A superhero is trying to leap over a tall building. The function f(x)=-16x^2+200x gives the superheroes height in feet as a function of time. The building is 612 feet high. Will the superhero make it over the building? Explain.
Vo = 200 Ft/s = Initial velocity.
g = 2*(-16) = -32 Ft/s^2.
hmax = (V^2-Vo^2)/2g = (0-(200)^2)/-64 =
625 Ft.
Yes, because his max. ht. is greater
than the ht. of the building.
To determine whether the superhero will make it over the building, we need to find the time at which the superhero's height is equal to or greater than the height of the building.
Given that the height of the building is 612 feet, we need to find the time when f(x) is equal to or greater than 612.
So, we set up the equation:
-16x^2 + 200x ≥ 612
Next, we need to solve this quadratic inequality.
Let's subtract 612 from both sides of the equation:
-16x^2 + 200x - 612 ≥ 0
Now, let's solve for x by factoring the quadratic expression.
-16x^2 + 200x - 612 = 0
Dividing the entire equation by -4 to simplify it:
4x^2 - 50x + 153 = 0
Now, we can factor the quadratic equation:
(4x - 17)(x - 9) = 0
Setting each factor equal to zero:
4x - 17 = 0 or x - 9 = 0
Solving for x, we get:
x = 17/4 or x = 9
Since we are talking about time, x must be positive.
Therefore, the superhero will make it over the building when x = 9.
To verify this, we can substitute x = 9 into the original equation:
f(9) = -16(9)^2 + 200(9)
Simplifying:
f(9) = -16(81) + 1800
f(9) = -1296 + 1800
f(9) = 504
From the calculation, we can see that the superhero's height at time x = 9 is 504 feet, which is greater than the height of the building (612 feet).
Therefore, the superhero will not make it over the building.
To determine whether the superhero will make it over the building, we need to find the maximum height reached by the superhero. This can be done by finding the vertex of the quadratic function f(x) = -16x^2 + 200x.
The vertex of a quadratic function in the form f(x) = ax^2 + bx + c is given by the x-coordinate -b/2a. In this case, a = -16 and b = 200.
Using the formula, we can find the x-coordinate of the vertex:
x-coordinate of the vertex = -b/2a
= -200/(2 * -16)
= -200/-32
= 6.25
So, the superhero reaches the maximum height at t = 6.25 seconds.
To find the maximum height, substitute the x-coordinate of the vertex into the function:
f(6.25) = -16(6.25)^2 + 200(6.25)
= -16(39.0625) + 1250
= -625 + 1250
= 625
The superhero reaches a height of 625 feet at its maximum point.
Since the building is 612 feet high and the superhero's maximum height is 625 feet, the superhero will indeed make it over the building.