Let Q = (0,6) and R = (6,7) be given points in the plane. We want to find the point P=(x,0) on the x-axis such that the sum of distances PQ+PR is as small as possible. (Before proceeding with this problem, draw a picture!)To solve this problem, we need to minimize the following function of x:f(x) ??over the closed interval [a,b] where a=?? b=??We find that f(x) has only one critical number in the interval at x=??where f(x) has value??Since this is smaller than the values of f(x) at the two endpoints, we conclude that this is the minimal sum of distances.
Answers
To solve the problem, we need to find the value of x on the x-axis that minimizes the function f(x) = PQ + PR, where Q = (0,6) and R = (6,7) are given points in the plane.
Let P = (x,0) be the point on the x-axis. The distance between two points in the plane is given by the distance formula:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Using this formula, we can calculate the distances PQ and PR as follows:
PQ = √((x - 0)^2 + (0 - 6)^2) = √(x^2 + 36)
PR = √((x - 6)^2 + (0 - 7)^2) = √((x - 6)^2 + 49)
The function f(x) is the sum of distances PQ and PR:
f(x) = PQ + PR = √(x^2 + 36) + √((x - 6)^2 + 49)
To find the minimum value of f(x), we need to minimize this function over the closed interval [a,b].
Looking at the problem description, we can see that the point P lies between the x-coordinates of Q and R, which are 0 and 6, respectively. Therefore, the interval is [a,b] = [0,6].
To find the critical numbers of f(x), we need to find the values of x where the derivative of f(x) is equal to zero or does not exist. Let's find the derivative:
f'(x) = (1/2)(2x)/(√(x^2 + 36)) + (1/2)(2(x - 6))/(√((x - 6)^2 + 49))
= x/(√(x^2 + 36)) + (x - 6)/(√((x - 6)^2 + 49))
To simplify further, we can multiply the numerator and denominator of the second term by (√(x^2 + 36)):
f'(x) = x/(√(x^2 + 36)) + (√(x^2 + 36))/(√(x^2 + 36)) * (x - 6)/(√((x - 6)^2 + 49))
= x/(√(x^2 + 36)) + (√(x^2 + 36)(x - 6))/(√((x^2 + 36)((x - 6)^2 + 49)))
= (x(x^2 + 36) + (√(x^2 + 36)(x - 6)))/√((x^2 + 36)((x - 6)^2 + 49))
To find the critical number, we set f'(x) equal to zero and solve for x:
x(x^2 + 36) + (√(x^2 + 36)(x - 6)) = 0
Simplifying and rearranging the equation, we get:
x^3 - 6x^2 + 36x - 6√(x^2 + 36) = 0
Unfortunately, this equation does not have a simple algebraic solution. We can use numerical methods such as graphing or iteration to find an approximate solution.
In this case, we can use graphing software or a graphing calculator to plot the function f(x) and find the x-value where the function reaches its minimum.
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Related Questions
Remember combine like terms :)
14 - k + 3k - 6 =
Answers
Answer:
8 + 2kStep-by-step explanation:
14 - k + 3k - 6
= 14 - 6 - k + 3k
= 8 + 2k (Ans)
Answer:
2k+8
Step-by-step explanation:
I did combine like terms hope this was helpful
To make a mixture that tastes the same as the original recipe, how much pineapple juice should you mix with 1 cup of orange juice?
Answers
Recipes 1 and 2 have the same proportion of orange juice to pineapple juice, whereas recipe 3 has a different proportion.
The recipes that would taste the same are recipe 1 and 2. Recipe 3 would taste different.
Recipe 1: ratio of orange juice to pineapple juice = 4 : 6
2 : 3
Recipe 2: ratio of orange juice to pineapple juice = 6 : 9
2: 3
Recipe 3: ratio of orange juice to pineapple juice = 9 : 12
3 : 4
Thus, Recipes 1 and 2 have the same proportion of orange juice to pineapple juice, whereas recipe 3 has a different proportion.
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The Complete Question is:
Here are three different recipes for Orangy-Pineapple juice. Two of these mixtures taste the same and one tastes different.
Recipe 1: Mix 4 cups of orange juice with 6 cups of pineapple juice.
Recipe 2: Mix 6 cups of orange juice with 9 cups of pineapple juice
Recipe 3: Mix 9 cups of orange juice with 12 cups of pineapple juice
Which two recipes will taste the same, and which one will taste different? explain or show your reasoning.
