Let [tex]s_n[/tex] denote the number of seats in the [tex]n[/tex]-th row. [tex]s_n[/tex] is arithmetic, so
[tex]s_n=s_{n-1}+d[/tex]
for some constant [tex]d[/tex].
We're told [tex]s_5=22[/tex] and [tex]s_{10}=37[/tex], so that
[tex]s_{10}=s_9+d[/tex]
[tex]s_{10}=(s_8+d)+d=s_8+2d[/tex]
[tex]s_{10}=(s_7+d)+2d=s_7+3d[/tex]
and so on up to
[tex]s_{10}=s_5+5d\implies37=22+5d\implies5d=15\implies d=3[/tex]
The pattern continues:
[tex]s_5=s_4+3[/tex]
[tex]s_5=(s_3+3)+3=s_3+2\cdot3[/tex]
and so on up to
[tex]s_5=s_1+4\cdot3\implies22=s_1+12\implies\boxed{s_1=10}[/tex]
Answer:
8
Step-by-step explanation:
Which of the following is a solution of y - x < -3?
A. (6,2)
B. (2,6)
C. (2,-1)
Thanks
Answer:
Option A. (6,2)
Step-by-step explanation:
We have the following inequality:
[tex]y- x <-3[/tex]
Solving for y we have:
[tex]y<x-3[/tex]
The line that limits the region of inequality is
[tex]y = x-3[/tex]
Then the region of inequality are all values of y that are less than [tex]f (x) = x-3[/tex]
In other words, the points belonging to the inequality are all those that lie below the line.
To find out which point belongs to this region substitute inequality and observe if it is satisfied
A. (6,2)
[tex]2<6-3[/tex]
[tex]2<3[/tex] is satisfied
B. (2, 6)
[tex]6<2-3[/tex]
[tex]6<-1[/tex] it is not satisfied
C. (2, -1)
[tex]-1<2-3[/tex]
[tex]-1<-1[/tex] it is not satisfied
The answer is the option A
Determine whether each ordered pair is a solution of the given linear equation?
Answer:
(1,2) is the only solution to the given linear equation
Step-by-step explanation:
To see whether an ordered pair is a solution to an equation, the easiest thing to do would be to plug the pair into the equation and see if it equals the right hand side.
(1,2):
[tex]4*x+7*y=18\\4*(1)+7*(2)=18\\\therefore (1,2) \text{ is a solution}[/tex]
(8,0):
[tex]4*(8)+7*(0)=32\\\therefore (8,0) \text{ is not a solution}[/tex]
(0, -2):
[tex]4*(0)+7*(-2)=-14\\\therefore (0,-2) \text{ is not a solution}[/tex]
The pair (1,2) is a solution to the linear equation 4x+7y=18. However, the pairs (8,0) and (0,-2) are not solutions because they do not satisfy the equation.
Explanation:a. To determine if the ordered pair (1, 2) is a solution to the linear equation 4x + 7y = 18, we substitute the values of x and y into the equation: 4(1) + 7(2) = 18. This simplifies to 4 + 14 = 18, which is true. Therefore, (1, 2) is a solution to the given linear equation.
b. To determine if the ordered pair (8, 0) is a solution, we substitute the values of x and y into the equation: 4(8) + 7(0) = 18. This simplifies to 32 + 0 = 18, which is false. Therefore, (8, 0) is not a solution to the given linear equation.
c. To determine if the ordered pair (0, -2) is a solution, we substitute the values of x and y into the equation: 4(0) + 7(-2) = 18. This simplifies to 0 - 14 = 18, which is false. Therefore, (0, -2) is not a solution to the given linear equation.
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The complete question is here:
Determine whether each ordered pair is a solution of the given linear equation.
4 x+7 y=18 ;(1,2),(8,0),(0,-2)
a. Is (1,2) a solution to the given linear equation?
No
Yes
b. Is $(8,0)$ a solution to the given linear equation?
Yes
No
c. Is (0,-2) a solution to the given linear equation?
