The variable overhead rate variance is: c. $6,052 U.
Variable overhead rate varianceFirst step is to calculate variable component of the predetermined overhead rate using this formula
Variable component of the predetermined overhead rate= Budgeted variable manufacturing overhead/Budgeted hours
Let plug in the formula
Variable component of the predetermined overhead rate=$ 28,200/12,000 labor-hours
Variable component of the predetermined overhead rate=$2.35 per labor-hour
Second step is to calculate variable overhead rate variance using this formula
Variable overhead rate variance=(AH×AR)-(AH×SR)
Let plug in the formula
Variable overhead rate variance= $ 26,967- ( 8,900 labor-hours×$2.35 per labor-hour)
Variable overhead rate variance= $ 26,967- $20,915
Variable overhead rate variance= $6,052 U (Unfavorable)
Therefore the variable overhead rate variance is: c. $6,052 U.
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thank you for helping
Answer: TRS is congruent to CSE
Step-by-step explanation:
You can find congruent angles by locating the ones with the same number of lines. You want to start with the angles in the order of TRS.
Angle T has 1 line, so is congruent to C, which also has 1 line.
Angle R has 2 lines, so is congruent to S, which also has 2 lines.
Angle S has 3 lines, so is congruent to E, which also has 3 lines.
You want to keep these in the same order, so TRS, in the order of 1-2-3 lines, is congruent to CSE, also in the order of 1-2-3 lines (aka congruency markings).
1-cos(6x)=___?
A. 3sin(2x)
B. 2sin^2(3x)
C. 3cos(2x)
D. 2cos^2(3x)
The solution to 1 - cos(6x) is 2sin²(x).
Hence, option B) 2sin²(x) is the correct answer.
What is solution to 1 - cos(6x)?Given that; 1 - cos(6x) = ?
First, we rewrite using trig identity
1 - cos(2 × 3x)
Using the double angle identity, { cos2(x) = 1 - 2sin²(x) }
1 - ( 1 - 2sin²(x) )
Eliminate the parentheses
1 - 1 + 2sin²(x)
2sin²(x)
The solution to 1 - cos(6x) is 2sin²(x).
Hence, option B) 2sin²(x) is the correct answer.
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4. Find the number of paths between c and d in the graph of length
The number paths between c and d of length 2.
What are the number paths?A counting model is a number path. Rectangles symbolize numbers, and it is possible to count each rectangle. Like a ruler, a number line is a model of length. The length starting at zero is used to represent each integer.A number path offers a more reassuring representation of numbers, which is crucial since we want representations that constantly assist pupils in gaining confidence and correctly resolving issues.Start adding the first number in the number sentence and work your way to the right after that. Starting with the largest number in the number sentence, go to the left on the number line to subtract. Skip counting by the number you are multiplying by when you multiply.Find the number of paths between c and d in the graph of length:
The number of paths between c and d is 0.
There is the number of paths between c and d of length 2.
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PLEASE EXPLAIN!!
The ratio of my sister's age to my age is 10: 9.
The ratio of my brother's age to my age is 29 : 27.
I am over 40 years old but under 70 years old.
What is my age?
Sis : you : bro
10 : 9 : ?
? ? : 27 : 29
We can get the 9 at the top to 27, by multiplying the whole row by 3 to get:
30 : 27 :?
? : 27 : 29
Your age matches (in the ratios or relation to your sis & bro) now 27 yrs in both ratio. Creating a single ratio, whereby multiplying the top row by 3 to get you to 27 & bottom row by 1 to get you to 27. Knowing when you're 27 in both cases, you know how old your sis & bro is
So, replace the ? with the number above or below it, creating a single ratio, where you are in common with both, by being 27 in both cases.
The ratio of sis : you : bro is now -
The ratio of sis : you : bro is now - 30: 27 : 29
To find what your age is we need to increase the ratio in the same amount to find your maximum age under 70.
So, if we were to double your age to 54, by multiplying your previous age of 27 by 2. We've got to do the same for your sis & bro too, they're aging with you.
We get :
We get :60 : 54 : 58
(30 x 2) : (27 x2) : (29 x2)
Keep increasing the ratio, to the point you're not above 70 years. Now, I'll multiply all ages by 3.
