A superhero is trying to leap over a tall building. The function f(x)= -16x^2 + 200x gives the superhero's height in feet as a function of time. The building is 612 feet high will the super hero make it over the building?
To determine whether the superhero will make it over the building, we need to find the vertex of the quadratic function and compare the maximum height with the height of the building.
The equation of the function is given as f(x) = -16x^2 + 200x, where x represents time in seconds.
The vertex of a quadratic function in the form f(x) = ax^2 + bx + c can be found using the formula x = -b/2a.
For our function, a = -16 and b = 200, so the x-value of the vertex is:
x = -200 / (2 * -16)
x = -200 / -32
x = 6.25
Now, substitute this value of x back into the function to find the maximum height:
f(6.25) = -16(6.25)^2 + 200(6.25)
f(6.25) = -16(39.0625) + 1250
f(6.25) = -625 + 1250
f(6.25) = 625
The maximum height the superhero will reach is 625 feet.
Since the building is 612 feet high, the superhero will make it over the building because the maximum height is greater than the height of the building.
To determine if the superhero will make it over the building, we need to find the time at which the superhero reaches a height equal to or greater than the height of the building.
We have the function f(x) = -16x^2 + 200x, where f(x) represents the superhero's height in feet and x represents time.
Since the building is 612 feet high, we can set up the following inequality:
-16x^2 + 200x ≥ 612
To solve this inequality, we need to find the values of x that satisfy it.
Step 1: Rearrange the inequality by subtracting 612 from both sides:
-16x^2 + 200x - 612 ≥ 0
Step 2: Factor the quadratic equation:
-4(4x^2 - 50x + 153) ≥ 0
Step 3: Solve for x by factoring the quadratic equation inside the parentheses:
4x^2 - 50x + 153 = 0
The quadratic equation does not factor nicely, so we can solve it using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Here, a = 4, b = -50, and c = 153.
x = (-(-50) ± √((-50)^2 - 4(4)(153))) / (2(4))
x = (50 ± √(2500 - 2448)) / 8
x = (50 ± √52) / 8
x ≈ (50 ± 7.211) / 8
We have two potential values for x: x ≈ 8.651 and x ≈ 7.099.
Step 4: Determine if the superhero will make it over the building.
Since time cannot be negative, we discard the negative value of x. Therefore, the superhero will make it over the building at approximately x ≈ 8.651.
Hence, based on the calculations, the superhero will make it over the building.
Since you give no indication of a horizontal speed, I don't see how just jumping up and down will get him over the building. However, his maximum height will be at x=200/32=6.25, so plug that in and see whether it is high enough.
I suspect that you meant the f(x) gives the height at distance x from his starting point. Still, depending on how far away he jumped, he still might not make it over the building.