No
Yes
What is wrong with this “proof”? “Theorem” For every positive integer n, if x and y are positive integers with max(x, y) = n, then x = y. Basis Step: Suppose that n = 1. If max(x, y) = 1 and x and y are positive integers, we have x = 1 and y = 1. Inductive Step: Let k be a positive integer. Assume that whenever max(x, y) = k and x and y are positive integers, then x = y. Now let max(x, y) = k +1, where x and y are positive integers. Then max(x – 1, y – 1) = k, so by the inductive hypothesis, x – 1 = y – 1. It follows that x = y, completing the inductive step. Online Discussion Guidelines: Post your logical argument on the discussion forum. Read the logical argument of your peers. Reply the results posted by at least two of your peers.
The assumption of the inductive step is not correct. If [tex]\mathrm{max}(x,y)=2[/tex], for instance, it's entirely possible that [tex]x=1[/tex] and [tex]y=2[/tex].
Kendra is working on her financial plan and lists all of her income and expenses in the spreadsheet below.
What is Kendra’s net cash flow?
a.
$295
b.
$285
c.
$275
d.
$255
need an answer within 40 min
Answer:
a. $295
Step-by-step explanation:
Add the net pay and the interest. This is the total net income.
Then add all other amounts separately. These are the expenses.
Subtract the expenses from the total net income.
The answer is a. $295
Kendra’s net cash flow is $295 because the total net income is $2320 and total expenses are $2025 option (a) is correct.
What are expenses?It is defined as the money spends on the utility, the amount of money is required to buy something, in other words, it is the outflow of money from the sole earner's income or the money incurred by any organization.
It is given that:
Kendra is working on her financial plan and lists all of her income and expenses in the spreadsheet shown in the picture.
From the spreadsheet:
Add the interest to the net pay. The overall net income is shown here.
= 2300 + 200
= $2320
Next, add each additional sum separately. The costs are as follows.
= 800+120+90+45+95+80+275+520
= $2025
From the entire net income, deduct the costs.
= 2320 - 2025
= $295
Thus, Kendra’s net cash flow is $295 because the total net income is $2320 and total expenses are $2025 option (a) is correct.
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Which of the following reasons would complete the proof in lines 3 and 5
Answer:
Option B, two line perpendicular to the same line are parallel.
Step-by-step explanation:
Since DA and CB are perpendicular to AB, they are considered parallel. This is like a transversal line.
Answer:
Definition of right angles.
Step-by-step explanation:
A rectangle is a parallelogram whose opposite sides are parallel to each other. Also, these opposite sides measure the same and all four sides create right angles. In this sense the reason would complete the proof in lines 3 and 5 is the "Definition of right angles". This is because by definition the rectangle is a parallelogram with a right angle, since it is a parallelogram, its opposite is also a right angle. The other angles, which are supplementary to the previous two, add up to 180º. And since they are opposite, they are the same, therefore each of the four is a right angle.
[tex]\angle ABC = \angle BCD = \angle CDA = \angle DAB=90^{\circ}[/tex]
Hannah and Heather are sisters. Hannah's age is four less than twice Heather's age. The sum of their ages is 30. Which system of equations can be used to determine Hannah's age, x, and Heather's age, y?
A.
x + 30 = y
x = 2 - 4y
B.
x + y = 30
x - 2y = 4
C.
x + y = 30
x = 2y - 4
D.
x + y = 4
y = 30 - 2x
Let Hannah = X and Heather = Y.
The sum of their ages is 30, so you would have x + y = 30
Then Hannah's age is 4 less than 2 times Heather's age so X = 2y-4
The equations are :
X + Y = 30 and X = 2y - 4
The answer is C.
Find the area of an octagon with a radius of 11 units. Round to the nearest hundredth
Answer:
342.24 units²
Step-by-step explanation:
The area of one of the 8 triangular sections of the octagon is ...
A = (1/2)r²·sin(θ) . . . . . where θ is the central angle of the section
The area of the octagon is 8 times that, so is ...
A = 8·(1/2)·11²·sin(360°/8) = 242√2
A ≈ 342.24 units²
The area of the octagon should be 342.24 units²
Calculation of area of an octagon:Since the area of 1 of the 8 triangular sections of the octagon should be.
[tex]A = (1\div 2)r^2.sin(\theta)[/tex]
Here θ represent the central angle of the section
Since
The area of the octagon is 8 times
So,
[tex]A = 8.(1/2).11^2.sin(360\div 8) \\\\= 242\sqrt2[/tex]
A ≈ 342.24 units²
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Kevin sold a box of 28 books at a yard sale for a total of $54.64. He sold the paperback books for $1.68 each and sold the hardcover books for $2.44 each. Which system of equations can be used to determine the number of $1.68 paperback books, x, and the number of $2.44 hardcover books, y, that were sold at the yard sale?
