First find the critical points of f :
[tex]f(x,y)=2x^2+3y^2-4x-5=2(x-1)^2+3y^2-7[/tex]
[tex]\dfrac{\partial f}{\partial x}=2(x-1)=0\implies x=1[/tex]
[tex]\dfrac{\partial f}{\partial y}=6y=0\implies y=0[/tex]
so the point (1, 0) is the only critical point, at which we have
[tex]f(1,0)=-7[/tex]
Next check for critical points along the boundary, which can be found by converting to polar coordinates:
[tex]f(x,y)=f(10\cos t,10\sin t)=g(t)=295-40\cos t-100\cos^2t[/tex]
Find the critical points of g :
[tex]\dfrac{\mathrm dg}{\mathrm dt}=40\sin t+200\sin t\cos t=40\sin t(1+5\cos t)=0[/tex]
[tex]\implies\sin t=0\text{ OR }1+5\cos t=0[/tex]
[tex]\implies t=n\pi\text{ OR }t=\cos^{-1}\left(-\dfrac15\right)+2n\pi\text{ OR }t=-\cos^{-1}\left(-\dfrac15\right)+2n\pi[/tex]
where n is any integer. We get 4 critical points in the interval [0, 2π) at
[tex]t=0\implies f(10,0)=155[/tex]
[tex]t=\cos^{-1}\left(-\dfrac15\right)\implies f(-2,4\sqrt6)=299[/tex]
[tex]t=\pi\implies f(-10,0)=235[/tex]
[tex]t=2\pi-\cos^{-1}\left(-\dfrac15\right)\implies f(-2,-4\sqrt6)=299[/tex]
So f has a minimum of -7 and a maximum of 299.
Can someone please help I don't understand. Determine the domain and range of the following function. Record your answers in set notation.
Look at the screenshot!!!
Based on the image, which list of 3 points are collinear?
Answer:
Collinear occurs when the two points has the same gradient,
So, for this question any line that forms by any three points would be collinear.
Hence, EBF,DGC,MGN,BGA are all collinears
Step-by-step explanation:
HELP PLEASE PLEASE :(
Answer:
16
Step-by-step explanation:
It’s a ratio.
x/12=21/28
21x=12*28
21x=336
x=336/21
x=16
Find the value of x.
A. 22
B. 7.3
C. 3.6
D. 5.5
Answer:
x= 5.5
Step-by-step explanation:
(segment piece) x (segment piece) = (segment piece) x (segment piece)
x*4 = 11*2
4x = 22
Divide each side by 4
4x/4 = 22/4
x =5.5
Please answer ASAP PLEASE!
Answer/Step-by-step explanation:
The inequality, x ≤ 7, has solutions that includes values that is equal to 1 or less than 7.
This can be represented on a number line as shown in the number line graphed in the attachment below.
A full circle or shaded "o" indicates that the number 7 is included in the solution.
The arrow points from 7 to the left, telling us that the value of x are all numbers from 7 and below.
Let A= {1 , 2 , 3 , ... ... ...... , 10} and R = {(a, b): a ∈ A , b ∈ A and a + 2b = 10} Find the domain and range of R.
In domain and range of a relation, if R be a relation from set A to set B, then
• The set of all first components of the ordered pairs belonging to R is called the domain of R.
Thus, Dom(R) = {a ∈ A: (a, b) ∈ R for some b ∈ B}.
• The set of all second components of the ordered pairs belonging to R is called the range of R.
Thus, range of R = {b ∈ B: (a, b) ∈R for some a ∈ A}.
Therefore, Domain (R) = {a : (a, b) ∈ R} and Range (R) = {b : (a, b) ∈ R}
Sketch the graph of the following equations:
y-3x+5
y=-3x-5
In the multiplication below, each of A, B and
C represents a different digit. What is ABC?
A B C
X
3
В В В
Answer:
ABC = 148, 3*148 = 444
Step-by-step explanation:
We know that 111 = 3* 37, so all numbers of the form BBB has the factor 37.
So we need a multiple of 37 such thant when multiplied, we get three digits the same as the middle digit.
Try 4*37 = 148, 148*3 = 444, bingo, we got the right combination.
So ABC is 148.