I am less than 20. I am 2 inches more than a foot. Who am I?
Answers
EXPLANATION:
what is the measure of one exterior angle of a polygon with 26 number of sides? round to one decimal place
Answers
Answer: The sum of the exterior angles of any polygon is 360 degrees.
The measure of one exterior angle of a regular polygon with n sides is given by the formula:
360/n
For a polygon with 26 sides, the measure of one exterior angle is:
360/26 = 13.8 degrees (rounded to one decimal place)
Step-by-step explanation:
Finding the surface area of rectangle cut out with dimensions
Answers
The surface area of the rectangle cut out with dimensions 3cm x 2cm is 14cm².
The surface area of the larger rectangle is:
SA = 2(10 cm x 12 cm + 10 cm x 2 cm + 12 cm x 2 cm)
SA = 2(120 cm² + 20 cm² + 24 cm²)
SA = 288 cm²
The surface area of the cut-out rectangle is:
SA = 5 cm x 8 cm
SA = 40 cm²
The surface area of the remaining rectangle is:
SA = 288 cm² - 40 cm²
SA = 248 cm²
Therefore, the surface area of the rectangle after cutting out a 5 cm by 8 cm rectangle is 248 cm².
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Complete question:
Finding the surface area of rectangle cut out with dimensions 5 cm by 8 cm from a rectangle with dimensions 10 cm by 12 cm.
Ashton designs movie sets. For one scene, he has to hang a banner across an archway that takes the shape of a parabola modeled by the quadratic equation y= -x^2+ 6x. The banner is to be hung at an angle defined by the linear equation y-x-4, as shown in the figure
Aston needs to know how ling a banner to order. He can find the length using the distance formula if he knows the points in the archway at which ends of the banner are to be attached.
Using the graphing method, find the points at which the equation of the angle of the banner will intersect the equation of the archway
Answers
Answer: The banner must be 4.24 units long
a. Using the graphing method, find the points at which the equation of the angle of the banner will intersect the equation of the archway.
Sample answer:
Please see attached screenshot
Step-by-step explanation:
The two equations intersect at (1, 5) and (4, 8), so the banner should be hung at these two points.
b. Using the algebraic method, confirm whether the two points obtained by the graphing method are correct.
Sample answer:
Given these two equations:
y = -x2 + 6x (1)
y – x = 4 (2)
Modify equation (2) to get y = x + 4.
To find the points of intersection, equate the two equations:
-x2 + 6x = x + 4
x2 - 5x + 4 = 0
(x – 4)(x – 1) = 0
x = 4 or x = 1.
Substitute the values of x in equation (2) to get the values of y:
y = x + 4
= (4) + 4
= 8.
y = x + 4
= (1) + 4
= 5
The two points of intersection are (4, 8) and (1, 5). These are the same points obtained by graphing.
c. Using the distance formula find the distance between points A and B. How long a banner does Ashton need to order?
Sample answer:
The distance formula is
x1 = 4
y1 = 8
x2 = 1
y2 = 5
Substituting these values in the formula:
d = 4.24 units.
The distance between the two points of intersection of the archway and the banner is 4.24 units, so the banner must be 4.24 units long.
Find the greatest common factor of the following monomials 30c 8c6
Answers
Answer:
2c
Step-by-step explanation:
The greatest common factor is the largest integer factor two numbers have in common. A faster way to find it is to prime factorize the number, but in this case, it is easier to just list out the factors;
30c = 2 * 15 * c
8\(c^{6}\) = 2 * 4 * c * \(c^5\)
Greatest common factor;
2 c
at a football stadium 4% of the fans in the attendance were teenagers if there were 390 teenagers at the football stadium ,what was the total number of people at the stadium
Answers
The total number of people in the football stadium is 9750.
How to find the total number of people in the stadium?At the football stadium, 4 percent of the fans in the attendance were teenagers.
There are 390 teenagers at the football stadium. The total number of people in the stadium is as follows:
let
x = total number of people in the stadium
Hence,
390 = 4% of x
390 = 4 / 100 × x
390 = 4x / 100
cross multiply
39000 = 4x
divide both sides of the equation by 4
x = 39000 / 4
x = 9750
Therefore,
total number of people in the stadium = 9750
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in a soccer league with four teams, each team played four games with every other team. each team received $3$ points for a win, $1$ point for a tie, and no points for a loss. after all the games, the point tallies were as follows: team a won $22$ points, team b won $19$ points, team c won $14$ points, and team d won $12$ points. how many games ended in a tie?
