1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
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}})

You might be interested in
どうすれば2個の卵を壊さずに最も少ない回数で落とすことができますか?
Vikentia [17]

Answer:

知りません

Step-by-step explanation:

6 0
3 years ago
Anyone good at trigonometry?<br> I need a tutor or something cause I suck
SVEN [57.7K]

Answer:

I'll help if you need it.

Step-by-step explanation:

5 0
3 years ago
State the domain and range of the following function:
mojhsa [17]

Answer:

Domain = \{-3,0,2,1,6,3\}

Range = \{4,6,-2,-3,-7,2\}

Step-by-step explanation:

Given

\{(- 3,4), (0,6), (2, - 2), (1, - 3), (6, - 7), (3, 2)\}

Required

The domain and range

A function is represented as:

Function = \{(x_1,y_1),(x_2,y_2),....(x_n,y_n)\}

Where

x \to domain

y \to range

So, we have:

Domain = \{-3,0,2,1,6,3\}

Range = \{4,6,-2,-3,-7,2\}

7 0
3 years ago
Find circumference when pi is 22/7
sashaice [31]

Answer:

what is radius

Step-by-step explanation:

tell me what is the radius

8 0
3 years ago
What is 514 divided by 28
stira [4]

Answer:

18.36

Step-by-step explanation:

3 0
3 years ago
Read 2 more answers
Other questions:
  • Solve the equation below<br><br> 3h - 2( 4h - 5 ) = 10 - 5h
    14·2 answers
  • A car is traveling at a speed of miles per hour. What is the car's speed in kilometers per hour? How many kilometers will the ca
    6·1 answer
  • If the greatest value of n is 9, which inequality best shows all the possible values of n?
    5·1 answer
  • What is the range of the function f(x) = -2(6^x) +3?<br><br><br> Answer is B
    12·1 answer
  • 5k + (-7) = 13 what does k =
    9·2 answers
  • Which of the following is the function f(x) if f^-1(x) = 5x - 19?
    8·1 answer
  • Function f is defined by the equation f(x)=x2<br> What is f(3)
    10·1 answer
  • If g(x) = 3x² – 2x–5, what is the<br> value of g(-1) ?
    13·1 answer
  • Do ASAP OVER DUE points, Brainliest, thanks
    15·1 answer
  • 2. Circle C, has center (0,0) and radius 2.
    14·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!