A.
x + y = 28
1.68x + 2.44y = 54.64
B.
2.44x - 1.68y = 28
x + y = 54.64
C.
x + y = 28
2.44y = -1.68x + 54.64
D.
y = x - 54.64
1.68x + 2.44y = 28
Answer:
A.
x + y = 28
1.68x + 2.44y = 54.64
Step-by-step explanation:
Let x = paperback books and y = hardback books
x+y =28
We know that paperbacks cost 1.68 and hardback cost 2.44
1.68x + 2.44y = 54.64
We have 2 equations and 2 unknowns
x+y =28
1.68x + 2.44y = 54.64
The coordinates of A, B, and C in the diagram are A(p,4), B(6,1), and C(9,q). Which equation correctly relates p and q?
Hint: Since AB is perpendicular to BC , the slope of AB x the slope of BC = -1.
This is a PLATO math question, will give 15 pts
attached is the diagram
answer choices:
A.
p − q = 7
B.
q − p = 7
C.
-q − p = 7
D.
p + q = 7
Answer:
D. p + q = 7
Step-by-step explanation:
The slope of AB is ...
slope AB = (1 -4)/(6 -p) = 3/(p -6)
The slope of BC is ...
slope BC = (q -1)/(9 -6) = (q -1)/3
The product of these is -1, so we have ...
(slope AB)·(slope BC) = -1 = (3/(p -6))·((q -1)/3)
Multiplying by q -1 gives ...
-(q -1) = p -6 . . . . . . the factors of 3 in numerator and denominator cancel
1 = p + q -6 . . . . . . add q
7 = p + q . . . . . . . . add 6 . . . . matches choice D
Answer:
D. p + q = 7
Step-by-step explanation:
Just did this quiz on edmentum :P
OH ⊥ BC , OB=10, OH=6, EF=9, AB=2 Find: AE.
Answer:
AE = 3
Step-by-step explanation:
A biologist doing an experiment has a bacteria population cultured in a petri dish. After measuring, she finds that there are 11 million bacteria infected with the zeta-virus and 5.2 million infection-free bacteria. Her theory predicts that 50% of infected bacteria will remain infected over the next hour, while the remaining of the infected manage to fight off the virus in that hour. Similarly, she predicts that 80% of the healthy bacteria will remain healthy over the hour while the remaining of the healthy will succumb to the affliction Modeling this as a Markov chain, use her theory to predict the population of non-infected bacteria after 3 hour(s).
Answer:
11.4 million
Step-by-step explanation:
Let's define the variables i and i' to represent the number of infected bacteria initially and after 1 hour, and the variables n and n' to represent the number of non-infected bacteria initially and after 1 hour. The biologist's theory predicts ...
0.50i +0.20n = i'
0.50i +0.80n = n'
In matrix form, the equation looks like ...
[tex]\left[\begin{array}{cc}0.5&0.2\\0.5&0.8\end{array}\right] \left[\begin{array}{c}i&n\end{array}\right]=\left[\begin{array}{c}i'&n'\end{array}\right][/tex]
If i''' and n''' indicate the numbers after 3 hours, then (in millions), the numbers are ...
[tex]\left[\begin{array}{cc}0.5&0.2\\0.5&0.8\end{array}\right]^3 \left[\begin{array}{c}11&5.2\end{array}\right]=\left[\begin{array}{c}i'''&n'''\end{array}\right][/tex]
Carrying out the math, we find i''' = 4.8006 (million) and n''' = 11.3994 (million).
The population of non-infected bacteria is expected to be about 11.4 million after 3 hours.
Use the theorem below to answer the question. Suppose θ is an acute angle residing in a right triangle. If the length of the side adjacent to θ is a, the length of the side opposite θ is b, and the length of the hypotenuse is c, then cos(θ) = a c and sin(θ) = b c . If θ = 13° and the side adjacent to θ has length 4, how long is the hypotenuse? (Round your answer to three decimal places.)
Answer:
c ≈ 4.105
Step-by-step explanation:
You want to find c in ...
cos(13°) = 4/c
Multiply the equation by c/cos(13°) and evaluate.
c = 4/cos(13°) ≈ 4.105
Answer:
Solve the equation where the sine of 30 degrees is equal to YZ/50.