30: 27 : 29
(30 x 3) : ( 27 x 3) : (29 x 3)
We get:
We get:90 : 81 : 87
Right now, you're above the age of 70.
Thus, your maximum age must be 54, where you're under 70 years, but over 40 years.
Answer : 54 years
Hope this helps!
Distance between -11 and 4 on a number line?
Answer:
Distance = 15
Step-by-step explanation:
4 - (-11) = 4 + 11 = 15
11. Use the following information for problems a-b. The table shows the results of a school lunch survey. In the
survey, students were asked whether they have chores and a curfew
Using it's concept, the probabilities are given as follows:
a) C. 7/20.
b) D. 15/22.
What is a probability?A probability is given by the number of desired outcomes divided by the number of total outcomes.
For item a, we have that there are 60 students with no curfew, and of those, 21 have no chores, hence the probability is given by:
p = 21/60 = 7/20
Which means that option C is correct.
For item b, we have that there are 66 students with no shores, and of those, 45 have a curfew, hence the probability is given by:
p = 45/66 = 15/22
Which means that option D is correct.
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Find the coordinates of the point on the curve y = 2x² - x - 1 at which the gradient is 7
Answer:
See below
Step-by-step explanation:
The derivative of the function is the slope
2x^2 - x -1 take derivative ( f ' )
4x -1 equate to 7
7 = 4x -1
x = 2 sub into the original equation to find y = 5
f(x) = 2x^2 - x -10
What are the x-intercepts of the graph of f(x)? Show your work
Answer:
[tex](-2,0) \textsf{ and }\left(\dfrac{5}{2},0\right)[/tex]
Step-by-step explanation:
Given quadratic function:
[tex]f(x)=2x^2-x-10[/tex]
The x-intercepts of the graph are the points at which the curve crosses the x-axis, so when f(x) = 0.
To find these points, factor the quadratic equation, set it to zero, then solve for x.
To factor a quadratic in the form [tex]ax^2+bx+c[/tex], find two numbers that multiply to ac and sum to b.
[tex]\implies ac=2 \cdot -10 = -20[/tex]
[tex]\implies b=-1[/tex]
Therefore, the two numbers are -5 and 4.
Rewrite the middle term as the sum of these two numbers:
[tex]\implies 2x^2-5x+4x-10[/tex]
Factor the first two terms and the last two terms separately:
[tex]\implies x(2x-5)+2(2x-5)[/tex]
Factor out the common term (2x - 5):
[tex]\implies (x+2)(2x-5)[/tex]
Set the factored function to zero:
[tex]\implies (x+2)(2x-5)=0[/tex]
Apply the zero-product property:
[tex]\implies (x+2)=0 \implies x=-2[/tex]
[tex]\implies (2x-5)=0 \implies x=\dfrac{5}{2}[/tex]
Therefore, the x-intercepts of the graph are:
[tex](-2,0) \textsf{ and }\left(\dfrac{5}{2},0\right)[/tex]
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How do I write the following number in scientific notation?
788,000
What is the solution to the system of equations graft below
Answer:
( 2,-3)
Step-by-step explanation:
The solution to the system of equations is where the two graphs intersect
The two graphs intersect at x = 2 and y = -3
( 2,-3)
Patel is solving 8x2 + 16x + 3 = 0. Which steps could he use to solve the quadratic equation? Select three options. 8(x2 + 2x + 1) = –3 + 8 x = –1 Plus or minus StartRoot StartFraction 5 Over 8 EndFraction EndRoot x = –1 Plus or minus StartRoot StartFraction 4 Over 8 EndFraction EndRoot 8(x2 + 2x + 1) = 3 + 1 8(x2 + 2x) = –3
The options Patel has to solve the quadratic equation 8x² + 16x + 3 = 0 is x = –1 Plus or minus StartRoot StartFraction 5 Over 8 EndFraction EndRoot.
Quadratic equation8x² + 16x + 3 = 0
8x² + 16x = -3
8(x² + 2x) = -3
Using completing the square8(x² + 2x + 1) = -3 + 8
factorization8(x² + 1) = 5
(x² + 1) = 5/8
Taking the square root of both sides(x + 1) = ± √5/8
x = -1 ± √5/8
Therefore,
x = –1 Plus or minus StartRoot StartFraction 5 Over 8 EndFraction EndRoot
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Apply the rule (-x, y - 2) to the following point, what is the result?
