Question

Find the minimum distance from the point to the paraboloid \(z=x^{2}+y^{2}\). (5,0,0)

Short answer

After performing all the steps, the final answer - the minimum distance is calculated by substituting the obtained values of variables x and y into the equation for d. This result will be the shortest distance from the point (5,0,0) to the given paraboloid surface.

Step-by-step solution

(Write equation for a distance and substitute the function of z into it)

The distance between the given point (5,0,0) to any point (x,y,z) on the parabola is given by the Pythagorean theorem: \(d=\sqrt{(x-5)^{2}+y^{2}+(z-0)^{2}}\). Since the function of z (height at the point (x,y)) on our parabola is given by \(z=x^{2}+y^{2}\), we can substitute z into the distance formula and rewrite it: \(d=\sqrt{(x-5)^{2}+y^{2}+(x^{2}+y^{2})^{2}}\). Simplifying, we obtain \(d=\sqrt{(x-5)^{2}+2x^{2}+2y^{2}}\).

(Set derivative of d with respect to x and y to zero)

To find the minimum distance, we need to find the minimum of the function d. This happens when the derivative of d with respect to x and y is zero. Differentiating the expression for d (use the chain rule), the resulting expressions can be set to zero to find the x and y that gives the minimum distance: \(\frac{dd}{dx}=0\) and \(\frac{dd}{dy}=0\).

(Solve the system of equations)

Solving the system of equations obtained in step 2, we find the values of x and y that result in the minimum distance.

(Substitute x and y in the distance equation)

Once we find the values of x and y, we can substitute them back into the distance function d to find the minimum distance from the point (5,0,0) to the parabola.