The correct statements are:
Mailani saves $200 in a box under her bed.The y-intercept of the function is (0, 700).Explanation:
The function f(x) represents the total amount of money that Mailani has saved after x years. The first term 500 represents the initial amount she saved in the box under her bed, and the second term (1+0.04)^x represents the amount she deposited into the savings account, which earns an interest rate of 4% compounded annually. The third term +200 represents the additional amount she deposited every year into the savings account.
The range of the function is y>=500, since the initial amount saved is 500, and the function is increasing with time.The y-intercept of the function is (0, 700), not (0, 200), because the initial amount saved in the box is 500, and the additional amount deposited into the savings account is 200 per year, so after 0 years, the total saved amount is 500+200=700.The function does not have an asymptote at y=200. An asymptote is a line that a graph approaches but never touches. In this case, the function is increasing with time, so it does not approach any line as x gets larger.Learn more about asymptote here brainly.com/question/4084552
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Full question:
Mailani saves some money in a box under her bed. She deposits an additional amount into a savings account where it earns interest compounded annually. . This situation can be modeled by the function
[tex]f(x)= 500(1+0.04)^x+200[/tex]
Which statements are true? Select all that apply.
The range of the function is y>=0Mailani saves $200 in a box under her bed.The y-intercept of the function is (0, 200).The function has an asymptote at y = 200.Solve for x please and thank you
The value of x in the given equation of the quadrilateral is 20.
What is the value of x?
The value of x is determined by applying the property of a quadrilateral which states that adjacent angles of a quadrilateral are supplementary while the opposite angles are equal.
(2x + 14) + 126 = 180
We can simplify the left side of the equation by combining the terms inside the parentheses:
2x + 140 = 180
Next, we can isolate the variable x by subtracting 140 from both sides of the equation:
2x = 40
Finally, we can solve for x by dividing both sides of the equation by 2:
x = 20
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Possible rational zeros of f(x)=4x^5-2x^3+10
Possible rational zeros of the polynomial is
±1, ±1/2, ±1/4, ±2, ±5, ±5/2, ±5/4, ±10.
What is polynomial?Polynomials are particular type of algebraic expressions which consists of variables and coefficients. Various arithmetic operations such as addition, subtraction, multiplication, and also positive integer exponents for polynomial expressions can be performed on polynomial but not division by variable. An example of a polynomial is x-12 . Here it has two terms: x and -12.
Given polynomial p(x)= 4x⁵-2x³+10
The polynomial can be rewritten as p(x)= 4x⁵+ 0x⁴-2x³+0x²+0x+10
Comparing the polynomial with
p(x) = aₙ xⁿ + aₙ₋₁xⁿ⁻¹+ ---------- + a₁ x + a₀ we get the leading coefficient
aₙ= 4 and the constant term a₀= 10
So the possible zeros are= ±( factors of a₀)/ (factors of aₙ)
Factors of 10:
1, 2, 5, 10
Factors of 4:
1, 2, 4
Hence, Possible rational zeros:
±1, ±1/2, ±1/4, ±2, ±5, ±5/2, ±5/4, ±10
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There are 18 students in Ms. Avila's reading class. Ms. Avila will assign an equal number of pages for each student to read aloud from a book that contains a total of 45 pages. What is the total number of pages that each student will read aloud? Select one answer. A 2/5 B 2 1/2 C 27 D 63 also subscribe to friends channel it's called your local kirby guy.
Each student will read 2 1/2 pages aloud. The answer is option B, 2 1/2.
What is arithmetic sequence?
An arithmetic sequence is a sequence of numbers in which each term after the first is found by adding a fixed constant number, called the common difference, to the preceding term.
For example, the sequence 2, 5, 8, 11, 14, ... is an arithmetic sequence with a common difference of 3, since each term after the first is found by adding 3 to the preceding term.
Ms. Avila's reading class has 18 students and they are going to read aloud from a book that has 45 pages. To determine the total number of pages that each student will read, we need to divide the total number of pages by the number of students.
So, 45 divided by 18 equals 2 1/2 pages per student.
This means that each student will read 2 1/2 pages aloud.
Therefore, the answer to the question is option B, 2 1/2.