(-3,5)
O (-5,-5)
O (3, 3)
O (-5,3)
O (-3,7)
Answer:
[tex] - 3 \times - 1 = 3 \\ \\ 5 - 2 = 3[/tex]
Step-by-step explanation:
3,3
Answer: Second Choice. (3, 3)
Step-by-step explanation:
Given point
(x, y) = (-3, 5)
Given the rule for application
(-x, y - 2)
Determine the new x-value
x = -3
-x = - (-3) = 3
Determine the new y-value
y = 5
y - 2 = (5) - 2 = 3
Therefore, the result is [tex]\Large\boxed{(3,~3)}[/tex]
Hope this helps!! :)
Please let me know if you have any questions
A farmer determines that, on average, his chickens lay a total of 16 eggs each day. A random sample of 10 days was taken, and the mean number of eggs was 15.1 eggs. Let μ = the true mean number of eggs the chickens lay each day. Under the assumption that the true mean number of eggs is 16, 100 simulated means for samples of size 10 are shown in the dotplot. A dotplot titled mean number of eggs. A number line labeled simulated means of samples, n = 10, goes from 15.0 to 17.2. A sample mean of eggs of 15.1 or less only occurred twice. Using the dotplot, is there evidence that the chickens are laying fewer than 16 eggs? Yes, since a sample mean number of eggs of 15.1 eggs or less only occurred twice in simulated values, there is evidence that the true mean number of eggs is less than 16. Yes, since a sample mean number of eggs of 15.1 is less than the mean number of eggs of 16, there is evidence to prove that the true mean number of eggs is less than 16. No, since a sample mean number of eggs of 15.1 is only 0.9 eggs less than a mean number of eggs of 16, there is insufficient evidence that the true mean number of eggs is less than 16. No, since a sample mean number of eggs of 15.1 never occurred in the dotplot, it is not possible that a random sample of 10 eggs will have a mean number of eggs of 15.1. Therefore, there is insufficient evidence that the true mean number of eggs is less than 16.
Given the above information, it is correct to say Yes, since a sample mean number of eggs of 15.1 eggs or less only occurred twice in simulated values, there is evidence that the true mean number of eggs is less than 16. (Option A)
What is the explanation for the above?
Recall that the mean (statistically speaking) is a measure of central tendency of a probability distribution along median and mode.
You could also state that it is an expected value. Given that samples of actual value taken from the
Hence, since from 100 samples taken, the number of eggs of 15.1 or less occurred, then the true mean or actual mean is less than 16.
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an angle exceeds three times its complement by 10 find it. please explain properly i will mark as brilliant
Answer:
It's supplementary angle I think
Answer:
one angle is 20° and the other is 70°
Step-by-step explanation:
Complementary angles add up to 90°.
Here it is given that an angle exceeds 3 times its complement by 10. So, we are taking the complement as x. Since it exceeds its complement 3 times, we are multiplying the complement 3 times. so we get [tex]3x[/tex].
Now we also need to add 10, so we get the final equation [tex]3x+10[/tex], which is the second angle.
As of now, first angle = x and second angle = 3x+10.
Since they are complementary angles, both angles add up to 90°
⇒ 1st angle + 2nd angle = 90°
∴ [tex]x+3x+10 = 90[/tex] [since 3x and x are like expressions, we should add 3x + x = 4x]
⇒ [tex]4x = 90 - 10[/tex] [by transposing 10 from LHS to RHS]
⇒ [tex]4x = 80[/tex]
⇒ [tex]x = \frac{80}{4}[/tex] [by transposing 4 from LHS to RHS]
⇒ [tex]x = 20[/tex]
∴The complementary angle is 20° and the given angle is [(3 × 20) + 10)] = 70°
Determine the slope of the line passing through the two points in the graph below.
