Given point P moving along a curve y = x²-1 and point Q moving along a line y = x-3. Find the coordinate of point Q and distance between P and Q that makes the distance between two points have minimum value. Show your work, thank you!
Answers
Answer:
Part A
The coordinates of point Q that makes the distance between the two points have minimum value is (9/8, -15/8)
Part B
The distance between P and Q at the closest point is approximately 1.287 units
Step-by-step explanation:
From the details given in the question, we have;
The curve along which point P is moving is, y = x² - 1
The (straight) line along which point Q is moving is, y = x - 3
The domain of the path of point P = All real numbers
The range of the path of point P = -1 < x ≤ ∞
The domain of the path of point Q = All real numbers
The range of the path of point Q = All real numbers
From the graph of the functions created with Microsoft Excel, we have that from point (0, 0) and moving to the positive x-axis, the slope of the path of P increases from 0 to ∞, while the slope of the path Q is constant and equal to 1
Given that the path of P is convex relative to the path of P, the two paths will be closest when they have equal slopes
Therefore at the closest point, for the path of point P, we have;
d(x² - 1)/dx = 1 (the slope of the path of point Q)
2·x = 1
∴ x = 1/2 on the path of point P at the closest point the path of point P to the path of point Q
∴ y = x² - 1 = (1/2)² - 1 = -3/4
y = -3/4
Given that two paths are parallel at the closest point, distance between the two paths is the perpendicular distance from the line y = x - 3 to the point (1/2, (-3/4)), which is given as follows;
∴ The slope of the line representing the perpendicular distance between the two paths = -1
Therefore, the equation of the line in point and slope form is given as follows;
y - (-3/4) = -1 × (x - 1/2)
y = -x + 1/2 - 3/4 = -x - 1/4
y = -x - 1/4
At the coordinates of point Q that makes the distance between the two points have minimum value, we have;
-x - 1/4 = x - 3
∴ 2·x = 3 - 3/4 = 2 1/4 = 9/4
x = 9/8
y = x - 3 = 9/8 - 3 = -15/8
Therefore, the coordinates of point Q that makes the distance between the two points have minimum value = (9/8, -15/8)
Part B
The formula for the length between two points is
[tex]l =\sqrt{\left (y_{2}-y_{1} \right )^{2}+\left (x_{2}-x_{1} \right )^{2}}[/tex]
The distance, d, between P and Q at the closest point is given as follows;
d = √(((9/8) - (1/2))² + ((-15/8) - (-3/4))²) = √(53/32) ≈ 1.287
The distance between P and Q at the closest point, d ≈ 1.287
Related Questions
Simplify the expression:
[tex]\left(\frac{1}{2}\right)^{2}\cdot\frac{9}{11}+\frac{2}{3}[/tex]
Plz give steps.
Answers
Answer:
[tex]\frac{115}{132}[/tex]
Step-by-step explanation:
[tex](\frac{1}{2} )^{2} * \frac{9}{11} + \frac{2}{3}[/tex]
= [tex]\frac{1}{4} * \frac{9}{11} + \frac{2}{3}[/tex]
= [tex]\frac{9}{44} + \frac{2}{3}[/tex]
= [tex]\frac{27 + 88}{3(44)}[/tex]
= [tex]\frac{115}{132}[/tex]
Help me please it’s due today and I really need this
Answers
Answer:
J
Step-by-step explanation:
If you look closely, you can see the slope line is decreasing which means all the positive slopes are automatically wrong, and second the first point the line intersects with the graph is 6 down which means the answer is -6
Explain what sampling is and how it is used to estimate a population?
In 5-7 sentences please
I will give Brainliest
Answers
how to use pythagoreom therom
Answers
Answer:
a² + b²= c²
Step-by-step explanation:
In these pictures you will see how to do pythagoreom therom in different situations.
PLEASE HELP!
A camper is flying a kite at the end of a string 250 feet long. If they look up to
the kite at an angle of 43 degrees, what is the approximate height of the kite? (Round
your answer to the nearest foot.).
Answers
hypotenuse is 250 feet.
base angle is 43 degrees.
trignometery here I come!