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Simplify the expression 3(x + 2) + 2(x + 3)
Answer:
5x + 12
Step-by-step explanation:
3(x + 2) + 2(x + 3)
= 3x + 6 + 2x + 6 (distributing the coefficients)
= 5x + 12 (combining like terms)
Answer:5x+12
Step-by-step explanation:
3(x+2)= 3x+6
2)x+3) = 2x+6
3x+2x+6+6=5x+12
HOPE I HELPED :)
Find the value of x.
Answer:
Step-by-step explanation:
I am assuming the triangles are similar.
You can make a proportion between the two triangles and solve for x but you must match the sides properly.
RS ≈ MN
ST ≈ NO
Making a proportion between the sides:
[tex]\frac{x}{3}[/tex] = [tex]\frac{6}{4}[/tex]
cross multiplyand then divide
6 · 3 = 18
18 ÷ 4 = 4.5
x = 4.5
If you check the ratios - they are all the same.
For example 6 ÷ 4 = 1.5
7.5 ÷ 5 = 1.5
4.5 ÷ 3 = 1.5
this checks out
What square root best approximates the point on the graph?
A number line going from 0 to 9. A point is slightly to the right of 5.
StartRoot 5 EndRoot
StartRoot 15 EndRoot
StartRoot 28 EndRoot
StartRoot 53 EndRoot
Answer:
The correct option is C. The best approximate for the given graph is [tex]\sqrt{28}[/tex].
Step-by-step explanation:
From the given graph it is noticed that the point lies between 5 and 6.
Let the square root of x is the best approximate for the given graph.
[tex]5 < \sqrt{x} < 6[/tex]
Taking square on each side.
[tex]5^2 < (\sqrt{x})^2 < 6^2[/tex]
[tex]25 < x < 36[/tex]
It means the value of x lies between 25 to 36 and the graph represents the approximate value between [tex]\sqrt{25}[/tex] to [tex]\sqrt{36}[/tex].
Therefore best approximate for the given graph is [tex]\sqrt{28}[/tex]. Option C is correct.
If the area of a square equals 25 sq inches, what is the side?
Answer: 5
Step-by-step explanation:
If the area of a square equals 25 square inches, then the side length of the square can be found by taking the square root of the area. The formula for the area of a square is A = s^2, where A is the area and s is the length of a side of the square.
So, in this case, we have:
A = 25 sq inches
s = sqrt(A) = sqrt(25 sq inches) = 5 inches
Therefore, the side of the square is 5 inches.
Your friend incorrectly says that the reflection of EFG to its image E'F'G' is a reflection across the x-axis
a. What is your friend’s mistake
b. What is the correct description of the reflection
a. Your friend's mistake is that the reflection of EFG to its image E'F'G' is not a reflection across the x-axis.
b. The correct description of the reflection is a reflection across the line y = x.
What is The correct description of the reflectionThe correct description of the reflection are:
a. Your friend's mistake is that the reflection of EFG to its image E'F'G' is not a reflection across the x-axis.
b. The correct description of the reflection is a reflection across the line y = x.
This is because the line of reflection is the perpendicular bisector of the segment connecting each point to its image. In this case, the segment connecting E to E' has a slope of 1 (since it is a diagonal line), so the perpendicular bisector of this segment must have a slope of -1.
The line with slope -1 passing through the midpoint of this segment (which is the point on the line y = x) is the line of reflection. Therefore, the reflection of EFG to its image E'F' is a reflection across the line y = x, not the x-axis.
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What number is 250% of 4.2
Answer:
10.5
Step-by-step explanation:
250% = 2.5
What number is 250% of 4.2?
We Take
2.5 x 4.2 = 10.5
So, 10.5 is 250% of 4.2
Expression that represents a quadratic expression
we can conclude that the expression [tex]2x^2 + 3x -1[/tex] is the quadratic expression among the given options.
Explain about Quadratic equation?A quadratic expression is an algebraic expression in which the highest power of the variable is 2. The general form of a quadratic expression is:
[tex]ax^2 + bx + c[/tex]
where a, b, and c are constants and x is the variable. The coefficient a must be non-zero for the expression to be considered quadratic.
In the given options, we can see that only one expression has a degree of 2, which is the second term of the quadratic expression.
So, we can conclude that the expression [tex]2x^2 + 3x -1[/tex] is the quadratic expression among the given options.
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A garden is being built in the shape below. How many square yards of space will the garden occupy?
An electronic book has a file size of 2.4 megabytes what is the file size in megabytes of 16 of these electronic books?