Answers
Answer:
Step-by-step explanation:
Here, 4 teams given. Each teams has played every other teams 4 times.
so here we will name each team
team a = bulls
team b = lakers
team c = knickers
team c = warriors
So to find total number of games, we know that Bulls played with the other 3 teams by 4 times that is (4×3) = 12, Lakers played with the other two teams by 4 times that is (4×2) = 8, Knicks played with Warriors 4 times.
Therefore, total number of games = .
Total points given, for Bulls = 22, for Lakers = 19, for Knickers = 14, for Warriors = 12.
So sum of total points =
For a win a team earned 3 points and for a tie two teams win 1 point.
That means for tie total point = 1+1 = 2.
Let's take total number of game which ended as win played be x and total number of game which ended as a tie be y.
We have got total number of games is 24.
So we can write the equations as,
.......Equation 1
And also we have got total number of points is 67. So the equation is,
.......Equation 2
From equation 1, if we move x to the right side we will get,
Now let's substitute this value in equation 2, to get the value of x. By substituting the value we will get,
We will expand 2 now.
We will move 48 to the other side by subtracting it from both sides. We will get,
So the number of games which ended as a win = 19
The number of games which ended as a tie = 24 -19 =5
So we have got the required answers.
u can just check the answer and this answer is copy righted from its rightful owner
The gross weekly pay is $424. His earnings to date for the year total $11.700. What amount is deducted from her pay this week for social security, which is taxed at 6.4 percent?
A. $748.80
B. $0
C. $424
D. $25.44
E. $27.14
Answers
Answer:424
Step-by-step explanation:
i don't know how to explain it but this seems the most correct one, her weekly gross pay is 424 and taxes end up getting taken out of the paycheck not earnings.
HELP PLEASE!!
Which expression is equivalent to 9a + 12? (1 point)
3(3a + 2)
3(6a + 9)
3(3a + 4)
3(6a + 3)
Answers
3(3a+4)
This is because after doing the math 3x3 would turn out to be 9a, and 3x4 would turn out to be 12, making it equivalent to the expression 9a+12. I hope this helps
solve 3x > -12 and graph the solutions
Answers
Answer:
2
Step-by-step explanation:
3x>-12
3x/3>-12/3
x>-4
Sixteen batteries are tested to see if they last as long as the manufacturer claims. Four batteries fail the test. Two batteries are selected at random without replacement. Find the probability that both batteries pass the test.
Answers
The probability that both batteries pass the test is; 11/20
Solving probability questionsTotal Number of Batteries tested = 16
Number of Batteries that fail the Test = 4
Number of batteries selected at Random = 2
Probability that both batteries fail the test = (4/16) * (3/15)
Probability that both batteries passed the test = 12/16 * 11/15 = 11/20
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Find the product and the quotient. 1/2 x 1/5 and 1/5 divided by 2
Answers
Answer:
1/2×1/5=1/10
1/5÷2=1/5×1/2=1/10
Answer:
1/2 x 1/5 would be 1/10 and 1/5 divided by 2 would be 1/10
Step-by-step explanation:
A sector of a circle has a central angle to 10° and arc length 28 pi units what is the radius of the circle
Answers
Answer:
Radius = 504units
Step-by-step explanation:
Angle = 10°
Length of arc = 28π units
L = ∅/360 × 2πr
28π = 10/360 × 2πr
28π = 1/36 × 2πr
2πr = 28π × 36
π cancelled off.......
2r = 28 × 36
r = (28 × 36)/2
r = 28 × 18
r = 504units
The radius of circle is 504 unit.
What is sector?A sector is referred to as a component of a circle made up of the circle's arc and its two radii. It is a section of the circle made up of the arc's circumference and the radii of the circle at its ends. A slice of pizza or a pie can be used as an analogy for the form of a circle's sector.
We have,
arc length = 28π unit
Central angle = 10°
so, length of Arc
28 π= 2πr (10/360)
28 = 2r (1/36)
14 = r/36
r= 504 units
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Solve the expression 9 + (20 x one fourth) − 6 ÷ 2 using PEMDAS. (1 point)
8
11
13
16
Answers
Answer:
B) 11
Step-by-step explanation:
PEMDAS=parenthesis, exponents, multiplication, division, addition, and subtraction.