The slope of the points on the graph is 4/3
How to determine the slope?The points on the graph are
(x, y) = (4,3) and (1, -1)
Slope is calculated as:
m = (y2 - y1)/(x2 - x1)
This gives
m = (-1 - 3)/(1 - 4)
Evaluate
m = 4/3
Hence, the slope of the points on the graph is 4/3
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WHEN LOIS MARTIN WAS BORN, HER FATHER DEPOSITED $2000 IN A SAVINGS
ACCOUNT IN HER NAME AT A SAVINGS AND LOAN ASSOCIATION (S&L). AT THE TIME,
THE S&L PAID 6% INTEREST COMPOUNDED SEMI-ANNUALLY. AFTER 10 YEARS, THE
S&L CHANGED TO A RATE OF 6% COMPOUNDED QUARTERLY. WHAT WAS THE
VALUE OF THE ACCOUNT AFTER 18 YEARS WHEN THE MONEY WAS WITHDRAWN TO
HELP PAY FOR HER COLLEGE EXPENSES
The future value of the savings account, after 18 years when it was withdrawn to help pay for Lois Martin's college expenses, would be $5,816.85.
How are the future values determined?The future values of the savings account are calculated in two installments.
The first installment is for 10 years when the account earns 6% compounded semiannually.
Using the future value after 10 years, the second installment is for 8 years when the account earns 6% compounded quarterly.
Future values can be determined using the future value formula or an online finance calculator, as follows:
Data and Calculations:Investment of $2,000 for 10 years:N (# of periods) = 20 (10 years x 2)
I/Y (Interest per year) = 6%
PV (Present Value) = $2,000
PMT (Periodic Payment) = $0
Results:
FV = $3,612.22
Total Interest $1,612.22
Investment of $3,612.22 for 8 years:N (# of periods) = 32 (8 years x 4)
I/Y (Interest per year) = 6%
PV (Present Value) = $3,612.22 ($2,000 + $1,612.22)
PMT (Periodic Payment) = $0
Results:
FV = $5,816.85
Total Interest $2,204.63
Thus, the value of the account after 18 years was $5,816.85.
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A two-digit number has two less units than tens. The difference
between twice the number and the number reversed is 93. Find
the number.
The required two-digit number is 75.
What is arithmetic?In mathematics, it deals with numbers of operations according to the statements.
let the number in tenth place be x.
And has two fewer units than tens.
= x - 2
The two-digit number can be given as
= 10x + (x - 2)
= (11x - 2)
Reversing the digits
= (x - 2)10 + x
= 11x -20
The difference between twice the number and the number reversed is 93.
[tex]2(11x - 2) - (11x - 20) = 93\\22x-4-11x+20 = 93\\11x+16 = 93\\11x = 93-16\\11x = 77\\x = 7[/tex]
Now the required two-digit number is,
= 11x -2
= 11*7 - 2
= 77 -2
= 75
Thus the required two-digit number is 75.
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Problem What is the volume of this rectangular prism?
Answer:
[tex] \frac{27}{8} cm {}^{3} [/tex]
Is the volume of this right rectangular prism.
Step-by-step explanation:
Rectangular prism volume formula is:-
[tex]v = l \times w \times h \\ v = \frac{3}{2} \times \frac{3}{2} \times \frac{3}{2}apply \: fraction \: rule \\ v = \frac{27}{8} .... \frac{a}{b} \times \frac{c}{d} = \frac{a \times b}{c \times d} [/tex]Answer:
[tex]V=\frac{27}{8}\: cm^3[/tex]
Step-by-step explanation:
The formula to find the volume of a rectangular prism is:
V = length × height × width
Let us find now.
[tex]V = length \:* height \:* width\\\\V =\frac{3}{2} *\frac{3}{2} *\frac{3}{2} \\\\V=\frac{9}{4}*\frac{3}{2} \\\\ V=\frac{27}{8}\: cm^3[/tex]
8 -2 5\8 how do I solve this. Can u show me the steps
Answer:
= 5.375
Step-by-step explanation:
Use the algorithm method.
7 9 9 10
8 . 0 0 0
- 2 . 6 2 5
5 . 3 7 5
= 5.375
Answer:
5 3/8
Step-by-step explanation:
change it so that every number is a improper fraction and same denominator
64/8 - 21/8 = 43/8 = 5 3/8
Tami earned $20.64 in simple interest by investing a principal of $400 in a Treasury bill.
If the interest rate was 1.72%/a, for how many years did she have her investment?
