I know this question, has been asked, but I have 2 question.
First :
3a*2^2 + 2b*2 + c = 0
3a(1/2)^2 + 2b(1/2) + c = 0
How do the equations equal zero when, H'(1/2) < 0, H'(2) > 0, and H'(2) = 1/2
Second:
6a(5/4) + 2b = 0
3a*2^2 + 2b*2 + c = 0
3a(1/2)^2 + 2b(1/2) + c = 0
how do these three systems of equations produce H(d) = 4d^3 - 15d^2 + 12d + 1, I'm confused
The Question:
The given function H(d) = ad^3 + bd^2 + cd + e represents a roller coaster, where H(d) represents the height above the ground, and d represents the horizontal distance the roller coaster has travelled.
Make it so the following parameters are true:
1) The roller coaster must have a local maximum at the point when d =(1)/2
2) The roller coaster must have a local minimum when d = 2 , h = 0.5
3) The roller coaster must have a point of inflection at the point when d = 1.25
4) The roller coaster starts at a height of 1 m above the ground.
To find values of a, b, c, and e that satisfy the given conditions, we can use the equations you provided to solve for these variables.
First, let's deal with the first set of equations:
1) H'(1/2) < 0:
Take the derivative of H(d) with respect to d and evaluate it at d = 1/2. Set it less than zero.
H'(d) = 3ad^2 + 2bd + c
H'(1/2) = 3a(1/2)^2 + 2b(1/2) + c
Simplify and set it less than zero. This will give you the first equation.
2) H'(2) > 0:
Similar to the previous step, evaluate H'(d) at d = 2 and set it greater than zero. This will give you the second equation.
3) H'(2) = 1/2:
Evaluate H'(d) at d = 2 and set it equal to 1/2. This will give you the third equation.
Solve these three equations simultaneously to find values for a, b, and c.
For the second set of equations:
1) H(5/4) = 0:
Plug in d = 5/4 into H(d) and set it equal to 0. Simplify this equation.
2) H(2) = 0.5:
Plug in d = 2 into H(d) and set it equal to 0.5. Simplify this equation.
3) H(1/2) = 0:
Plug in d = 1/2 into H(d) and set it equal to 0. Simplify this equation.
Solve these three equations simultaneously to find the value of e.
Once you have values for a, b, c, and e, substitute them back into the original equation H(d) = ad^3 + bd^2 + cd + e to obtain the roller coaster function H(d) = 4d^3 - 15d^2 + 12d + 1.