What is the smallest distance between the origin and a point on the graph of:y=1/sqrt(2)*(x^2-8)

Answers

Answer 1

What we are being asked here is to simply minimize distance. Also, note that we can write
f
(
x
)
=
√
x
as
y
=
√
x
.
Now, what is this "distance?" How do we find it? Well, if you think back to Algebra I or Geometry, you'll remember that the distance between two points
(
x
1
,
y
1
)
and
(
x
2
,
y
2
)
is given by:
√
(
y
2
−
y
1
)
2
+
(
x
2
−
x
1
)
2
. For example, the distance between the points
(
4
,
0
)
and
(
0
,
3
)
would be:
√
(
3
−
0
)
2
+
(
4
−
0
)
2
=
√
9
+
16
=
√
25
=
5
Ok, so what is
(
x
1
,
y
1
)
and
(
x
2
,
y
2
)
in our example?
(
x
1
,
y
1
)
is simple - it's just the point given in the problem,
(
4
,
0
)
. Because we don't know what
x
2
is, we'll just call it
x
for now. As for
y
2
, we don't know that either; and since
y
=
√
x
, we'll call it
√
x
.
Our formula then becomes:
√
(
√
x
−
0
)
2
+
(
x
−
4
)
2
=
√
(
√
x
2
)
+
x
2
−
8
x
+
16
=
√
x
+
x
2
−
8
x
+
16
=
√
x
2
−
7
x
+
16
We are being asked to minimize this distance, which we'll call
s
to make the following calculations easier. To minimize something, we have to take its derivative, so let's start there:
s
=
√
x
2
−
7
x
+
16
=
(
x
2
−
7
x
+
16
)
1
2

d
s
d
x
=
(
2
x
−
7
)
⋅
1
2
(
x
2
−
7
x
+
16
)
1
2
→
Using power rule and chain rule
d
s
d
x
=
2
x
−
7
2
√
x
2
−
7
x
+
16
Now we set this equal to
0
and solve for
x
:
0
=
2
x
−
7
2
√
x
2
−
7
x
+
16
0
=
2
x
−
7
x
=
7
2
This is known as the critical value, and it represents the
x
-value for which the function is minimized. All we need to do now is find the corresponding
y
-value, using the definition of
y
:
y
=
√
x
. Substituing
7
2
for
x
:
y
=
√
7
2

y
≈
1.87
And voila, the
y
-value. We can now say that the minimum distance between
f
(
x
)
=
√
x
and the point
(
4
,
0
)
(the place where these two are closest) occurs at
(
7
2
,
1.87
)
. For a little extra fun, we can use the distance formula to see what the actual distance between the points is:
s
=
√
(
1.87
−
0
)
2
+
(
7
2
−
4
)
2
≈
1.8
units

Related Questions

PLEASE HELP MEEE!A MILLION THANKS IF YOU ANSWER ME CORRECTLY!PLEASE NO LINKS!

Answers

Answer: Christian bought 256 fluid ounces of juice. I don't know about the unit rate.

Step-by-step explanation:

Answer:

307 fl oz

Step-by-step explanation:

HELP.....................

Answers

Let me first ask this :

how do u find the midnumber of 1 and 3 ?

( 1 + 3 ) ÷ 2 =

4 ÷ 2 =

_________________________________

It is the exact way we do to find the midpoint of a line ; have a look :

The midpoint of AB is P thus :

( Point A + Point B ) ÷ 2 = Point P

Suppose that we have point B as ( x , y ) then :

[ ( 2 , 3 ) + ( x , y ) ] ÷ 2 = ( 6 , - 1 )

Now we add the same components of points A and B together, that is, we add the first component A to the first component B and replace the first component, and so we add the second component A to the second component B. And replace the second component :

[ ( 2 + x , 3 + y ) ] ÷ 2 = ( 6 , - 1 )

( 2 + x/2 , 3 + y/2 ) = ( 6 , - 1 )

Well now we have a draw so the first components of both sides of the draw must be equal Also, the second components of the two sides of the equation must be equal , thus :

2 + x/2 = 6

Multiply both sides by 2

2 × ( 2 + x )/2 = 2 × 6

2 + x = 12

Subtract both sides 2

2 - 2 + x = 12 - 2

x = 10

3 + y/2 = - 1

Multiply both sides by 2

2 × ( 3 + y )/2 = 2 × ( - 1 )

3 + y = - 2

Subtract both sides 3

3 - 3 + y = - 2 - 3

y = - 5

________________________________

And finally, after all the hardship and misery, we realized that

B = ( 10 , - 5 )

Which mean Option C is the correct answer.