Solve 180 – (30 + 90) = 60 to find the measure of angle Z. Then use the cosine of Z to write and solve an equation.
Use the 30°-60°-90° triangle theorem to find that YZ = 50/2.
Step-by-step explanation:
These are the sample answers, use how you would like :)
Write the equation of the parabola that has the vertex at point (5,0) and passes through the point (7,−2).
The vertex form of the equation of a parabola is
f(x) = a( x − h)^2 + k
where (h,k) is the vertex of the parabola. In this case, we are given that (h,k) = (5,0). Hence,
f(x) = a( x − 5)^2 + 0
=a( x − 5)^2
Since we also know the parabola passes through the point (7,−2), we can solve for a because we know that f(7) = −2.
a( 7 − 5)^2 = -2
a(2)^2 = -2
4a = -2
a = -1/2
Thus, the given parabola has equation
f(x) = -1/2(x − 5)^2
Please help!!
What is the value of x? Enter your answer in the box. x = NOTE: Image not drawn to scale. Triangle G E H with segment E D such that D is on segment G H, between G and H. Angle G E D is congruent to angle D E H. E G equals 44.8 millimeters, G D equals left parenthesis x plus 4 right parenthesis millimeters, D H equals 35 millimeters, and E H equals 56 millimeters.
Answer:
x = 24
Step-by-step explanation:
The segments on either side of an angle bisector are proportional:
(x +4)/44.8 = 35/56
x +4 = 44.8·(35/56) = 28 . . . . multiply by 44.8
x = 24 . . . . . subtract 4
Answer:
The value of x is 24.
Step-by-step explanation:
Given information: In ΔGHE, ED is angle bisector, EG=44.8 millimeters, GD=(x+4) millimeters, DH=35 millimeters, and EH=56 millimeters.
According to the angle bisector theorem, an angle bisector divide the opposite side into two segments that are proportional to the other two sides of the triangle.
In ΔGHE, ED is angle bisector, By using angle bisector theorem, we get
[tex]\frac{GD}{DH}=\frac{EG}{EH}[/tex]
[tex]\frac{x+4}{35}=\frac{44.8}{56}[/tex]
Multiply both the sides by 35.
[tex]x+4=\frac{44.8}{56}\times 35[/tex]
[tex]x+4=28[/tex]
Subtract 4 from both the sides.
[tex]x=28-4[/tex]
[tex]x=24[/tex]
Therefore the value of x is 24.
Please help this is my last question
Answer:
y = 90°
Step-by-step explanation:
The angle y subtends an arc of 180°, so its measure is 180°/2 = 90°. (We know the arc is 180° because the end points of it are on a diameter, so it is half a circle.)
Which transformation is a isometry?
Answer:
A. The two triangles.
Step-by-step explanation:
Isometry can be divided into two words: iso = same and metry = measure
So, isometry means "same measure".
In this case, that means the transformation didn't change the measures of the object.
In B, they kept the same shape, but not the same side.
In C, you can see the figure has been transformed,.
A bit if not A them it’s C (sorry I tried)
3x+2y=8
Find the slope and y-intercept?
Please show work and how to find the slope and y-intercept in algebraic (:
The slope is 3
I am not that shure on the y intercept
Need help ASAP!! (Geometry) *Attachments*
Answer:
option b 5√2
option a 20
option c 4
option d 3/4
option b 3
Step-by-step explanation:
To solve all these questions we will use trigonometry identity which says that
sinΔ = opposite / hypotenuse
1)
sin45 = 5 / KL
Kl = [tex]\frac{5}{\frac{\sqrt{2}}{2} }[/tex]
KL = 5√2
2)
TanФ = opp / adj
Ф = Tan^-1(5/14)
Ф = 19.65
≈ 20
3)
sin(30) = rq / 8
rq = sin(30)(8)
rq = 4
4)
tan(Ф) = 6/8
5)
angle 3
You are asked to draw a triangle with the side lengths of 6 inches and 8 inches. What is the longest whole number length that you're third side can be?
Answer:
13
Step-by-step explanation:
The longest side must be less than the sum of the shorter sides:
c < a + b
c < 6 + 8
c < 14
So the largest whole number length is 13.