First start with the parenthesis:
20 x 1/4=5
9 + (5) - 6 ÷ 2
We don't have exponents or multiplication, so go onto division:
-6 ÷ 2 = -3
9+5-3
Finally addition and subtraction:
9+5-3=11
Hope this helps!
Can someone please help me ASAP?? It’s due tomorrow!! I will give brainliest If It’s correct.
Answers
Answer: To match the shapes produced by the slice through the triangular prism, we need to consider the orientation of the slice relative to the prism. Here are the matching options:
A. Perpendicular to the base: Rectangle
B. Parallel to the base: Triangle with dimensions equal to the base
C. Diagonal from vertex to vertex: Triangle with unknown dimensions
Integrate by hand the following functions: adr b) (42³-2r+7) dz Upload Choose a File
Answers
The integral of (42³ - 2r + 7) dz is equal to (42³ - 2r + 7)z + C.
To integrate the function (42³ - 2r + 7) dz, we treat r as a constant and integrate with respect to z. The integral of a constant with respect to z is simply the constant multiplied by z:
∫ (42³ - 2r + 7) dz = (42³ - 2r + 7)z + C
where C is the constant of integration.
Note: The integral of a constant term (such as 7) with respect to any variable is simply the constant multiplied by the variable. In this case, the variable is z.
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pls, help me it has been a missing assignment for too long :/
Answers
Answer:
Snail A: 0, 5, 10, 15, 20, 25, 30
Snail A travels at a rate of 30 inches in 1 hour or 30 inches per hour.
Snail B: 0, 3, 6, 9, 12, 15, 18
Snail B travels at a rate of 18 inches in 1 hour or 18 inches per hour.
Snail C: 0, 6, 12, 18, 24, 30, 36
Snail C travels at a rate of 36 inches in 1 hour or 36 inches per hour.
Snail D: 0, 4, 8, 12, 16, 20, 24
Snail D travels at a rate of 24 inches in 1 hour or 24 inches per hour.
PLEASE HELP ME PLEASEEE I'll give brainliest if correct please help me...
Answers
Answer:
60 mph
50 wpm
$36.25 dollars an hour
Hope this helps :)
Answer:
120 miles in 2 hours = 60 miles per hour.
150 words per 3 min = 50 words per min
145 dollars in 4 hours = 36.25 dollars per hour
Step-by-step explanation:
Hope this helps! :)
please help_________
Answers
Answer:
\(11\sqrt{797}\pi\)
Step-by-step explanation:
\(A=\pi r \sqrt{h^2+r^2}\)
the radius is 22/2 = 11 in
h is 26 in
substitute: \(\pi \left(11\right)\sqrt{\left(26^2\right)+\left(11^2\right)}\)
= \(11\sqrt{797}\pi\)
Answer:
286π in²
Step-by-step explanation:
The lateral area (A) of the cone is calculated as
A = πrs ( r is the radius and s the slant height )
Here r = 22 ÷ 2 = 11 and s = 26 , then
A = π × 11 × 26 = 286π in²
The graph represents the system of inequalities shown below. Determine whether the statement that follows the
graph is true or false.
y greater than or equal to 6
5x+3y ≤ 15
f(x,y) - 12x+3y
Answers
Assuming that x ≥ 0 and y ≥ 0, the situation is infeasible: True.
What are the rules for writing an inequality?In Mathematics, these rules are generally used for writing and interpreting an inequality or system of inequalities that are plotted on a coordinate plane:
The line on a graph is a solid line when the inequality symbol is (≥ or ≤).The inequality symbol is greater than or equal to (≥) when a solid line is shaded above.The inequality symbol is less than or equal to (≤) when a solid line is shaded below.In this context, we can logically deduce that the given system of inequalities 5x+3y ≤ 15 and y ≥ 6 would not have a feasible solution set if they are subjected to x ≥ 0 and y ≥ 0:
Assuming the ordered pair (1, 2), we have:
y ≥ 6
2 ≥ 6 (False).
5x+3y ≤ 15
5(1) +3(2) ≤ 15
5 + 6 ≤ 15
11 ≤ 15 (True).
Therefore, it is not a feasible solution.