Find the intersection point for the following liner function f(x)= 2x+3 g(x)=-4x-27
Answer:
( -5,-7)
Step-by-step explanation:
f(x)= 2x+3 g(x)=-4x-27
Set the two functions equal
2x+3 = -4x-27
Add 4x to each side
2x+3+4x = -4x-27+4x
6x+3 = -27
Subtract 3
6x+3 - 3 = -27-3
6x = -30
Divide each side by 6
6x/6 = -30/6
x =-5
Now we need to find the output
f(-5) = 2(-5) +3 = -10+3 = -7
Answer:
Step-by-step explanation:
big burgewr
Plzzz help me on this question
This is Additional mathematics IGCSE
Answer:
[tex] \alpha = 7[/tex]
Step-by-step explanation:
[tex]a(vector) = 4i - 2j[/tex]
[tex]b(vector) = \alpha i + 2j[/tex]
[tex]ab(vector) = ( \alpha - 4)i \: + 4j[/tex]
Now,
Let K * ab (unit vector) = ab (vector)
(0.4 * k) j = 4 j Thus, K = 10[tex](0.3 \times k)i = ( \alpha - 4)i[/tex]Solving further :
[tex] \alpha = 7[/tex]
Write down the answers to a,b,c,d
Answer:
(A) 1
(B) -2
(C) 3.5
(D) -0.5
Step-by-step explanation:
We can treat each thermometer like a vertical number line and read the values on each.
A is right on 1.
B is right on -2.
C is in the middle of 3 and 4, so 3.5
D is in the middle of 0 and -1, so -0.5
Hope this helped!
In a triangle, the sum of two angles equals the third, Find the measure of the third angle.
A.45 degree
B.60 degree
C.90 degree
D.30 degree
Answer:
C.90 degree
Step-by-step explanation:
45 + 45 + 90 = 180
90 = 45 + 45
=
Graphing an integer function and finding its range for a given...
The function h is defined as follows for the domain given.
h(x) = 2 -2x, domain = {-3, -2, 1, 5}
Write the range of h using set notation. Then graph h.
Answer:
Step-by-step explanation:
● h(x) = 2-2x
The domain is {-3,-2,1,5}
● h(-3) = 2-2×(-3) = 2+6 = 8
● h(-2) = 2 -2×(-2) = 2+4 = 6
● h(1) = 2-2×1 = 2-2 = 0
● h(5) = 2-2×5 = 2-10 = -8
The range is {-8,0,6,8}
there are 5 discs, 6 jump ropes, 3 balls, and 12 pieces of sidewalk chalk in a bin. If two items are drawn at random without replacement, what is the probability that both items removed are not jump ropes?
Answer: 0.584
Step-by-step explanation:
We have:
5 discs
6 jump ropes
3 balls
12 pieces of sidewalk.
5 + 6 + 3 + 12 = 26
If all of them have exactly the same probability of being removed, then:
in the first selection, we do not want to remove a jump rope, so we can remove one disc, one ball or one piece of sidewalk.
The total number of those objects is:
5 + 3 + 12 = 20.
Then the probability of removing one of those objects is:
P1 = 20/26 = 0.769
Now in the second selection, we have the same situation, but now we have 25 objects in total, and because in the previous selection we removed one ball, or one disc, or one piece of sidewalk, the total number of these things now is 19.
So the probability of removing another object of that set is:
P2 = 19/25 = 0.76
The joint probability is equal to the product of the individual probabilities, so we have:
P = P1*P2 = 0.769*0.76 = 0.584
graph the solution set to the inequality
Graphed using the given range equation. The shaded area is the possible range, extending to infinity, infinity from 0, -1.
A basketball player scored 33 points during a game by shooting 1-point free throws, 2-point field goals, and 3-point field goals. The player scored 17 times. She scored 3 more 2-point field goals than 1-point free throws. The system of equations below represents the situation, where x is the number of 1-point free throws, y is the number of 2-point field goals, and z is the number of 3-point field goals. x + y + z = 17 x + 2y + 3z = 33 y – x = 3
Answer:
No. of 1 pt free throws = 5, No. of 2 pt goals = 8, No. of 3 pt goals = 4
Step-by-step explanation:
Equations : x + y + z = 17 [ Total times taken to score ]
1x + 2y + 3z = 33 [ Total Score ]
Also, y = x + 3
Putting the value of 'y' in both equations :
x + (x + 3)+ z = 17 → 2x + 3 + z = 17 → 2x + z = 14 (i)
1x + 2 (x + 3) + 3z = 33 → x + 2x + 6 + 3z = 33 → 3x + 3z = 27 (ii)
Solving these equations :
From (i), z = 14 - 2x
Putting this value in (ii), 3x + 3(14 - 2x) = 27 → 3x + 42 - 6x = 27
42 - 3x = 27 → 3x = 15 → x = 5
y = x + 3 = 5 + 3 → y = 8
z = 17 - x - y → z = 17 - 5 - 8 = 17 - 13 → z = 4
Answer:
4
Step-by-step explanation:
Allied Corporation is trying to sell its new machines to Ajax. Allied claims that the machine will pay for itself since the time it takes to produce the product using the new machine is significantly less than the production time using the old machine. To test the claim, independent random samples were taken from both machines. You are given the following results.