The number of years she had her investment is after 3 years
How to determine the years of investment?From the question, the given parameters are:
Principal Amount, P = $400Interest Rate, r = 1.72%Simple Interest, I = $20.64The number of years (T) is calculated from the following simple interest formula
I = PRT
Substitute the given parameters in the above equation
20.64 = 400 * 1.72% * t
Express 1.72% as decimal without percentage
20.64 = 400 * 0.0172 * t
Evaluate the product
20.64 = 6.88 * t
Divide both sides by 6.88
20.64/6.88 = 6.88/6.88 * t
Evaluate the quotient
3 = t
Rewrite the equation as
t = 3
This means that she had her investment after 3 years
Hence, the number of years she had her investment is after 3 years
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when a fraction of 17 is taken away from 17 what remains exceeds one third of seventeen by six Using symbolic language
Fraction is a topic that deals with expressing the relationship between two numbers or terms in the form of a ratio. So that the required symbolic language required in the question is: 17 - [tex]\frac{x}{17}[/tex] = [tex]\frac{17}{3}[/tex] + 6
Thus the value of x is 90[tex]\frac{2}{3}[/tex].
Fraction is a topic that deals with expressing the relationship between two numbers or terms in the form of a ratio. Some types of fractions are mixed fractions, proper fractions, and improper fractions.
Thus to express the given question in a symbolic language, let the fraction of 17 taken away be represented by x.
So that;
i. a fraction of 17 is taken away from 17 can be expressed as 17 - [tex]\frac{x}{17}[/tex].
ii. remains exceeds one-third of seventeen by six can be expressed as [tex]\frac{17}{3}[/tex] + 6
Therefore the required symbolic language to the question is:
17 - [tex]\frac{x}{17}[/tex] = [tex]\frac{17}{3}[/tex] + 6
So that,
[tex]\frac{289 - x}{17}[/tex] = [tex]\frac{17 + 18}{3}[/tex]
cross multiply to have
3(289 - x) = 17(17 + 180)
867 - 3x = 595
3x = 867 - 595
=272
x = [tex]\frac{272}{3}[/tex]
= 90[tex]\frac{2}{3}[/tex]
x = 90[tex]\frac{2}{3}[/tex]
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07* Calculate the limits
(a) lim x→∞ (3x^7/2+7x^-1/2) / (x²-x^1/2)
infinity
-infinity
3
1
The limit is
[tex]\displaystyle \lim_{x\to\infty} \frac{3x^{7/2} + 7x^{-1/2}}{x^2 - x^{1/2}} = \lim_{x\to\infty} \frac{3x^{3/2} + 7x^{-5/2}}{1 - x^{-3/2}} = \boxed{\infty}[/tex]
since the degree of the numerator exceeds the degree of the denominator.
Using the quote, "Those of us on the margins wonder if our stories matter" what is the significance of the quoted text?
The significance of the quoted text is to illustrate the less privileged in an overwhelmingly crowded and disordered chronological reality.
What is a quote?Quote means to cite something as a form of proof. It has several other senses as a verb and a noun. It should be noted that to quote something or someone is to repeat the exact words they said or to recite the exact words written in a book.
Great speakers often quote other inspiring people when making speeches. When you quote, you include the words and ideas of others in your text exactly as they have expressed them.
You signal this inclusion by placing quotation marks (“ ”) around the source author's words and providing an in-text citation after the quotation.
In this case, the significance of the quoted text is to illustrate the less privileged in an overwhelmingly crowded and disordered chronological reality. A quotation is the repetition of a sentence, phrase, or passage from speech or text that someone has said or written.
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The question is below
The events E1 and E2 are not independent events.
How to determine if the events are independent?The two events are given as:
E1 and E2 such that
P(E1) = 0.27, P(E2) = 0.40 and P(E1 U E2) = 0.58.
The two events E1 and E2 are independent if the following equation is true
P(E1 and E2) = P(E1) * P(E2)
Where
P(E1 and E2) = P(E1) + P(E2) - P(E1 U E2)
Substitute the known values in the above equation
P(E1 and E2) = 0.27 + 0.40 - 0.58
Evaluate the sum in the above equation
P(E1 and E2) = 0.67 - 0.58
Evaluate the difference in the above equation
P(E1 and E2) = 0.09
Substitute P(E1 and E2) = 0.09 in P(E1 and E2) = P(E1) * P(E2)
P(E1) * P(E2) = 0.09
Substitute P(E1) = 0.27 and P(E2) = 0.40 in the above equation
0.27 * 0.40 = 0.09
Evaluate the product
0.108 = 0.09
The above is false because 0.108 and 0.09 are not equal
Hence, the events E1 and E2 are not independent events.