Since we must find the height, aka adjacent side of right triangle, and we have the given hypotenuse
We shall use the sine!
sinθ is sin43
sin43 = A/H = x/250
multiply both sides by 250
250 (sin43) = x
DEGREE MODE on graphing calculator! [because the θ is in degrees mode]
x ≈ 171
Answer: x ≈ 171 feet
NEED HELP ASAP
A ticket broker is selling seats to a rock concert. If the broker's commission is a $5.00 base fee and $0.50 extra for every ticket (t), the total cost for the broker's services is calculated by the expression below.
$0.50t + $5.00
How much does the broker earn by selling 12 tickets?
$5.50
$10.50
$11.00
$15.50
Answers
Answer:
11.00
Step-by-step explanation:
"T" is the number of tickets sold so take 0.50*12 and you would get 6.00
then take 6+5=11, then 11 is the answer
Which equation represent the nth term of the arithmetic sequence?
Answers
Answer:
D. aₙ = 5n - 11Step-by-step explanation:
The equation of the nth term of the arithmetic sequence is: [tex]a_n =a_1+d(n-1)[/tex]
d = 9 - 4 = 5
a₁ = -6
Therefore:
[tex]a_n =-6+5(n-1)\\\\a_n =-6+5n-5\\\\a_n =5n-11[/tex]
3+b^3+b^2 in standard form
Answers
a hot air balloon travels 18 miles in 3 hours. At this rate how many miles will the hot air balloon travel in 3/4 hour
Answers
Answer:
4 miles
Step-by-step explanation:
Answer:
4.5 miles
Step-by-step explanation:
18 miles 3 hours = 6 miles/hour
6 x (3/4) = 4.5 miles
If there are 60 seconds in an minute, 60 minutes in an hour, 24 hours in a day, and 7 days in a week, how many seconds are in a week? Hint: Use a calculator to solve this problem.James and Kelly have $10 to spend at a grocery store. If they get 3 avocados which cost $1 each, 4 tomatoes which cost a total of $2 for all 4, and two loaves of bread which cost $2 each, how much change will James and Kelly receive after their purchase?
Answers
Answer:
Part a) 60 seconds in a minute, 60 minutes in an hour, 24 hours in a day, and 7 days in a week, how many seconds are in a week
60x60=3600 x 24= 86400 x 7= 604800
So 604,800 seconds in a week
Part b) James and Kelly have $10 to spend at a grocery store. If they get 3 avocados which cost $1 each, 4 tomatoes which cost a total of $2 for all 4, and two loaves of bread which cost $2 each, how much change will James and Kelly receive after their purchase?
So 3 x 1= $3
+ $2
2 x 2= $4
3+2+4= $9
10-9= $1
So they'll get back $1
Help me plz this is to much
Answers
9514 1404 393
Answer:
1 triangle: 2 m²2 triangles: 4 m²large rectangle: 32 m²small rectangle: 3 m²total area: 39 m²cost of tile: $487.50Step-by-step explanation:
You will run across many examples of poorly edited curriculum materials. This is one of them. We presume that every instance of "inches" should be an instance of "meters". The difference between in and m is not always obvious in a diagram, and m apparently got misinterpreted here.
__
The area of a triangle is given by the formula ...
A = 1/2bh
where b is the base and h is the height perpendicular to the base.
The triangles in this figure have a base of 4 m, and a height of 1 m. Using this formula, we find the area of 1 triangle to be ...
A = 1/2(4 m)(1 m) = 2 square meters . . . . area of 1 triangle
__
The triangles are the same size, so the area of both together is ...
2(2 m²) = 4 square meters . . . . area of both triangles together
__
The large rectangle is 8 m long and 4 m high. Its area is given by the formula ...
A = LW
where the length is L and the width is W. Using this formula, we find the area of the large rectangle to be ...
A = (8 m)(4 m) = 32 m² . . . . area of large rectangle
__
The small rectangle has a length of 3 m and a width of 1 m. Its area is ...
A = (3 m)(1 m) = 3 m² . . . . area of small rectangle
__
The total area is the sum of the areas of the pieces:
total area = 2 triangles + large rectangle + small rectangle
total area = 4 m² + 32 m² + 3 m² = 39 square meters
_____
At $12.50 per square meter, the cost of covering the patio is ...
