Page 66 - Calc for the Clueless

inches

The dimensions to give the maximum strength are inches wide and inches deep.

Example 8—

2
Find the shortest distance from y = 2x to the point (2,0).
Although we will draw the picture, it is not truly needed here. Distance means the distance formula. A trick that
can always be used is this: instead of using the distance formula (which involves a square root), we can use the
square of the distance formula (since the distance is a minimum if the square of the distance is the minimum of
the square of all the distances).

From the picture, we see that two solutions are possible.

2
2
2
Let the square of the distance be H = (x - 2) + (y - 0) = x - 4x + 4 + y . But on the curve, y = 2x. Therefore
2
2

The closest points are . To find the exact minimum distance, substitute in the distance formula.

The next is a standard-type problem, and many similar kinds are found in all calculus books.

Example 9—

2
Find the dimensions of the largest rectangle (in area) that can be inscribed in the parabola y = 12 - x , with two
2
vertices of the rectangle on the x axis and two vertices on the curve y = 12 - x .
2
Since y = 12 - x is symmetric with respect to the y axis, the inscribed rectangle is also symmetric to the y axis.
From the picture, the height of the rectangle is y and the base is x- (-x) = 2x.