Answer:
38.4 megabytes for 16 electronic books
Step-by-step explanation:
2.4 megabytes per electronic book
2.4 megabytes x 16 electronic books = 38.4 megabytes
Current research indicates that the distribution of the life expectancies of a certain protozoan is normal with a mean of 46 days and a standard deviation of 10.5 days. Find the probability that a simple random sample of 49 protozoa will have a mean life expectancy of 47 or more days.
a) 0.7475
b) 0.5379
c) 0.4525
d) 0.2525
e) 0.5050
0.2525 is the probability that a simple random sample.
What is the simple definition of probability?
A probability is a number that expresses the possibility or likelihood that a specific event will take place. Probabilities can be stated as proportions with a range of 0 to 1, or as percentages with a range of 0% to 100%. Probability is a measure of how likely or likely-possible something is to happen.
Since the distribution of the life expectancies of a certain protozoan is normal, we would apply the formula for normal distribution which is expressed as
z = (x - µ)/σ
Where
x = life expectancies of the certain protozoan.
µ = mean
σ = standard deviation
n = number of samples
From the information given,
µ = 46 days
σ = 10.5 days
n = 49
The probability that a simple random sample of 49 protozoa will have a mean life expectancy of 47 or more days is expressed as
P(x ≥ 47) = 1 - P(x < 47)
For x = 47
z = (47 - 46)/(10.5/√49) = 0.67
Looking at the normal distribution table, the probability corresponding to the z score is 0.0.75
P(x ≥ 47) = 1 - 0.75 = 0.25
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At some moment, Julia measured the angle between the minute and the hour hands of an anlog clock. In exactly one hour, Julia measured the angle again and got the same result. Find all possible angles.
Answer: 165 degrees
Step-by-step explanation:
Let's start off and name the angle between the hour and minute hand x, representing the number of degrees between the hands. Next, let's look at the minute hand first. After exactly an hour, the minute hand does not change its position(since after one hour, it comes back to where it was previously). As for the hour hand however, its position differs by exactly "1 hour tick", which is equivalent to 30 degrees (360 / 12 hours). What that implies is that x would have to equal x + 30, which is clearly impossible. However, we can think about the problem in a different way: by looking at the acute angle only between the hands, that would indeed be possible. Assuming that x + 30 is greater than 180(since if it wasn't, that wouldn't be possible), we can write the following equation:
x = 360 - (x+30)
, with the right hand side coming from the fact that the obtuse angle of x + 30 must have an acute counterpart, of which both add up to 360.
Then, we can simplify to:
x = 330 - x
or:
2x = 330
x = 165.
This gives us our only answer of 165 degrees
Find the exact value of the real number y. y = arcsin ( square root 3/2)
Find the slope of a line parallel to the line whose equation is 3x-y=-1. fully simplify your answer 
Answer:
To find the slope of a line parallel to 3x - y = -1, we need to first rearrange this equation into slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. Starting with 3x - y = -1, we can add y to both sides to get: 3x = y - 1 Then, we can add 1 to both sides to get: 3x + 1 = y This is now in slope-intercept form, with y = 3x + 1, so we can see that the slope of the line is 3. Since we want to find the slope of a line parallel to this one, we know that it must have the same slope of 3. Therefore, the answer is: Slope = 3.
NEED HELP ASAP
Find f(2) for the piece-wise function
f(x) =
-2x+1 if x ≤ 1
-x+2
if x > 1
f(2) = [?]
The function f(2) from the piecewise function has a value of 0
From the question, we have the following parameters that can be used in our computation:
The piecewise function
The x value of 2 belongs to the domain x > 2
So, we have
f(x) = -x + 2
Substitute the known values in the above equation, so, we have the following representation
f(2) = -2 + 2
Evaluate
f(2) = 0
Hence, the value iss 0
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What is the area of ABC if AD = 20, DB = 15, and CD = 9?
Answer:
the area is 1000
Step-by-step explanation:
This one is for the boys with the booming system
Top down, AC with the cooler system
find the minors of the matrix A 2*2
Answer: Given a 2x2 matrix A:
A = [a11 a12]
[a21 a22]
The minor of any element a_ij is the determinant of the 1x1 matrix obtained by deleting the i-th row and j-th column from A. In other words, the minor of a_ij is given by:
M_ij = det(A_ij)
where A_ij is the matrix obtained by deleting the i-th row and j-th column from A.