And we're done...

Mr. La Fleur pays $130 to rent a car for a week. He also pays $5 for every 100 miles that he drives. Write an equation in slope intercept form that shows the relationship between the total cost of renting the car(y) and the number of miles that are driven (x).

Answers

Step-by-step explanation:

y+0.05x

0.05 because

5$ for 100

5$/100=0.05(for a single mile)

What is the Domain and Range of the following Graph?

Answers

Answer:

This is a square function with its maximum at (-3, 2).

The domain is:

x ∈ [-3, + ∞)

The range is:

y ∈ (- ∞, 2]

The line segment, AB has a point C as its midpoint. If A is at (2,2) and C is at (5,4) what are the coordinates of the endpoint B

A:(-1,0)
B:(3.5,3)
C:(5,4)
D:(8,6)

Answers

Answer:

( 8 , 6 )

Step-by-step explanation:

A = ( 2 , 2 ) and C = ( 5 , 4 )

→ Form an equation

[tex]\frac{2+x}{2} =5[/tex] and [tex]\frac{2+y}{2}=4[/tex]

→ Now solve them

2 + x = 10 and 2 + y = 6

→ Simply find the values

x = 8 and y = 6

Let the coordinates of B be ( x , y ). Now , Let A ( 2 , 2 ) be ( x₁ , y₁ ) & B ( x , y ) be ( x₂ , y₂ )

SOLUTION :

All you need to apply is the formula of Midpoint i.e [tex] \sf{ \bold{( \frac{x_{1} + x_{2}}{2} \: , \frac{y_{1} + y_{2}}{2} )}}[/tex]

Plug the known values :

[tex] \large{ \rightarrow{ \tt{ \: (5 \: ,4 \: ) = ( \frac{2 + x}{2} \: , \frac{2 + y}{2} )}}}[/tex]

Equating corresponding values :

[tex] \large{ \tt{5 = \frac{2 + x}{2} }}[/tex]

[tex] \large{ \rightarrow{ \tt{10 = 2 + x}}}[/tex]

[tex] \large{ \rightarrow{\tt{10 - 2 = x}}}[/tex]

[tex] \boxed{ \large{ \rightarrow \tt{x = 8}}}[/tex]

Value of x = 8

[tex] \large{ { \tt{4 = \frac{2 + y}{2} }}}[/tex]

[tex] \large{ \rightarrow{ \tt{8 = 2 + y}}}[/tex]

[tex] \large{ \rightarrow{ \tt{8 - 2 = y}}}[/tex]

[tex] \boxed{ \large{ \rightarrow{ \tt{y = 6}}}}[/tex]

Value of y = 6

[tex] \large{ \boxed{ \tt{OUR \: FINAL \: ANSWER : \: D. \: (8 \:, 6 \: ) }}}[/tex]

:) Hope I helped! Let me know if you have any questions regarding my answer! ~

Nadia drew a square in her notebook Each side measured 2.5 centimeters

Answers

Answer:

The perimeter of the square is 10 cm.

The area of the square is 6.25 cm.

Step-by-step explanation:

To find perimeter, add all the sides.

2.5 + 2.5 + 2.5 + 2.5 = 10

To find area, multiply the length by the width. Since Nadia drew a square, all the sides will have the same measurement (2.5 cm, in this case). So, simply multiply 2.5 by 2.5, knowing that the length and width of the square are the same size.

2.5 × 2.5 = 6.25

The area of Square is 6.25 cm² and Perimeter of Square is 10 cm.

what is Area and Perimeter of Square?

The Area of Square is the product if its side

i.e., Area of square = side x side

and, perimeter of square = 4 x side

Given:

Side of Square= 2.5 cm

then , area of Square= side x side

= 2.5 x 2.5

= 6.25 cm²

and, Perimeter of Square= 4 x side

= 4 x 2.5

= 10 cm

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Solve for the value of m.
73°(6m-1)º

Answers

Step-by-step explanation:

6m-1+73=180 (linear pair)

6m+72=180

6m=180-72

6m=108

m=108/6

m=18°

I need help plz..plzz

Answers

Answer:

hope it helps ...........

the distance between points S and T is 61 units. point S is located at (-6,8) and point T is at (x,-3). what could be the value of x?

