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.
I've tried each part separately, and I can get it to work but as soon as I put the parts together it seems to fall apart, I'm confused
so, just what did you do? Too bad you don't show anything we can work with.
H(d) = ad^3 + bd^2 + cd + e
H'(d) = 3ad^2 + 2bd + c
H"(d) = 6ad + 2b
So, we need
H"(1/2) < 0
H"(2) > 0
H(2) = 1/2
H"(5/4) = 0
H(0) = 1
Putting all that into equations, we need to solve
6a(5/4) + 2b = 0
e = 1
3a*2^2 + 2b*2 + c = 0
3a(1/2)^2 + 2b(1/2) + c = 0
or,
15a+4b = 0
12a+4b+c = 0
3a+4b+4c = 0
Solving those, we get
H(d) = 4d^3 - 15d^2 + 12d + 1
H'(d) = 6(2x^2-5x+2)
H"(d) = 6(4x-5)
H has a local maximum at (1/2, 15/4)
H has a local minimum at (2,-3)
H has a point inflection at (5/4, 3/8)
H(0) = 1
To satisfy all the given parameters, we need to find the values of a, b, c, and e that meet each requirement. Let's go through the steps to achieve this:
1) The roller coaster must have a local maximum at the point when d = 1/2:
To have a local maximum at d = 1/2, we need the derivative of the function to be zero at that point.
Taking the derivative of the given function H(d), we get: H'(d) = 3ad^2 + 2bd + c.
For a local maximum, the derivative should be zero, so:
0 = 3a(1/2)^2 + 2b(1/2) + c
0 = 3a/4 + b/2 + c
2) The roller coaster must have a local minimum when d = 2, h = 0.5:
Similarly to the previous step, we need the derivative to be zero at d = 2 for a local minimum. So:
0 = 3a(2)^2 + 2b(2) + c
0 = 12a + 4b + c
3) The roller coaster must have a point of inflection at d = 1.25:
For a point of inflection, the second derivative should be zero at that point. Taking the derivative of the derivative, we get:
H''(d) = 6ad + 2b
Setting d = 1.25 and equating it to zero, we have:
0 = 6a(1.25) + 2b
0 = 7.5a + 2b
4) The roller coaster starts at a height of 1 m above the ground:
To meet this requirement, we have H(0) = 1, since H(d) represents the height above the ground. Substituting 0 for d in the function, we get:
H(0) = a(0)^3 + b(0)^2 + c(0) + e
1 = e
Now, we have a system of equations that need to be solved simultaneously to find the values of a, b, c, and e. Substituting e = 1 into the equations from steps 1, 2, and 3, we have:
1) 0 = 3a/4 + b/2 + c
2) 0 = 12a + 4b + c
3) 0 = 7.5a + 2b
Solving this system of equations should give you the values of a, b, and c that meet all the given parameters.
To meet all the given parameters, we need to find the coefficients a, b, c, and e such that the function H(d) satisfies the conditions. Let's go step by step to solve this problem.
1) The roller coaster must have a local maximum at the point when d = 1/2.
To find the local maximum, we need to find the critical points of the function H(d). Critical points occur when the derivative of the function is zero or undefined. So, let's find the derivative of H(d):
H'(d) = 3ad^2 + 2bd + c.
To find the critical points, we set the derivative equal to zero and solve for d:
3ad^2 + 2bd + c = 0.
Since we want a local maximum, we need a concave-down parabola. This means that the coefficient of the d^2 term (2b) must be negative. Let's assume b = -1. Then the above equation becomes:
3ad^2 - 2d + c = 0.
Now, substituting d = 1/2 into the above equation, we get:
3a(1/2)^2 - 2(1/2) + c = 0.
Simplifying, we have:
3a/4 - 1 + c = 0.
3a/4 + c = 1.
This equation gives us a relationship between a and c. We can choose any value for a and then solve for c. Let's choose a = 4:
3(4)/4 + c = 1.
c = 1 - 3.
c = -2.
So, we have a = 4 and c = -2, which gives us the equation:
H(d) = 4d^3 - d^2 - 2d + e.
2) The roller coaster must have a local minimum when d = 2, h = 0.5.
To find the coefficient e, we substitute d = 2 and H(d) = 0.5 into the equation:
4(2)^3 - (2)^2 - 2(2) + e = 0.5.
Simplifying, we get:
32 - 4 - 4 + e = 0.5.
24 + e = 0.5.
e = 0.5 - 24.
e = -23.5.
So, we have e = -23.5, which gives us the equation:
H(d) = 4d^3 - d^2 - 2d - 23.5.
3) The roller coaster must have a point of inflection at the point when d = 1.25.
To find the coefficient b, we substitute d = 1.25 into the equation:
4(1.25)^3 - (1.25)^2 - 2(1.25) - 23.5.
Simplifying, we get:
9.765625 - 1.5625 - 2.5 - 23.5.
-17.796875.
So, the roller coaster does not meet the condition of having a point of inflection at d = 1.25 with the current coefficients.
To summarize, we have the following equation for the roller coaster:
H(d) = 4d^3 - d^2 - 2d - 23.5.
Unfortunately, this equation does not meet all the given parameters. You might need to adjust the values of the coefficients a, b, c, and e to satisfy the conditions of the roller coaster.