New Machine Old Machine
Sample Mean 25 23
Sample Variance 27 7.56
Sample Size 45 36
As the statistical advisor to Ajax, would you recommend purchasing Allied's machine? Explain.
Answer:
z(s) is in the acceptance region. We accept H₀ we did not find a significantly difference in the performance of the two machines therefore we suggest not to buy a new machine
Step-by-step explanation:
We must evaluate the differences of the means of the two machines, to do so, we will assume a CI of 95%, and as the interest is to find out if the new machine has better performance ( machine has a bigger efficiency or the new machine produces more units per unit of time than the old one) the test will be a one tail-test (to the left).
New machine
Sample mean x₁ = 25
Sample variance s₁ = 27
Sample size n₁ = 45
Old machine
Sample mean x₂ = 23
Sample variance s₂ = 7,56
Sample size n₂ = 36
Test Hypothesis:
Null hypothesis H₀ x₂ - x₁ = d = 0
Alternative hypothesis Hₐ x₂ - x₁ < 0
CI = 90 % ⇒ α = 10 % α = 0,1 z(c) = - 1,28
To calculate z(s)
z(s) = ( x₂ - x₁ ) / √s₁² / n₁ + s₂² / n₂
s₁ = 27 ⇒ s₁² = 729
n₁ = 45 ⇒ s₁² / n₁ = 16,2
s₂ = 7,56 ⇒ s₂² = 57,15
n₂ = 36 ⇒ s₂² / n₂ = 1,5876
√s₁² / n₁ + s₂² / n₂ = √ 16,2 + 1.5876 = 4,2175
z(s) = (23 - 25 )/4,2175
z(s) = - 0,4742
Comparing z(s) and z(c)
|z(s)| < | z(c)|
z(s) is in the acceptance region. We accept H₀ we did not find a significantly difference in the performance of the two machines therefore we suggest not to buy a new machine
The very hight dispersion of values s₁ = 27 is evidence of frecuent values quite far from the mean
Which expression is equivalent to 5y^3/(5y)^-2
Answer:
5^3 y^5
125 y^5
Step-by-step explanation:
5y^3/(5y)^-2
Distribute the exponent in the denominator
5y^3/(5 ^-2 y^-2)
A negative exponent in the denominator brings it to the numerator
5y^3 5 ^2 y^2
Combine like terms
5 * 5^2 * y^3 5^2
We know that a^b * a^c = a^(b+c)
5^(1+2) * y^( 3+2)
5^3 y^5
125 y^5
please help. urgent. calculate the value of: E= x^3+1/x^3 if 1/x+x=4
(picture below)
Answer:
52
Step-by-step explanation:
1. We see if we cube both sides of the second equation, then it looks more similar to the first equation (E = x^3 + 1/x^3)
(1/x + x)^3 = 4^3
1/x^3 + 3/x + 3x + x^3 = 64
2. Now we rearrange, because we see x^3 + 1/x^3 in the first equation in this equation
x^3 + 1/x^3 + 3x + 3/x = 64
3. We look at 3x + 3/x and see that it looks like the second equation, so we try factoring it
x^3 + 1/x^3 + 3(x + 1/x) = 64
we know x + 1/x = 4, so 3(x + 1/x) = 12
x^3 + 1/x^3 +12 = 64
4. Now we subtract and get our answer
x^3 + 1/x^3 = 52
A researcher surveys middle-school students on their study habits. She finds that in a random sample of 28 middle-school students, the mean amount of time that they spend working on the computer each night is 2.4 hours with a standard deviation of 0.92 hours. She uses the sample statistics to compute a 95% confidence interval for the population mean - the the mean amount of time that all middle-school students spend working on the computer each night. What is the margin of error for this confidence interval
Answer:
The margin of error is [tex]E = 1.96 * \frac{ 0.92}{\sqrt{28 } }[/tex]
Step-by-step explanation:
From the question we are told that
The sample size is [tex]n = 28[/tex]
The sample mean is [tex]\= x = 2.4 \ hr[/tex]
The standard deviation is [tex]\sigma = 0.92 \ hr[/tex]
Given that the confidence level is 95% the the level of significance can be evaluated as
[tex]\alpha = 100 -95[/tex]
[tex]\alpha = 5 \%[/tex]
[tex]\alpha = 0.05[/tex]
Next we obtain the critical value of [tex]\frac{\alpha }{2}[/tex] from the normal distribution table,the value is [tex]Z_{\frac{\alpha }{2} } = Z_{\frac{0.05}{2} } = 1.96[/tex]
Generally the margin of error is mathematically represented as
[tex]E = Z_{\frac{\alpha }{2} } * \frac{\sigma }{\sqrt{n} }[/tex]
substituting values
[tex]E = 1.96 * \frac{ 0.92}{\sqrt{28 } }[/tex]
[tex]E = 0.3408[/tex]
State whether each ratio forms a proportion.
