Question
The suspension cables between the two towers of the Mackinac Bridge in Michigan form a parabola that can be modeled by y = 0.000098x2 - 0.37x 552, where x and y are measured in feet. Estimate and interpret the minimum value of the function.
Answers
GPT 3.5
The minimum value of the function can be found by setting the derivative of the function equal to zero and solving for x:
dy/dx = 0.000196x - 0.37
0.000196x - 0.37 = 0
0.000196x = 0.37
x = 0.37 / 0.000196
x ≈ 1887 feet
To find the corresponding y-coordinate, we can plug this value back into the original equation:
y = 0.000098(1887)^2 - 0.37(1887) + 552
y ≈ 267.58 feet
Therefore, the minimum value of the function occurs at approximately (1887, 267.58).
Interpretation: The minimum value of approximately 267.58 feet represents the lowest point of the parabolic shape formed by the suspension cables between the two towers of the Mackinac Bridge. This point indicates the point at which the cables dip the lowest between the towers.
dy/dx = 0.000196x - 0.37
0.000196x - 0.37 = 0
0.000196x = 0.37
x = 0.37 / 0.000196
x ≈ 1887 feet
To find the corresponding y-coordinate, we can plug this value back into the original equation:
y = 0.000098(1887)^2 - 0.37(1887) + 552
y ≈ 267.58 feet
Therefore, the minimum value of the function occurs at approximately (1887, 267.58).
Interpretation: The minimum value of approximately 267.58 feet represents the lowest point of the parabolic shape formed by the suspension cables between the two towers of the Mackinac Bridge. This point indicates the point at which the cables dip the lowest between the towers.