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Missing information:
The question is incomplete and the complete question is shown in the attached picture.
let b=b1,b2 and c=c1,c2 be bases for ℝ2. find the change-of-coordinates matrix from b to c and the change-of-coordinates matrix from c to b. b1= −3 −4 , b2= 2 3 , c1= 1 4 , c2= 3 4
Answers
The change-of-coordinates matrix from c to b is:
Q = [−2/5 3/5; 4/5 4/5]
How to find the change-of-coordinates matrix from b to c?To express the basis vectors of c in terms of the basis vectors of b.
This can be done by solving the equation [c1, c2] = [b1, b2]P, where P is the change-of-coordinates matrix from b to c.
We can write out the equation in components as follows:
1 3 = -3x + 2y
4 4 -4x + 3y
Solving this system of equations, we get x = -4/5 and y = 3/5. Therefore, the change-of-coordinates matrix from b to c is:
P = [−4/5 3/5; 2/5 1/5]
To find the change-of-coordinates matrix from c to b, we need to express the basis vectors of b in terms of the basis vectors of c.
This can be done by solving the equation [b1, b2] = [c1, c2]Q, where Q is the change-of-coordinates matrix from c to b.
We can write out the equation in components as follows:
-3 2 = 1x + 3y
-4 3 4x + 4y
Solving this system of equations, we get x = -2/5 and y = 3/5. Therefore, the change-of-coordinates matrix from c to b is:
Q = [−2/5 3/5; 4/5 4/5]
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Answers please help me
Answers
Answer:
Rita: r - 5= 2c, or r - 5= 2(36), or r - 5 = 72
Cora: (r - 5)/2 = 36
Step-by-step explanation:
Rita had r apps (let r = Rita original apps)She deleted 5 of them so r-5 She now has twice the number of apps as Cora so r-5 = 2c (let c = Cora's total apps) Cora's total apps are 36 so Rita's equation is r-5= 2(36) which means Rita equation for her number of apps is r-5= 72. As for Cora her equation is c= 36 but in relation to Rita's number of apps it's (r-5)/2 = 3680% of grade 8 students, that is, 60 students want a new ice-cream shop in the
cafeteria. The total number of grade 8 students are?
Answers
Answer:
48
Step-by-step explanation:
60 students X 80%=48
20% is 12 students
40% is 24 students
60% is 36 students
so on and so forth.
Answer:
48.
Step-by-step explanation:
.80 x 60 = 48
analysts should not worry about the users' perceptions of the new system during acceptance testing. true or false?
Answers
False. Analysts should definitely worry about the users' perceptions of the new system during acceptance testing. Acceptance testing is the final stage of software development where the system is tested by end-users to ensure that it meets the specified requirements and is ready for deployment.
During acceptance testing, analysts need to ensure that users find the system easy to use, efficient, and effective in performing the intended tasks. If the users perceive the system to be difficult to use or inefficient, it can lead to user resistance, decreased productivity, and increased costs. Therefore, analysts need to gather feedback from users during acceptance testing and make necessary adjustments to ensure that the system meets the user's expectations. By doing so, analysts can ensure a successful deployment and adoption of the new system.
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What is 3x+4=10 solve for x
Please help me
Answers
Answer:
x=2
Step-by-step explanation:
3x+4=10
3x=10-4
3x=6
x=6/3
x=2
please also explain how
(3) How do you cut a 14 inch pizza into three pieces of equal area using two parallel cuts? (Assume the cuts are placed symmetrically from the center.)
Answers
To cut a 14-inch pizza into three equal area pieces using two parallel cuts placed symmetrically from the center, each piece will have an area of 150.
As we need to cut a 14-inch pizza into three pieces of equal area using two parallel cuts, we have to follow the steps given below
:Step 1: Cut the pizza with a line that goes through the center of the pizza and marks its diameter. This cut separates the pizza into two equal halves.
Step 2: The second cut needs to be made parallel to the first cut and needs to be at a distance of approximately 1/3 the diameter of the pizza from the first cut.
Step 3: Then, the pizza will be separated into three equal area pieces as required. As we have to cut the pizza into three equal area pieces, the area of each piece will be 150 square inches.
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Help ASAP! Use the graph to answer the question
Answers
Answer:second 1
Step-by-step explanation:
Answer:
The blue line models the data points better than the red line. It is because the data points are all close to the line.
Fill in the blank.
(5x)2=___ x2
Answers
Answer:
5 is the answer
because it is commutative property