($12.50/m²)(39 m²) = $487.50 . . . . cost of tile
Question 7 of 12 Solve: 3x- 8 = 6(1 - 2x) + 16 O A. x = 2 O B. x = 5 O C. x= -2 O D. There are infinitely many solutions.
Answers
Answer:
A. x =2
Step-by-step explanation: Rearrange terms 3x - 8 = 6 ( 1 - 2x ) + 16 ---> 3x - 8 = 6 ( -2x + 1 ) +16
Distribute
Add the numbers
Add 8 to both sides of the equation
Simplify
Add 12x to both sides of the equation
Simplify
Divide both sides of the equation by the same term
Simplify -------> x=2
please help me, I’ve been on this for 30 mins + I’m an idiot
Answers
I think it's 3/7?
hope this helps
Solve for x. Round to the nearest tenth.
Answers
good luckkkk
Answer:
x=10.4
Step-by-step explanation:
tan(60°)=18/x
tan(60°)x=18
x=18/tan(60°)
x=10.4
Plz answer the question if u want 30 points.
A marketing company is conducting a survey about snack foods. The surveyors will call a number of households in one city to question them about their snacking preferences. Which strategy should the company use in order to have an accurate collection of survey responses relating to the preferences of the general population of the city?
select and call phone numbers at random
select and call phone numbers at random
offer money to people to answer a survey
offer money to people to answer a survey
call customers of a single busy supermarket
call customers of a single busy supermarket
call people who work at restaurants or grocery stores
Answers
2x[tex]2x^{2}-3x-5[/tex]
Answers
Answer:
2x²-3x-5
=2x²-3x-5
=2x²-5x+2x-5
=2x(x-5)+2(x-5)
=(x-5)(2x+2)
Answer:
(x + 1)(2x - 5)
Step-by-step explanation:
Assuming you require the expression factorised.
2x² - 3x - 5
Consider the factors of the product of the x² term and the constant term which sum to give the coefficient of the x- term.
product = 2 × - 5 = - 10 and sum = - 3
The factors are + 2 and - 5
Use these factors to split the x- term
2x² + 2x - 5x - 5 ( factor the first/second and third/fourth terms )
= 2x(x + 1) - 5(x + 1) ← factor out (x + 1) from each term
= (x + 1)(2x - 5) ← in factored form
Here is a graph of the relationship between the height and volume of some cylinders that all have the same radius, R. An equation that represents this relationship is V =Rºh (use 3.14.as an approximation for 7). What is the radius of these cylinders?
Answers
Answer:
Radius is 1 unitStep-by-step explanation:
The relationship is:
V = πr²hWe have points on the graph:
(9, 28.26) and (18, 56.52)Use one of the points, substitute into equation and solve for r:
28.26 = 3.14*9*r²r² = 28.26/28.26r² = 1r = 1which list shows these numbers in order from least to greatest
Answers
Answer:
B) -5, -2.4, 20%, sqrt(9), 3pi
Step-by-step explanation:
sqrt(9) = 3
3pi = 3 * 3.14 = 9.42
20% = 0.2
Answer: B) -5, -2.4, 20%, sqrt(9), 3pi
i think it is "b" but im not 100% sure because im learning this right now too. i hope this helped :)
Refer to the diagram below. What is the scale factor used to create the new shape.
Answers
Answer:
The correct answer is A: 4/5
Step-by-step explanation:
We would divide 24/30 and get 4/5
PLEASE HELP!! 50 POINTS
Two Quadratic functions are represented below
f(x)=-x^2+4x-3
a. In two or more complete sentences, explain how you can determine the maximum value of the
quadratic functions, f(x) and g(x).
b. Which function has the greater maximum value, f(x) or g(x)?
Answers
You can tell that the maximum value that [tex]g[/tex] attains is roughly [tex]4[/tex].
Rewriting [tex]f[/tex] in vertex form: [tex]f(x)=-(x-2)^2+1[/tex]. This tells us that the maximum of [tex]f[/tex] is [tex]1[/tex].
So [tex]g[/tex] has the greater maximum.