Independent Events A and BHow to solve for x
We have:
P(A) = x
P(B) = x + 0.2
P(A n B) = 0.15
For independent events, we have:
P(A) * P(B) = P(A n B)
So, we have:
x * (x + 0.2) = 0.15
Expand
x^2 + 0.2x - 0.15 = 0
Using a graphing calculator, we have:
x = 0.3
Hence, the value of x is 0.3
The value of P(A U B)
This is calculated using
P(A U B) = P(A) + P(B) - P(A n B)
Substitute the known values in the above equation
P(A U B) = x + x + 0.2 - 0.15
Evaluate the like terms
P(A U B) = 2x + 0.05
Substitute 0.3 for x
P(A U B) = 2 * 0.3 + 0.05
Evaluate
P(A U B) = 0.65
Hence, the value of P(A U B) is 0.65
The value of P(A' U B')
This is calculated using
P(A' U B') = P(A') + P(B') - P(A' n B')
Where
P(A') = 1 - x
P(B') = 1 - x - 0.2 = 0.8 - x
P(A' n B') = P(A') * P(B')
P(A' n B') = (1 - x) * (0.8 - x)
Substitute the known values in the equation P(A' U B') = P(A') + P(B') - P(A' n B')
P(A' U B') = (1 - x) + (0.8 - x) - (1 - x) * (0.8 - x)
Substitute 0.3 for x
P(A' U B') = (1 - 0.3) + (0.8 - 0.3) - (1 - 0.3) * (0.8 - 0.3)
Evaluate
P(A' U B') = 0.85
Hence, the value of P(A' U B') is 0.85
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- Describe the slope of each line. Then find the slope.
13
Company A has 800 employees, and it decides to grant each of the employees 50 share options as
part of its new rewards plan. The options are exercisable over 5 years and subject to a 3-year
service condition. The fair value of each option at the grant date is $16. The company estimates
that 80% of its employees will meet the service condition required for receiving the options.
Calculate the total share-based payment expense for Company A assuming that 80% of the
employees actually meet the service condition.
$512,000
$853,333
$341,333
$170,667
Option A. The total share expense that the company would share would be given as 512,000
What is meant by share expense?These are the necessary expenses that are needed for the smooth functioning of a particular business that are not within the confinement of the O and M agreement. It has to do with shared facilities.
How to solve for the share expenseThe total number of the employees that are knwon to satistfy condition are given as
800 * 0.8
= 640
The options that are estmated that would be exercised
This is given as the employees * share option
= 640×50
=32000.
The total shae for the company would be gotten as
= 32000× $16
This gives us $512,000.
Hence it can be concluded that the total share of the company if they have 80 percent meeting the condition is $512000.
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Complete the table of inputs and outputs for the function.
f(x) = -5(x + 7)
X
-9
01
0
f(x)
0
-60
Answer:
10, -7, -35, 5
Step-by-step explanation:
f(9) = -5(-9 + 7) = 10
0 = -5(x + 7)
0 = x + 7
x = -7
f(0) = -5(0 + 7) = -35
-60 = -5(x + 7)
12 = x + 7
x = 5
Draw a line between the points representing wicket 3 and wicket 9 on the diagram. Then, find the
area of the rectangle. Show your work and round your answer to the nearest hundredth.
Based on the calculations, the area of a rectangle to the nearest hundredth is equal to 1143.92 ft².
How to calculate the area of a triangle?Mathematically, the area of a triangle can be calculated by using this formula:
Area = 1/2 × b × h
Where:
b represents the base area.h represents the height.By drawing a line between the points representing wicket 3 and wicket 9, we can logically deduce that wicket 2 forms a perpendicular bisector. Thus, the distance between the points representing wicket 3 and wicket 9 is given by:
Distance = 1/2 × 36.77
Distance = 36.77/2
Distance = 18.39 ft.
For the height of this triangle, we would apply Pythagorean's theorem:
h² = 25.02² - 18.39²
h² = 626.0004 - 338.1921
h² = 287.8083
h = √287.8083
h = 16.97 ft.