For example, the minor of a11 is given by:
M_11 = det([a22])
= a22
Similarly, we can find the minors of the other elements:
M_12 = det([a21])
= a21
M_21 = det([a12])
= a12
M_22 = det([a11])
= a11
Therefore, the minors of the matrix A are:
M = [a22 a21]
[a12 a11]
Step-by-step explanation:
find the value of x
Answer:
The triangles are similar by AA since vertical angles are congruent.
[tex] \frac{8}{12} = \frac{x}{7} [/tex]
[tex]12x = 56[/tex]
[tex]x = \frac{14}{3} = 4 \frac{2}{3} [/tex]
A TV cable company has 6400 subscribers who are each paying $28 per month. It can get 160 more subscribers for each $0.50 decrease in the monthly fee. What rate will yield maximum revenue, and what will this revenue be?
Answer:
Step-by-step explanation:
Here is a step-by-step explanation of the solution and how it leads to the maximum revenue:
Let's start with the current revenue generated by the TV cable company, which is the product of the number of subscribers and the monthly fee:
Current revenue = Number of subscribers * Monthly fee
Current revenue = 6400 * $28
Current revenue = $179,200
To increase the revenue, the TV cable company can decrease the monthly fee by $0.50, which will result in an increase in the number of subscribers by 160 for each $0.50 decrease. Let's calculate the additional revenue generated by each $0.50 decrease in the monthly fee:
Additional revenue per decrease in monthly fee = Increase in subscribers * Decrease in monthly fee
Additional revenue per decrease in monthly fee = 160 * $0.50
Additional revenue per decrease in monthly fee = $80
The TV cable company can continue to decrease the monthly fee by $0.50 until the revenue stops increasing. Let's calculate the number of $0.50 decreases in monthly fee required to reach the maximum revenue:
Number of $0.50 decreases in monthly fee = (Total revenue - Current revenue) / Additional revenue per decrease in monthly fee
Number of $0.50 decreases in monthly fee = ($20,000,000 - $179,200) / $80
Number of $0.50 decreases in monthly fee = 248,525
The corresponding decrease in monthly fee is:
Decrease in monthly fee = Number of $0.50 decreases in monthly fee * Decrease in monthly fee
Decrease in monthly fee = 248,525 * $0.50
Decrease in monthly fee = $124,262.50
The corresponding increase in subscribers is:
Increase in subscribers = Number of $0.50 decreases in monthly fee * Increase in subscribers
Increase in subscribers = 248,525 * 160
Increase in subscribers = 39,764,000
The total number of subscribers at the maximum revenue point is:
Total subscribers = 6400 + Increase in subscribers
Total subscribers = 6400 + 39,764,000
Total subscribers = 39,770,400
The monthly fee at the maximum revenue point is:
Monthly fee = Current revenue / Total subscribers
Monthly fee = $179,200 / 39,770,400
Monthly fee = $0.0045
The maximum revenue is:
Maximum revenue = Total subscribers * Monthly fee
Maximum revenue = 39,770,400 * $0.0045
Maximum revenue = $178,966.80
Therefore, the TV cable company can generate maximum revenue of $178,966.80 per month by decreasing the monthly fee by $0.50 for each 160 additional subscribers.
Claim: The mean pulse rate (in beats per minute) of adult males is equal to 69 bpm. For a random sample of 154 adult
males, the mean pulse rate is 69.5 bpm and the standard deviation is 11.4 bpm. Find the value of the test statistic.
The value of the test statistic is
(Round to two decimal places as needed.)
According to the question the value of the test statistic [tex]0.55[/tex]
What is standard deviation?Data variation from the mean is referred to as "standard deviation." If the standard deviation appears low, the data tend to cluster around the mean; on the other hand, if it appears large, the data are widely spread.
The data deviate from the mean value is indicated by the standard . It is useful for contrasting huge datasets with distinct ranges but the same mean.
The other, though, is without a doubt more spread. The standard deviation reveals the degree of data dispersion. It describes how far away from the mean each number obtained is.
We must compute the t-score, which is represented by: in order to determine the value of the test statistic.