1) 6:3, 18:9 2) 3:4, 30:40 3) 14/18,28/36 4) 2/5,5/2
Answer: Please Give Me Brainliest, Thank You!
#1, #2, #3 do, but #4 doesn't
Step-by-step explanation:
#1
18/9=2
6/3=2
#2
30/3=10
40/4=10
#3
28/14=2
36/18=2
Suppose that 200 students are randomly selected from a local college campus to investigate the use of cell phones in classrooms. When asked if they are allowed to use cell phones in at least one of their classes, 40% of students responded yes. Using these results, with 95% confidence, the margin of error is 0.068. How would the margin of error change if the sample size increased from 200 to 400 students?
Answer:
It would change to 0.04802
Step-by-step explanation:
from this question we have that n became 400
40% of 400
= 160
p* = 160/400
= 0.4
1 - p* =
= 1 - 0.4
= 0.6
at confidence level,
1 - 0.95
= 0.05
alpha/2 = 0.025
z= 1.96
margin of error. E
= 1.96 x √[(0.4 x 0.6)/400]
= 1.96 x 0.0245
= 0.04802
M.E = 0.04802
In a random sample of 20 NBA basketball games the mean number of points scored by the home team was 100.4 with a standard deviation of 4.86.
Create and interpret a 95% confidence interval for the true mean number of points scored by an NBA basketball team at home.
You and your friend were watching a LA Lakers game where they were not playing at home. They only scored 98 points. Your friend says, "Wow, I bet if they were playing at home they would have scored a lot more points." Do you agree or disagree with your friend? Support your detailed answer.
Answer:
The 95% confidence interval is [tex]98.27 < \mu < 102.53[/tex]
This interval means that there 95% confidence that the true mean is within this interval
Yes i would agree with my friend because the lower and the upper limit 95% confidence interval for mean points scored at home is greater than 98 points
Step-by-step explanation:
From the question we are told that
The sample size is n = 20
The sample mean is [tex]\mu = 100.4[/tex]
The standard deviation is [tex]\sigma = 4.86[/tex]
Given that the confidence level is 95% then the level of significance is mathematically evaluated as
[tex]\alpha = 100 - 95[/tex]
[tex]\alpha = 5\%[/tex]
[tex]\alpha = 0.05[/tex]
Next we obtain the critical value of [tex]\frac{ \alpha }{2}[/tex] from the normal distribution table, the value is
[tex]Z_{\frac{ \alpha }{2} } = 1.96[/tex]
Generally the margin of error is mathematically represented as
[tex]E = Z_{\frac{\alpha }{2} } * \frac{\sigma}{ \sqrt{n} }[/tex]
substituting values
[tex]E = 1.96 * \frac{ 4.86 }{ \sqrt{20 } }[/tex]
[tex]E = 2.13[/tex]
The 95% confidence interval is mathematically represented as
[tex]\= x - E < \mu < \= x + E[/tex]
substituting values
[tex]100.4 - 2.13 < \mu < 100.4 + 2.13[/tex]
[tex]98.27 < \mu < 102.53[/tex]
(Algebra)
Plz help me ASAP!! I’ll be so grateful!
