Answer: x = 2r.
- A x = r
- B x = 2r
- C x = 3r
- D x = r/2
Correct answer: B. x = 2r
Explanation: Total area A = x<sup>2</sup> + pi r<sup>2</sup> with 4x + 2 pi r = 36. Substituting and differentiating with respect to x, then setting dA/dx = 0, gives the condition x = 2r, meaning the side of the square equals the diameter of the circle at the minimum.
The tangent at a point touches the curve with slope f'(a); the normal is the line perpendicular to the tangent at that same point, with slope -1/f'(a) - together they describe the curve's local direction and the line "straight into" the curve.
Concept context
Use derivatives to study rate of change, increasing and decreasing functions, tangents and normals, and maxima and minima, with classic optimization problems.