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Marat540 [252]
3 years ago
5

Find the point (,) on the curve =8 that is closest to the point (3,0). [To do this, first find the distance function between (,)

and (3,0) and minimize it.]
Mathematics
1 answer:
ELEN [110]3 years ago
3 0

Question:

Find the point (,) on the curve y = \sqrt x that is closest to the point (3,0).

[To do this, first find the distance function between (,) and (3,0) and minimize it.]

Answer:

(x,y) = (\frac{5}{2},\frac{\sqrt{10}}{2}})

Step-by-step explanation:

y = \sqrt x can be represented as: (x,y)

Substitute \sqrt x for y

(x,y) = (x,\sqrt x)

So, next:

Calculate the distance between (x,\sqrt x) and (3,0)

Distance is calculated as:

d = \sqrt{(x_1-x_2)^2 + (y_1 - y_2)^2}

So:

d = \sqrt{(x-3)^2 + (\sqrt x - 0)^2}

d = \sqrt{(x-3)^2 + (\sqrt x)^2}

Evaluate all exponents

d = \sqrt{x^2 - 6x +9 + x}

Rewrite as:

d = \sqrt{x^2 + x- 6x +9 }

d = \sqrt{x^2 - 5x +9 }

Differentiate using chain rule:

Let

u = x^2 - 5x +9

\frac{du}{dx} = 2x - 5

So:

d = \sqrt u

d = u^\frac{1}{2}

\frac{dd}{du} = \frac{1}{2}u^{-\frac{1}{2}}

Chain Rule:

d' = \frac{du}{dx} * \frac{dd}{du}

d' = (2x-5) * \frac{1}{2}u^{-\frac{1}{2}}

d' = (2x - 5) * \frac{1}{2u^{\frac{1}{2}}}

d' = \frac{2x - 5}{2\sqrt u}

Substitute: u = x^2 - 5x +9

d' = \frac{2x - 5}{2\sqrt{x^2 - 5x + 9}}

Next, is to minimize (by equating d' to 0)

\frac{2x - 5}{2\sqrt{x^2 - 5x + 9}} = 0

Cross Multiply

2x - 5 = 0

Solve for x

2x  =5

x = \frac{5}{2}

Substitute x = \frac{5}{2} in y = \sqrt x

y = \sqrt{\frac{5}{2}}

Split

y = \frac{\sqrt 5}{\sqrt 2}

Rationalize

y = \frac{\sqrt 5}{\sqrt 2} *  \frac{\sqrt 2}{\sqrt 2}

y = \frac{\sqrt {10}}{\sqrt 4}

y = \frac{\sqrt {10}}{2}

Hence:

(x,y) = (\frac{5}{2},\frac{\sqrt{10}}{2}})

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3 0
3 years ago
A rectangular poster is 3 times as long as it is wide. A rectangular banner is 5 times a long as it is wide. Both the banner and
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Answer:

The answer to your question is

a) Poster             width = 3 in

                            lenght = 9 in

b) banner

                           width = 2 in

                           lenght = 10 in

Step-by-step explanation:

a) For the poster

   width = x

   lenght = 3x

Perimeter of a rectangle = 2base + 2 height

                                         = 2(x) + 2(3x) = 24

                                         2x + 6x = 24

                                         8x = 24

                                           x = 24/8

                                           x = 3

    width = 3 in

     lenght = 3(3) = 9 in

b) For the banner

    width = y

    lenght = 5y

Perimeter of a banner = 2 base + 2 height

                                        2 (y) + 2(5y) = 24

                                        2y + 10y = 24

                                        12y = 24

                                           y = 24/12

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     width = 2 in

     lenght = 5(2) = 10 in

7 0
3 years ago
At present the sum of Monica's age and daughter's age is 44 years. . After 2 years Monica's age will be three times that if her
sergeinik [125]

Answer:

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Step-by-step explanation:

Let Monica's age = x

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x + y = 44   -------> equ 1

After 2 years:

Monica's age = x + 2

Her daughter's age = y + 2

x + 2 = 3 * (y+2)

x + 2 = 3y + 6

      x = 3y + 6 -2

      x = 3y + 4

Put x value in equ 1:

    3y + 4 + y = 44

            4y = 44 - 4

             4y = 40

               y = 40/4

               y = 10

x = 44 -10

x = 34

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