[tex]t = x -[/tex] μ[tex]/(s/\sqrt{n}[/tex]
s= sampling standard deviation, n= sample size,[tex]x=[/tex] sampling rate μ = population mean
[tex]x = 69.5 bpm,[/tex] μ[tex]= 69 bpm, s = 11.4 bpm, n = 154[/tex]
[tex]t = (69.5- 69)/ 11.4/\sqrt{154} \\t = 0.5/ (11.4/12.4)\\t = 0.5/ 0.917\\t = 0.546[/tex]
Therefore the value of the test statistics is [tex]0.55[/tex]
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What's the area of the park in square units
What is the measurement of angle A?
(look at pic!!)
The measure of angle A is 59°
What is cyclic quadrilateral?A cyclic quadrilateral is a quadrilateral which has all its four vertices lying on a circle. It is also sometimes called inscribed quadrilateral. The circle which consists of all the vertices of any polygon on its circumference is known as the circumcircle or circumscribed circle.
The sum of opposite angle in a cyclic quadrilateral is 180. This means, 121+<A = 180°
therefore,
= angle A = 180-121
= 59°
therefore the measure of angle A is 59°
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A large passenger airplane is making the 9,537-mile trip from New York to Singapore. During this flight, the plane will use 47,685 gallons of fuel, or 5 gallons per mile of flight. The function F(d) represents the amount of fuel, in gallons, consumed by the airplane on this trip after d miles of flight.
What is the domain of F(d)?
The domain of F(d) is: {d ∈ R | d ≥ 0}
What is Domain:The domain is a mathematical concept that refers to the set of all possible input values (independent variables) for which a function is defined. In other words, the domain of a function is the range of values that can be used as input to the function.
Here we have
A large passenger airplane is making the 9,537-mile trip from New York to Singapore. During this flight, the plane will use 47,685 gallons of fuel or 5 gallons per mile of flight.
The function F(d) represents the amount of fuel, in gallons, consumed by the airplane on this trip after d miles of flight.
Therefore, we can express F(d) as:
F(d) = 5d
The domain of F(d) represents the set of all valid values of d for which the function is defined.
In this case, d represents the distance flown by airplane, and it must be a non-negative number since the plane cannot fly a negative distance.
Therefore,
The domain of F(d) is: {d ∈ R | d ≥ 0}
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(08.01) Two lines, A and B, are represented by the following equations: Line A: 3x + 3y = 12 Line B: x + y = 4 Which statement is true about the solution to the set of equations? (4 points) Question 2 options: 1) It is (12, 4). 2) There are infinitely many solutions. 3) It is (4, 12). 4) There is no solution.
Answer:
Step-by-step explanation:
The solution to the set of equations can be found by solving the system:
3x + 3y = 12
x + y = 4
We can simplify the second equation by solving for y:
y = 4 - x
Substituting this expression for y into the first equation, we get:
The solution to the set of equations can be found by solving the system:
3x + 3y = 12
x + y = 4
We can simplify the second equation by solving for y:
y = 4 - x
Substituting this expression for y into the first equation, we get:
3x + 3(4 - x) = 12
3x + 12 - 3x = 12
12 = 12
This is a true statement, which means that the system is consistent and has infinitely many solutions. Therefore, the correct answer is:
There are infinitely many solutions.
Answer:
Infinitely many solutions.Step-by-step explanation:
To find:-
The correct option from the given ones .Answer:-
We are here given that there are two linear equations, namely,
[tex]\begin{cases} 3x+3y = 12 \\ x+y = 4 \end{cases}[/tex]
These can be rewritten as ,
[tex]\begin{cases} 3x+3y - 12 =0\\ x+y -4=0 \end{cases}[/tex]
Before we precede we must know that,
Conditions for solvability :-
If there are two linear equations namely,
[tex]\begin{cases} a_x + b_1y+c_1 = 0 \\ a_2x+b_2y+c_2=0\end{cases}[/tex]
Then ,
Case 1 :-
If we have,
[tex]\longrightarrow \boxed{ \dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2} } \\[/tex]
Then , the lines are coincident and there are infinitely many solutions .
Case 2 :-
If we have,
[tex]\longrightarrow \boxed{ \dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}\neq\dfrac{c_1}{c_2} } \\[/tex]
Then, the linear equations are inconsistent and have no solutions , thus the lines are parallel .
