This is a calculus 3 problem. Please explain each step clearly, no cursive writing. Find the shortest distance from the point (- 1, – 2, 5) to the cone z =x² + y². Pleaseround to the nearest hundredth.1.9403

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Description: This is a calculus 3 problem. Please explain each step clearly, no cursive writing. Find the shortest distance from the point (- 1, – 2, 5) to the cone z =x² + y². Pleaseround to the nearest hundredth.1.9403

Distance of a point from cone is d=(x-x1)2+(y-y1)2+(z-z1)2 from…

Answered: Find the shortest distance from the…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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This is a calculus 3 problem. Please explain each step clearly, no cursive writing.

Find the shortest distance from the point (- 1, – 2, 5) to the cone z =
x² + y². Please
round to the nearest hundredth.
1.9403

Expert Solution

Step 1

Distance of a point from cone is

$d = \sqrt{(x - x}$ ₁)2+(y-y₁)2+(z-z₁)2

from point (-1,-2,5)

$d = \sqrt{{(x + 1)}^{2} + {(y + 2)}^{2} + {(z - 5)}^{2}} d^{2} = {(x + 1)}^{2} + {(y + 2)}^{2} + {(z - 5)}^{2}$

Let $d^{2} = f$

$f = {(x + 1)}^{2} + {(y + 2)}^{2} + {(z - 5)}^{2} f = x^{2} + 2 x + 1 + y^{2} + 2 y + 4 + z^{2} - 10 z + 25 f_{x} = 2 x + 2 f_{y} = 2 y + 2$

Put,

$f_{x} = 0 and 2 x + 2 = 0 \Rightarrow x = - 1 \dots (1) and f_{y} = 0 2 y + 2 = 0 y = - 1 \dots (2)$

Point (-1,-1) is a critical point of f(x,y).

Put x=-1 and y=-1 in the equation of cone, we get

$z = \sqrt{x^{2} + y^{2}} z = \sqrt{{(1)}^{2} + {(1)}^{2}} z = \sqrt{2}$

we get $(- 1, - 1, \sqrt{2})$

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