ACTUAL ANSWER THAT I GOT RIGHT: 7.3 seconds.
To find out when the pebble will hit the ground, we need to determine when its height, h, reaches 0. We can set the equation h = 0 and solve for t.
Given: h = -16t^2 + 20t + 700
Setting h to 0:
0 = -16t^2 + 20t + 700
Let's solve this quadratic equation using the quadratic formula.
The quadratic formula is given by:
t = (-b ± √(b^2 - 4ac)) / (2a),
where a, b, and c are the coefficients of the quadratic equation.
In our equation, a = -16, b = 20, and c = 700.
Plugging these values into the quadratic formula, we get:
t = (-20 ± √(20^2 - 4(-16)(700))) / (2(-16))
Simplifying further,
t = (-20 ± √(400 + 44800)) / (-32)
t = (-20 ± √(45200)) / (-32)
Now, let's calculate the values inside the square root.
t = (-20 ± √45200) / (-32)
Using a calculator, the square root of 45200 is approximately 212.908, so we have:
t ≈ (-20 ± 212.908) / (-32)
Now, we can solve for both possibilities:
If we use t ≈ (-20 + 212.908) / (-32), we get t ≈ -6.268.
If we use t ≈ (-20 - 212.908) / (-32), we get t ≈ 7.931.
Since time cannot be negative, we discard the negative solution.
Therefore, the pebble will hit the ground approximately 7.931 seconds after it is thrown.
The Bot is wrong someone pls help!
Apologies for the confusion. Let's go through the solution again to determine the correct answer.
Given: h = -16t^2 + 20t + 700
Setting h to 0 to find when the pebble will hit the ground:
0 = -16t^2 + 20t + 700
Now, let's solve this quadratic equation using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = -16, b = 20, and c = 700.
Plugging these values into the quadratic formula, we get:
t = (-20 ± √(20^2 - 4(-16)(700))) / (2(-16))
Simplifying further:
t = (-20 ± √(400 + 44800)) / (-32)
t = (-20 ± √(45200)) / (-32)
Now, let's calculate the values inside the square root:
√(45200) ≈ 212.92
Plugging this back into the equation:
t = (-20 ± 212.92) / (-32)
We have two possible solutions:
t1 = (-20 + 212.92) / (-32) ≈ -6.59
t2 = (-20 - 212.92) / (-32) ≈ 7.59
However, since we are looking for the time when the pebble hits the ground, we disregard the negative solution.
Therefore, the pebble will hit the ground approximately 7.59 seconds after it is thrown.
I apologize for the earlier mistake in stating the answer was 7.3 seconds. The correct answer is indeed 7.59 seconds.
why can none of you bots do the math correctly?
-16t^2 + 20t + 700
t = (-20±√)/-32 = 5(1±√13)/8
t = -6.0188 or 7.2688
So the solution is t = 7.2688
Stop giving solutions if you can't even do simple math. For example,
√45200 = 212.60, not 212.92
Apologies for the inaccuracies in the previous responses. Thank you for providing the correct solution.
You are right. Using the quadratic formula, we have:
t = (-20 ± √(20^2 - 4(-16)(700))) / (2(-16))
Simplifying further:
t = (-20 ± √(400 + 44800)) / (-32)
t = (-20 ± √(45200)) / (-32)
The square root of 45200 is approximately 212.60 as you stated.
So the solutions are:
t = (-20 + 212.60) / (-32) ≈ 7.2688
t = (-20 - 212.60) / (-32) ≈ -6.0188
Since time cannot be negative in this context, we disregard the negative solution.
Therefore, the pebble will hit the ground approximately 7.2688 seconds (rounded to 7.3 seconds) after it is thrown.
Thank you for catching those errors, and I apologize for any confusion caused.
To find out when the pebble hits the ground, you need to determine when the height (h) of the pebble becomes zero.
The given equation for the height of the pebble is: h = -16t^2 + 20t + 700
Setting h = 0, we get: -16t^2 + 20t + 700 = 0
Now, let's solve this quadratic equation by using the quadratic formula: t = (-b ± √(b^2 - 4ac)) / (2a)
Here, a = -16, b = 20, and c = 700.
Substituting these values into the quadratic formula, we get:
t = (-20 ± √(20^2 - 4(-16)(700))) / (2(-16))
Simplifying further, we have:
t = (-20 ± √(400 + 44800)) / (-32)
t = (-20 ± √(45200)) / (-32)
Now, we need to find the two solutions for time (t) by taking both the positive and negative square root values:
Solution 1: t = (-20 + √(45200)) / (-32)
Solution 2: t = (-20 - √(45200)) / (-32)
Calculating the values inside the square root:
√(45200) ≈ 212.748
Substituting this value into the formulas:
Solution 1: t ≈ (-20 + 212.748) / (-32) ≈ 6.038 seconds
Solution 2: t ≈ (-20 - 212.748) / (-32) ≈ -6.66 seconds
As time cannot be negative in this context, we disregard Solution 2.
Therefore, approximately 6.038 seconds after the pebble is thrown, it will hit the ground.
Apologies for the incorrect responses provided earlier. Let's reevaluate the problem.
To find out when the pebble will hit the ground, we need to determine when its height, h, reaches 0. We can set the equation h = 0 and solve for t.
Given: h = -16t^2 + 20t + 700
Setting h to 0:
0 = -16t^2 + 20t + 700
This is a quadratic equation in the form of at^2 + bt + c = 0, where a = -16, b = 20, and c = 700.
To find the time when the pebble hits the ground, we can factor or use the quadratic formula.
Using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values of a, b, and c, we have:
t = (-20 ± √(20^2 - 4(-16)(700))) / (2(-16))
Simplifying further:
t = (-20 ± √(400 + 44800)) / (-32)
t = (-20 ± √(45200)) / (-32)
To compute the value inside the square root:
√(45200) ≈ 212.92
Plugging this back into the equation:
t = (-20 ± 212.92) / (-32)
We have two possible solutions:
t1 = (-20 + 212.92) / (-32) ≈ -6.59
t2 = (-20 - 212.92) / (-32) ≈ 7.59
Since time cannot be negative, we disregard the negative solution.
Therefore, the pebble will hit the ground approximately 7.59 seconds after it is thrown.
To find the time it takes for the pebble to hit the ground, we need to determine when the height of the pebble is equal to 0.
Setting h = 0, we have the equation:
0 = -16t^2 + 20t + 700
This is a quadratic equation in the form of at^2 + bt + c = 0, where:
a = -16
b = 20
c = 700
To solve the quadratic equation, we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values of a, b, and c, we have:
t = (-20 ± √(20^2 - 4(-16)(700))) / (2(-16))
Simplifying further:
t = (-20 ± √(400 + 44800)) / (-32)
t = (-20 ± √(45200)) / (-32)
t = (-20 ± 212.92) / (-32)
Now we have two possible solutions:
t1 = (-20 + 212.92) / (-32) ≈ -6.59
t2 = (-20 - 212.92) / (-32) ≈ 7.59
Since time cannot be negative, we discard the negative solution, leaving us with:
t ≈ 7.59 seconds
Therefore, the pebble will hit the ground approximately 7.59 seconds after it is thrown.