Find the point on the parabola shown below that is closest to the point(8, 0). E

Question

Find the point on the parabola shown below that is closest to the point(8, 0). Express the square of the distance between the parabola and the point(8, 0) in terms of x only. S(x) = find the coordinates of the point on the parabola to two decimal places. (____________ middot ___________) The cross-section of a tunnel is a rectangle of height h surmounted by a semicircular roof section of radius r. (See figure below). Suppose the cross-sectional area is A ft^2. Express the perimeter in terms of r only. the constant A will also appear in your answer.

Explanation / Answer

Ans)

To minimize the distance between (x, y) and (8, 0), on the curve y = x^2.

What is the distance between (x, y) and (8, 0)? We determine this using the distance formula.

D = sqrt( (x2 - x1)^2 + (y2 - y1)^2 )

Let (x1, y1) = (8, 0) and (x2, y2) = (x, y). Then

D = sqrt( (x - 8)^2 + (y - 0)^2 )
D = sqrt( (x - 8)^2 + y^2 )

Remember that, since y = x^2, we can replace every instance of y in that formula with x^2.

D = sqrt( (x - 8)^2 + (x^2)^2 )
D = sqrt( (x - 8)^2 + x^4 )

We want to minimize this by making dD/dx = 0.
We *could* differentiate this directly, but when we have to deal with derivatives with roots followed by the chain rule. I will instead square both sides, and then differentiate implicitly.

D^2 = (x - 8)^2 + x^4

Differentiate implicitly with respect to x,

2D (dD/dx) = 2(x - 8) + 4x^3

Make dD/dx = 0, to get

0 = 2(x - 8) + 4x^3

Solve for x.

0 = 2x - 16 + 4x^3
0 = 4x^3 + 2x - 16
0 = 2x^3 + x - 8

We have a cubic that we need to factor. Fortunately for us, if p(x) = 2x^3 + x - 8, we have roots

x1 = 1.4826

x2 = -0.7413-1.4658 i

x3 = -0.7413+1.4658 i

lets ignore x2 and x3 as they are not possible.

But we want the y-coordinate too.
Since y = x^2, then y = 1.4826^2 = 2.1981,
so the closest point is (1.4826, 2.1981).

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