Answer:
y > 1
Step-by-step explanation:
-2(7 + y) > -8(y + 1)
-14 -2y > -8y -8
-2y +8y > -8 +14
6y > 6
6y/6 > 6/6
y > 1
An ice cream store makes 144 quarts of ice cream in 8 hours. How many quarts could be made in 12 hours?
Hey there! I'm happy to help!
We know that the ice cream store makes 144 quarts in eight hours. What about in one hour? Let's divide this by eight to find out.
144/8=18
So, they make 18 quarts every hour. We want to figure out how many can be made in 12 hours. So, we just multiply 18 by 12!
18(12)=216
Therefore, 216 quarts of ice cream could be made in 12 hours.
Have a wonderful day! :D
The ice cream store will make 216 quarts of ice cream in 12 hours.
What is division?Division is breaking a number up into an equal number of parts.
Given that, An ice cream store makes 144 quarts of ice cream in 8 hours.
Since, they make 144 quarts of ice cream in 8 hours
Therefore, in 1 hour they will make = 144/8 = 18 quarts
So, in 12 hours = 18x12 = 216 quarts.
Hence, The ice cream store will make 216 quarts of ice cream in 12 hours.
For more references on divisions, click;
https://brainly.com/question/21416852
#SPJ2
Use a definition, postulate, or theorem to find the value of x in the figure described. Point E is between points D and F. If DE = x − 3, EF = 6x + 5, and DF = 8x − 3, find x. Select each definition, postulate, or theorem you will use. A)definition of segment bisector B)definition of midpoint C)Linear Pair Theorem D)Segment Addition Postulate The solution is x =?
Answer:
Option (D)
x = 5
Step-by-step explanation:
Since point E is in the mid of the segment DF,
Therefore, by the Segment addition postulate,
DF = DE + EF
Since DF = (8x - 3), DE = (x - 3) and EF = (6x + 5)
By substituting these values in the given postulate,
(8x - 3) = (x - 3) + (6x + 5)
8x - 3 = (x + 6x) + (5 - 3)
8x - 3 = 7x + 2
8x - 7x = 3 + 2
x = 5
Therefore, x = 5 will be the answer.
Answer:
x=6 and D
Step-by-step explanation:
How do i do this equation
-3(-2y-4)-5y-2=
Answer:
combined like terms and then follow the order of operations.
Step-by-step explanation:
The table shows data collected on the relationship between time spent playing video games and time spent with family. The line of best fit for the data is ý = -0.363 +94.5. Assume the line of best fit is significant and there is a strong linear relationship between the variables.
Video Games (Minutes) Time with Family (Minutes)
40 80
55 75
70 69
85 64
Required:
a. According to the line of best fit, what would be the predicted number of minutes spent with family for someone who spent 36 minutes playing video games?
b. The predicted number of minutes spent with family is:_________
Answer:
81.432 minutes
Step-by-step explanation:
Given the following :
Video Games (Mins) - - - Time with Family(Mins)
40 - - - - - - - - - - - - - - - - - - - 80
55 - - - - - - - - - - - - - - - - - - - 75
70 - - - - - - - - - - - - - - - - - - - 69
85 - - - - - - - - - - - - - - - - - - - 64
Best fit line:
ý = -0.363x +94.5
For someone who spent 36 minutes playing video games, the predicted number of minutes spent with family according to the best fit line will be:
Here number of minutes playing video games '36' is the independent variable
ý is the dependent or predicted variable ;
94.5 is the intercept
ý = -0.363(36) +94.5
ý = −13.068 + 94.5
ý = 81.432 minutes
Which is about 81 minutes to the nearest whole number.
BRAINLIEST ANSWER GIVEN! Find the equation of the line passing through the pair points (-8,6) (-9,-9). The equation of the line in the form is Ax+By=C.
Answer:
y=15x+126
Step-by-step explanation:
the slope is
15 because -8-(-9) is 1 and 6-(-9) is 15 and y is over x so slope 15
To find y intercept start from -8,6 and add 15 to the y value every time you add one to the x value
you will add 8 times and you get 126 as the intercept
Please answer this correctly without making mistakes
Answer:
105/4 or 26.25 mi
Step-by-step explanation:
hillsdale to fairfax 8 7/8 = 71/8
fairfax to yardley = 17 3/8 = 139/8
71/8 + 139/8 = 105/4 or 26 2/8