According to legend, in 1626 Manhattan Island was purchased for trinkets worth about $24. If the $24 had been invested at a rate of 6% interest per year, what would be its value in 2006? Compare this with a total of $802.4 billion in assessed values for Manhattan in 2006.
Answer:
Its value in 2006 would be $99, 183, 639, 920
Step-by-step explanation:
Use the compounding interest formula for this one, which looks like this in its standard form:
[tex]A(t)=P(1+r)^t[/tex]
Our P is the initial amount of $24, the r in decimal form is .06, and the time between 2006 and 1626 in years is 380. Fitting this into our formula we get:
[tex]A(t)=24(1+.06)^{380}[/tex] or
[tex]A(t)=24(1.06)^{380}[/tex]
First raise the 1.06 to the power of 380 on your calculator and then multiply in 24.
As for the second part of the question, I'm not quite sure how it's supposed to be answered. But maybe you can figure that out according to what the unit normally asks you to do.
If you know the radius, r, of a circle, what do you multiply r by to get the circle's CIRCUMFERENCE?
A) 2
B) π
C) 2π
D) 3π
E) 4π
Answer:
C
Step-by-step explanation:
When you have the radius of a circle, you find the circumference by multiplying the radius by 2PI.
The answer is C.
1. A baseball is thrown into the air with an upward velocity of 25 ft/sec. It’s height (in feet) after t seconds can be modeled by the function h(t) = -16t^2 + 25t + 5. Algebraically determine how long will it take the ball to reach its maximum height? What is the ball’s maximum height?
2. A company that sells digital cameras has found that their revenue can be remodeled by the equation R(p) = -5p^2 + 1230p, where p is the price of the camera in dollars. Algebraically determine what price will maximize the revenue? What is the maximum revenue?
We can use the concept of derivative to find this result, but since the problem states we must use algebraic procedures, then we solve this as follows:
Step 1: Write the original equation:[tex]h(t)=-16t^2+25t+5[/tex]
Step 2: Common factor -16:[tex]h(t)=-16(t^2-\frac{25}{16}t-\frac{5}{16})[/tex]
Step 3: Take half of the x-term coefficient and square it. Add and subtract this value:X-term: [tex]-\frac{25}{16}[/tex]
Half of the x term: [tex]-\frac{25}{32}[/tex]
After squaring: [tex](-\frac{25}{32})^2=\frac{625}{1024}[/tex]
[tex]h(t)=-16(t^2-\frac{25}{16}t-\frac{5}{16}+\frac{625}{1024}-\frac{625}{1024}) \\ \\ h(t)=-16(t^2-\frac{25}{16}t+\frac{625}{1024}-\frac{5}{16}-\frac{625}{1024}) \\ \\ h(t)=-16(t^2-\frac{25}{16}t+\frac{625}{1024}-\frac{945}{1024}) \\ \\[/tex]
Step 4: Write the perfect square:[tex]h(t)=-16[(t-\frac{25}{32})^2-\frac{945}{1024}] \\ \\ \boxed{h(t)=-16(t-\frac{25}{32})^2-\frac{945}{64}}[/tex]
Finally, the vertex of this function is:
[tex](\frac{25}{32},\frac{945}{64})[/tex]
So in this vertex we can find the answer to this problem:
The ball will reach its maximum height at [tex]t=\frac{25}{32}s=0.78s[/tex]
The ball maximum height is [tex]H=\frac{945}{64}=14.76ft[/tex]
2. Algebraically determine what price will maximize the revenue? What is the maximum revenue?Also we will use completing squares. We can use the concept of derivative to find this result, but since the problem states we must use algebraic procedures, then we solve this as follows:
Step 1: Write the original equation:[tex]R(p)=-5p^2+1230p[/tex]
Step 2: Common factor -5:[tex]R(p)=-5(p^2-246p)[/tex]
Step 3: Take half of the x-term coefficient and square it. Add and subtract this value:X-term: [tex]-246[/tex]
Half of the x term: [tex]-123[/tex]
After squaring: [tex](-123)^2=15129[/tex]
[tex]R(p)=-5(p^2-246p+15129-15129)[/tex]
Step 4: Write the perfect square:[tex]R(p)=-5[(x-123p)^2-15129] \\ \\ R(p)=-5(x-123p)^2+75645[/tex]
Finally, the vertex of this function is:
[tex](123,75645)[/tex]
So in this vertex we can find the answer to this problem:
The price will maximize the revenue is [tex]p=123 \ dollars[/tex]
The maximum revenue is [tex]R=75645[/tex]
What is the length of the base of an isosceles triangle if the center of the inscribed circle divides the altitude to the base into the ratio of 12:5 (from the vertex to the base), and the length of a leg is 60 cm?