Answers

The distance between two points is the number of units between them

The possible value of x is -66

The given parameters are:

[tex]\mathbf{S = (-6,8)}[/tex]

[tex]\mathbf{T = (x,-3)}[/tex]

[tex]\mathbf{ST = 61}[/tex]

Distance ST is calculated using:

[tex]\mathbf{d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}}[/tex]

So, we have:

[tex]\mathbf{d = \sqrt{(-6 - x)^2 + (8 - -3)^2}}[/tex]

[tex]\mathbf{d = \sqrt{(-6 - x)^2 + (8 +3)^2}}[/tex]

[tex]\mathbf{d = \sqrt{(-6 - x)^2 + 11^2}}[/tex]

Substitute 61 for d

[tex]\mathbf{ \sqrt{(-6 - x)^2 + 11^2} = 61}[/tex]

Square both sides

[tex]\mathbf{ (-6 - x)^2 + 11^2 = 61^2}[/tex]

[tex]\mathbf{ (-6 - x)^2 + 121 = 3721}[/tex]

Subtract 121 from both sides

[tex]\mathbf{ (-6 - x)^2 = 3721 - 121}[/tex]

[tex]\mathbf{ (-6 - x)^2 = 3600}[/tex]

Take square roots of both sides

[tex]\mathbf{ -6 - x = 60}[/tex]

Add 6 to both sides

[tex]\mathbf{ - x = 60 + 6}[/tex]

[tex]\mathbf{ - x = 66}[/tex]

Multiply both sides by -1

[tex]\mathbf{ x = -66}[/tex]

Hence, the possible value of x is -66

Answers

Answer: Points A, D, and G are collinear, meaning they all lie on the same line.

Step-by-step explanation:

Which of the following situations are equal to 90%? Select all that apply.

A) 18 out of 24
B) 18 out of 20
C) 36 out of 40
D) 54 out of 60
E) 81 out of 90

Answers

It’s the letter c

40>36=4% 94%-4%=90

△ABC ~ △ADF.
What is the length of side AC?

Answers

yo you got this dude just keep trying!!!

help i forgot how to do this

Answers

Answer:

a=3/5 or 0.6

Step-by-step explanation:

a-8/5-2/3a=-7/5

Step 1: Multiply both sides

15a-24-10a=-21

Step 2: Combine like terms

5a-24=-21

Step 3: Calculate and divide both sides by 5

5a=3

Solution:

a=3/5 or 0.6

if u know any of these please let me know

Answers

number nine is A :) i hope that goes well ! :)

Luisa took a job in which her salary can be modeled by the exponential function
y = 36.000(1.025)", where x is the number of years after she is hired.
Damon took a job with a starting salary of $38.500 and yearly raises of $800.
Select EACH correct statement about the scenario.
The difference in the starting salaries is $1,600.
The difference in the starting salaries is $2,500.
The difference in the starting salaries is $2,000.
Luisa will make more money in the first 8 years.
Damon will make more money in the first 8 years.
They will make the same amount of money in the

Answers

Answer:

Step-by-step explanation:

38,500-36,000=2,500

B. The difference in the starting salaries is $2,500.

D. Luisa will make more money in the first 8 years.

The required scenario are,
(1) The difference in the starting salaries is $2,500.
(2) Damon will make more money in the first 8 years.

Given that,
Luisa took a job in which her salary can be modeled by the exponential function y = 36.000(1.025)", where x is the number of years after she is hired. Damon took a job with a starting salary of $38.500 and yearly raises of $800.

What is simplification?

The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.

here,
According to the question,
Difference in starting salary = 38500 - 36000 = $2500
Luisa's salary after 8 years = $43862.50
Damon's salary after 8 years = $45108.89

Thus, the required scenario is,
(1) The difference in the starting salaries is $2,500.
(2) Damon will make more money in the first 8 years.

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Pls I need to hand this up soon! Thanks

Answers

330 fish balls 600 meatballs

Answer:

405 fishballs and 405 meatballs

Step-by-step explanation:

Please see the attached pictures for the full solution.

this doesn’t make any sense I swear she didn’t teach us this and I’m so confused

Answers

I think it’s the first choice

The answer is 2
10/5=5

Solve for h.