Answer:
g(X)
Step-by-step explanation:
please help, this is very urgent!!! no links please, and also if ur answer is correct I'll give brainliest!!!!
Answers
Answer:
I) 616
II) 4.53
Step-by-step explanation:
For the first one, just plug in 7 as R.
A = 4 x (22/7) x 7^2
A = 12.57142857 x 49
A = 616
For the second, one, put 452 4/7 as A.
452 4/7 also is 258.285
258.285 = 4 x (22/7) x r^2
258.285 = 12.57142857 x r^2
Divide both sides by 12.57142857
20.55372408 = r^2
Square root 20.55372408 for R.
4.533621519 = R
OR
4.53 = R
Match each technique with its purpose.
reversal
to directly mimic something for comedy
ridicule
to present something as the opposite of
what is expected or normal
to criticize a subject by making it seem
ridiculous
parody
Answers
Parody: to directly mimic something for comedy
Reversal: to present something as the opposite
The terms are matched is Exaggeration: Makes something seem too good or too bad.
What are Literary terminology?Literary terminology refers to the techniques, styles, and formats used by writers and speakers to skillfully emphasize, decorate, or enhance their composition.
Reversal: Changes circumstances suddenly
Exaggeration: Makes something seem too good or too bad
Parody: Mimics a subject directly
Ridicule: Makes a subject appear to be silly
Satire: Exposes human weakness
Hence, the sentences are matched above according to Literary terminology. It can refer to the playful techniques that comedians use to make us laugh, or the witty tricks that Wordsmith uses to shape new words and phrases in their composition.
Therefore, the terms are matched is Exaggeration: Makes something seem too good or too bad.
Learn more about the Literary terms here:
brainly.com/question/19034685.
#SPJ7
The graph of an equation is shown below:
Based on the graph, which of the following represents a solution to the equation?
Answers
Answer:
(-2, -3)
Step-by-step explanation:
solutions are all the points where the line intersects
I will mark you brainlist!
A. 24.12
B. 50.24
C. 58.14
D. 96.02
Answers
What is the factored form of the polynomial 3x-3y
Answers
Answer:
Step-by-step explanation:
3x - 3y = 3(x-y)
During the 2 9-weeks reporting, a student's grade increased from 80 to 96. By what percentage did the grade increase?
32%
20%
16%
12%
Answers
The first term of a geometric sequence is 4, and the common ratio is 5. What is the 5th term of the sequence?
a.2,500
b.12,500
c.1,024
d.500
Answers
Answer:
A) 2,500
Step-by-step explanation:
Soo.....
A1 = 4
A2 = 4*5 = 20
A3 = 20*5 = 100
A4 = 100*5 = 500
A5 = 500*5 = 2,500
The perimeter of the triangle is 29 cm.
a. Write an equation and solve for x.
b. Use the value of x to find the lengths of the sides of the triangle.
c. What kind of triangle is this: scalene, isosceles, or equilateral? Explain how you know using the calculated side lengths, not the picture
2x+2
3x - 1
x + 4
Answers
Answer:
Step-by-step explanation:
Equation
2x + 2 + 3x - 1 + x +4 = 29 Collect like terms on the left
6x + 5 = 29 Subtract 5 from both sides
6x = 29 - 5
6x = 24 Divide by 6
x = 24/6
x = 4
Length of the Sides.
2x + 2 = 2*4+2 = 10
3x - 1 = 3*4 - 1 = 12 - 1 = 11
x + 4 = 4 + 4 = 8
Type of Triangle
Scalene.
Simplify fully 8x/12x2y
Answers
Answer:
2/3xy.
Step-by-step explanation:
8x/12x^2y
The 8 and 12 simplify to 2/3 and x and x^2 simplify to 1/x so we have:
= 2/3xy.
Answer please I’m stew
Answers
Answer:
1st box=18 days
2nd box= 4 miles
Step-by-step explanation:
1st box is the number of dots
=18 days
2nd box is the dot that is on the max value
= 4 miles
Second box: 4
The number of days on this chart are represented by the number of dots.
Each dot represents one day and the miles Zac ran. The highest number on the chart with a dot is 4.
Please let me know if you have any questions.