Answer:
50 cm
Step-by-step explanation:
Consider isosceles triangle AEF in the attachment. Point B is the center of the incircle, and it divides altitude AC into segments having the ratio 12:5.
Triangle ABD is similar to triangle AEC by AA similarity. (Angle A is the same for both right triangles. Then the ratio of hypotenuse to short leg will be the same for each. In triangle ABD, that ratio is 12:5, as given by the problem statement. Since we know AE = 60 cm, also from the problem statement, we know that ...
AB/BD = AE/EC
12/5 = 60 cm/EC
so ...
EC = (60 cm)·(5/12) = 25 cm
Base length EF is twice that, or 50 cm.
kong took 15% fewer seconds that Nolan took to complete his multiplication timed test. Kong took 85 seconds.
How many seconds did Nolan take?
Answer:
Answer: Nolan took 97.75 Seconds
Step-by-step explanation:
85 x .15 = 12.75
85 + 12.75 = 97.75
A circle has a radius of 119.3 millimeters. What is the circumference of this circle? Use 3.14 for pi. Round your final answer to the nearest tenth.
Answer:
0.74 m
Step-by-step explanation:
The distance around a circle on the other hand is called the circumference (c).
A line that is drawn straight through the midpoint of a circle and that has its end points on the circle border is called the diameter (d) .
Half of the diameter, or the distance from the midpoint to the circle border, is called the radius of the circle (r).
The circumference of a circle is found using this formula:
C=π⋅d
or
C=2π⋅r
Given r = 119.3 mm
So, C = 2 * 3.14 * 119.3 = 749.2 mm = 0.74 m
Complete the identity
Answer: [tex]cos(\pi-x)=-cos(x)[/tex]
Step-by-step explanation:
We need to apply the following identity:
[tex]cos(A - B) = cos A*cos B + sinA*sin B[/tex]
Then, applying this, you know that for [tex]cos(\pi-x)[/tex]:
[tex]cos(\pi-x)=cos(\pi)*cos(x)+sin(\pi)*sin(x)[/tex]
We need to remember that:
[tex]cos(\pi)=-1[/tex] and [tex]sin(\pi)=0[/tex]
Therefore, we need to substitute these values into [tex]cos(\pi-x)=cos(\pi)*cos(x)+sin(\pi)*sin(x)[/tex].
Then, you get:
[tex]cos(\pi-x)=(-1)*cos(x)+0*sin(x)[/tex]
[tex]cos(\pi-x)=-1cos(x)+0[/tex]
[tex]cos(\pi-x)=-cos(x)[/tex]
A soccer ball is kicked off from the ground in an arc defined by the function, h(x)=-8x^2+64x. At what point does the ball hit the ground?
(0,4) , (0,8) , (4,0) , (8,0)
Answer:
(8,0)
Step-by-step explanation:
The equation that models the path traced by the ball is
[tex]h(x)=-8x^2+64x[/tex]
To find the point at which the ball hit the ground, we must equate the function to zero.
[tex]-8x^2+64x=0[/tex]
Factor;
[tex]-8x(x-8)=0[/tex]
[tex]-8x=0,(x-8)=0[/tex]
This implies that;
x=0,x=8,
At x=0, the ball was not yet kicked.
So we take x=8, to be the time the ball hit the ground.
We substitute x=8 into the function to get;
[tex]h(8)=-8(8)^2+64(8)=0[/tex]
Hence the point at which the ball hit the ground is (8,0)
The equations that must be solved for maximum or minimum values of a differentiable function w=f(x,y,z) subject to two constraints g(x,y,z)=0 and h(x,y,z)=0, where g and h are also differentiable, are gradientf=lambdagradientg+mugradienth, g(x,y,z)=0, and h(x,y,z)=0, where lambda and mu (the Lagrange multipliers) are real numbers. Use this result to find the maximum and minimum values of f(x,y,z)=xsquared+ysquared+zsquared on the intersection between the cone zsquared=4xsquared+4ysquared and the plane 2x+4z=2.