Area of triangle = 1/2 × b × h
Area of triangle = 1/2 × 36.77 × 16.97
Area of triangle = 311.99 ft².
For area of the rectangle, we have:
Mathematically, the area of a rectangle can be calculated by using this formula;
Area = LW
Area = 36.77 × 31.11
Area = 1143.92 ft².
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Find the maximum value of s = xy + yz + xz where x+y+z=9.
From the constraint, we have
[tex]x+y+z=9 \implies z = 9-x-y[/tex]
so that [tex]s[/tex] depends only on [tex]x,y[/tex].
[tex]s = g(x,y) = xy + y(9-x-y) + x(9-x-y) = 9y - y^2 + 9x - x^2 - xy[/tex]
Find the critical points of [tex]g[/tex].
[tex]\dfrac{\partial g}{\partial x} = 9 - 2x - y = 0 \implies 2x + y = 9[/tex]
[tex]\dfrac{\partial g}{\partial y} = 9 - 2y - x = 0[/tex]
Using the given constraint again, we have the condition
[tex]x+y+z = 2x+y \implies x=z[/tex]
so that
[tex]x = 9 - x - y \implies y = 9 - 2x[/tex]
and [tex]s[/tex] depends only on [tex]x[/tex].
[tex]s = h(x) = 9(9-2x) - (9-2x)^2 + 9x - x^2 - x(9-2x) = 18x - 3x^2[/tex]
Find the critical points of [tex]h[/tex].
[tex]\dfrac{dh}{dx} = 18 - 6x = 0 \implies x=3[/tex]
It follows that [tex]y = 9-2\cdot3 = 3[/tex] and [tex]z=3[/tex], so the only critical point of [tex]s[/tex] is at (3, 3, 3).
Differentiate [tex]h[/tex] again and check the sign of the second derivative at the critical point.
[tex]\dfrac{d^2h}{dx^2} = -6 < 0[/tex]
for all [tex]x[/tex], which indicates a maximum.
We find that
[tex]\max\left\{xy+yz+xz \mid x+y+z=9\right\} = \boxed{27} \text{ at } (x,y,z) = (3,3,3)[/tex]
The second derivative at the critical point exists
[tex]$\frac{d^{2} h}{d x^{2}}=-6 < 0[/tex] for all x, which suggests a maximum.
How to find the maximum value?Given, the constraint, we have
x + y + z = 9
⇒ z = 9 - x - y
Let s depend only on x, y.
s = g(x, y)
= xy + y(9 - x - y) + x(9 - x - y)
= 9y - y² + 9x - x² - xy
To estimate the critical points of g.
[tex]$&\frac{\partial g}{\partial x}[/tex] = 9 - 2x - y = 0
[tex]$&\frac{\partial g}{\partial y}[/tex] = 9 - 2y - x = 0
Utilizing the given constraint again,
x + y + z = 2x + y
⇒ x = z
x = 9 - x - y
⇒ y = 9 - 2x, and s depends only on x.
s = h(x) = 9(9 - 2x) - (9 - 2x)² + 9x - x² - x(9 - 2x) = 18x - 3x²
To estimate the critical points of h.
[tex]$\frac{d h}{d x}=18-6 x=0[/tex]
⇒ x = 3
It pursues that y = 9 - 2 [tex]*[/tex] 3 = 3 and z = 3, so the only critical point of s exists at (3, 3, 3).
Differentiate h again and review the sign of the second derivative at the critical point.
[tex]$\frac{d^{2} h}{d x^{2}}=-6 < 0[/tex]
for all x, which suggests a maximum.
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Simplify: 9x + 2(x - 3) - 2x + 10
Answer: 9x+4
Step-by-step explanation: We take out the parentheses by multiplying both values by 2 (2 times x and 2 times 3) to get 2x-6. Now our equation is 9x+2x-6-2x+10. We can see that there is a +2x and a -2x so we just take both of them out as they cancel out each other. Now our equation is 9x -6+10. we add 10 to -6 to get +4. The result is 9x+4.
Answer:
9x + 4
Step-by-step explanation:
9x + 2x - 6 - 2x + 10
Rearranging,
=> 9x + 2x - 2x + 10 - 6
=> 9x + 4