19 =
h
4
+ 16

h =

Answers

Answer:

h=3/4

Step-by-step explanation:

19=h4+16

19-16=h4+16-16

3=h4

3/4=h4/4

3/4=h

h=3/4

Answer:

h = 3/4 or as a decimal it would be 0.75

Step-by-step explanation:

19 = h × 4 + 16

Reorder the terms

Use the communative property to reorder the terms

19 = 4h + 16

Move the terms

Move the variable to the left - hand side and change its side

19 - 4h = 16

Move the constant to the right - hand side and change its sign

-4h = 16 - 19

Calculate

calculate the difference of 16 minus 19

-4h = -3

Divide both sides of the eqaution by -4

Therefore your answer is h = 3/4 or if you want a decimal simplify 3/4 by dividing 3 and 4, which gives you 0.75

According to financial reports, the average household credit card debt in 2010 was $7768 and in 2012 was $7117. What is the rate of change of the credit card debt per year? If this trend were to continue, how much would you expect the average household credit card debt to be in 2016?

Answers

Answer: 1953

Step-by-step explanation: You need to subtract 7768 and 7117 to find the average change in 2 years; you get 651, divide this by 2 to find the amount of change in 1 year and you'll get 325.5, now if we take 325.5 times 6 (the amount of years until 2016) you will find 1953 as the answer.

Explain why:
expressing 0.78 as a fraction is (78/100).

Answers

Answer:

Because in 0.78 78 is in the hundreths place so you would place 78/100

Step-by-step explanation:

please explain how to use cancellation method with examples. I'm really confused...

Answers

Answer:

Cancellation it enables us to eliminate or get rid of one of the variables, so we can solve a more simplified equation.

Some textbooks refer to the elimination method as the addition method or the method of linear combination.

This is because we are going to combine two equations with addition!

Step 1

First, we align each equation so that like variables are organized into columns.

Step 2

Second, we eliminate a variable.

If the coefficients of one variable are opposites, you add the equations to eliminate a variable, and then solve.

If the coefficients are not opposites, then we multiply one or both equations by a number to create opposite coefficients, and then add the equations to eliminate a variable and solve.

Step 3

Thirdly, we substitute this value back into one of the original equations and solve for the other variable.

Example of how to apply the elimination method for solving systems of equations is attached.

Tìm cực trị của hàm số y = 2x3 - 6x + 2.

Answers

Step-by-step explanation:

câu này 2x3 hay 2^3 vậy bạn

Hi please I need some help

Answers

Answer:

Step-by-step explanation:

because 20÷1/4=5

so 5 ounces

and then the 3 ounces for yourself so 8 ounces taken away

20 - 8 = 1212 ounces left

The table shows a proportional relationship. What is the value of y when x = 29.4?

x

0.7

3.5

8.4

24.5

y

0.4

2

4.8

14

Answers

Answer:y=16.8

Step-by-step explanation:

proportional means that x/y= constant

so there the constant is 1.75

if x=29.4 then 29.4/y =1.75

y = 29.4/1.75

PLEASE HELP ASAP!!!!!!!!
<3

Answers

Answer:

Step-by-step explanation:

1. 9.3521x10^12

2. 8.07x10^-3

3. 8x10^8

4. 5.2x10^4

5. 4.8x10^4

6. 8.369x10^9

7. 9x10^14

8. 2x10^2

9. 2.44x10^-3

10. 9x10^4

Can anyone help please

Answers

Answer:

x=83

Step-by-step explanation:

Can someone help Find the value of f(-8)?

Answers

Answer:

Step-by-step explanation:

When y =f(-8) the answer is y = -2

£800 is divided between Natasha, Richard & Stephen so that Natasha gets twice as much as Richard, and Richard gets three times as much as Stephen. How much does Richard get?

Answers

Answer: Natasha = 480; Richard = 240, Stephen = 80

Step-by-step explanation:

Let N, R, and S stand for the amounts Natasha, Richard and Stephen receive. The problem tells us that:

N = 2R, and

R = 3S

We know that N+R+S = 800

Since N=2R and R=3S, N = 6S

N+R+S = 800

6S+3S+S=800

10S=800

S=80

R=3S or 240

N=2R or 480

Conclusion:

N gets 480

R gets 240,

and S gets 80

That adds to 800.

. . . AND the last conclusion is that I need to meet Natasha. Please send Stephen home.

25. 2x + 3y = 4
- 3x + 2y = 6
Demtermine if each of the lines are perpendicular

Answers

Answer:

the lines are perpendicular

Step-by-step explanation:

Simplify the expression:
-7n + -2 - 6

Answers

Answer:

-7n + -8

= -7n - 8

= -(7n + 8)

Step-by-step explanation:

brainliest plz

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