The Lagrangian is
[tex]L(x,y,z,\lambda,\mu)=x^2+y^2+z^2+\lambda(4x^2+4y^2-z^2)+\mu(2x+4z-2)[/tex]
with partial derivatives (set equal to 0)
[tex]L_x=2x+8\lambda x+2\mu=0\implies x(1+4\lambda)+\mu=0[/tex]
[tex]L_y=2y+8\lambda y=0\implies y(1+4\lambda)=0[/tex]
[tex]L_z=2z-2\lambda z+4\mu=0\implies z(1-\lambda)+2\mu=0[/tex]
[tex]L_\lambda=4x^2+4y^2-z^2=0[/tex]
[tex]L_\mu=2x+4z-2=0\implies x+2z=1[/tex]
Case 1: If [tex]y=0[/tex], then
[tex]4x^2-z^2=0\implies4x^2=z^2\implies2|x|=|z|[/tex]
Then
[tex]x+2z=1\implies x=1-2z\implies2|1-2z|=|z|\implies z=\dfrac25\text{ or }z=\dfrac23[/tex]
[tex]\implies x=\dfrac15\text{ or }x=-\dfrac13[/tex]
So we have two critical points, [tex]\left(\dfrac15,0,\dfrac25\right)[/tex] and [tex]\left(-\dfrac13,0,\dfrac23\right)[/tex]
Case 2: If [tex]\lambda=-\dfrac14[/tex], then in the first equation we get
[tex]x(1+4\lambda)+\mu=\mu=0[/tex]
and from the third equation,
[tex]z(1-\lambda)+2\mu=\dfrac54z=0\implies z=0[/tex]
Then
[tex]x+2z=1\implies x=1[/tex]
[tex]4x^2+4y^2-z^2=0\implies1+y^2=0[/tex]
but there are no real solutions for [tex]y[/tex], so this case yields no additional critical points.
So at the two critical points we've found, we get extreme values of
[tex]f\left(\dfrac15,0,\dfrac25\right)=\dfrac15[/tex] (min)
and
[tex]f\left(-\dfrac13,0,\dfrac23\right)=\dfrac59[/tex] (max)
This problem involves using Lagrangian multipliers to optimize a function with two constraints. The maximum and minimum points can be found by solving the Lagrange equations, which are derivatives of the function and constraints. These points can be confirmed by checking the positive or negative value of the second-order derivative.
Explanation:To find the maximum and minimum values of the function f(x,y,z)=x²+y²+z² subject the cone z²=4x²+4y² and the plane 2x+4z=2, we use Lagrange multipliers. We have two constraint functions here, given by the cone and the plane equations.
The first step is to set up the Lagrange equations, with and as Lagrange multipliers. From w=gradientf=gradientg+gradienth, we have three equations: 2x=*8x+2, 2y=*8y+0, 2z=*4. The second step is to solve these three equations, together with the original constraints g(x,y,z)=0 and h(x,y,z)=0.
Solving these equations will give you specific values for x, y, and z that correspond to the maximum and minimum points. To determine if a point is a maximum or minimum, one can compute the second-order partial derivatives and organize them into the Hessian matrix.
Learn more about Lagrange Multipliers here:https://brainly.com/question/32544889
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I really need help with these three questions, urgent!
6.
Intersecting chords:
RT x ST = PT x TQ
2 x 6 = 3 x TQ
12 = 3TQ
TQ = 12/3
TQ = 4
7.
AD = 90 -BE = 90-18 = 72
ADE = 180. DE = 180 - AD = 180-72 = 108
AE = 180. AB = 180-18 = 162
DE = 108
BD = BE +DE = 18 + 108 = 126
DAB = 72 + 162 = 234
ADE = 90 degrees
8.
AB^2 = BC* (BC +x)
8^2 = 2 * (2 +x)
64 = 4 + 2x
60 = 2x
X = 60/2
X = 30
If A = {x | x is an even integer}, B = {x | x is an odd integer find a u b
Answer:
A U B = {x|x is an integer}
Step-by-step explanation:
If A={x|x is an odd integer} then:
A = 1, 3, 5, 7, 9, 11, etc.
(Negative odd integers are also included)
and B= {x|x is an even integer}
A = 2, 4, 6, 8, 10, 12, 14, etc.
(Negative even integers are also included)
We have that a u b is going to be the set of all integer numbers. That is to say:
A U B = {x|x is an integer}