[tex]\rule{200}2[/tex]
So here with respect to angle standard form of pair linear equations, we have;
[tex]a_1 = 3 [/tex] , [tex]b_1 = 3 [/tex] , [tex] c_1 = -12 [/tex][tex]a_2= 1 [/tex] , [tex]b_2 = 1 [/tex] , [tex] c_2 = -4 [/tex]Hence here we have,
[tex]\longrightarrow \dfrac{a_1}{a_2} = \dfrac{3}{1} \\[/tex]
[tex]\longrightarrow \dfrac{b_1}{b_2}=\dfrac{3}{1} \\[/tex]
[tex]\longrightarrow \dfrac{c_1}{c_2}=\dfrac{-12}{-4}=\dfrac{3}{1} \\[/tex]
Therefore we can clearly see that,
[tex]\longrightarrow \boxed{ \dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2} =\boxed{\dfrac{3}{1}}} \\[/tex]
Hence there are infinitely many solutions and the lines are coincident .
Complete the statement below about the two figures.
Choices:
the Two figures are (Not similar/similar) because 2/5
(Equals/ does not equal) 3/6
The Two figures are (Not similar) because 2/5 ( does not equal) 3/6.
What is similar rectangle?
If two rectangles are similar, their corresponding sides are proportionately equal.
Now comparing the length and width of the two rectangles then.
=> 2/5 = 3/6
Which is not equally proportional.
Hence the both rectangles are similar because 2/5 does not equal 3/6 .
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A water desalination plant can produce 2.46x10^6 gallons of water in one day. How many gallons can it produce in 4 days?
Write your answer in scientific notation.
The desalination plant can produce [tex]9.84*10^6[/tex] gallons of water in four day in scientific notation.
We must divide the daily production rate by the number of days in order to get the total volume of water that a desalination plant can generate in four days:
4. days at a rate of [tex]2.46*10^6[/tex] gallons equals [tex]9.84*10^6[/tex] gallons.
Hence, in four days, the desalination plant can produce [tex]9.84 * 10^6[/tex] gallons of water.
Large or small numbers can be conveniently represented using scientific notation, especially when working with measurements in science and engineering. A number is written in scientific notation as a coefficient multiplied by 10 and raised to a power of some exponent. For example, [tex]2.46*10^6[/tex] denotes 2,460,000, which is 2.46 multiplied by 10 to the power of 6.
The solution in this case is [tex]9.84*10^6[/tex], or 9,840,000, which is 9.84 multiplied by 10 to the power of 6. Large numbers can be written and compared more easily using this format, and scientific notation rules can be used to conduct computations with them.
In conclusion, the desalination plant has a four-day capacity of [tex]9.84 * 10^6[/tex] gallons of water production.
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A rectangular placemat is 18 inches long and 12 inches wide. What is the area of this tablecloth in square inches?
a. 216
b. 60
c. 54
d. 900
In a rectangle, there are 2 pairs of congruent sides. Therefore, the area of a rectangle can be found using:
[tex]\text{A}=\text{l} \times \text{w}[/tex]
We know that the length is 18 and the width is 12, so we can multiply 18 in for l, and 12 in for w.
[tex]\text{A}=\text{18} \times \text{12}[/tex]
Multiply the numbers together
[tex]\text{A}=216[/tex]
So, the area is 216 inches.
To find the area of a rectangular placemat, you multiply the length by the width. In this case, the length is 18 inches and the width is 12 inches. Therefore, the area of the tablecloth is:
Area = Length × Width
Area = 18 inches × 12 inches
Area = 216 square inches
So, the correct answer is option a. 216.
Consider the following function.
g(x, y) = e^(− 4x^2 + 7y^2 + 14√8y
(a) Find the critical point of g.
If the critical point is (a, b) then enter 'a,b' (without the quotes) into the answer box.
(b) Using your critical point in (a), find the value of D(a, b) from the Second Partials test that is used to classify the critical point.
(c) Use the Second Partials test to classify the critical point from (a).
The critical point of the function g is (0,0), the value of D(0,0) is -6272, and the critical point is a saddle point.
(a) To find the critical point of g, we need to find the partial derivatives of g with respect to x and y, and set them equal to zero:
[tex]∂g/∂x = -8xe^(-4x^2+7y^2+14√8y) = 0[/tex]
[tex]∂g/∂y = 14ye^(-4x^2+7y^2+14√8y) + 14√8e^(-4x^2+7y^2+14√8y) = 0[/tex]
From the first equation, we get x = 0. Substituting this value into the second equation, we get:
[tex]14ye^(7y^2+14√8y) + 14√8e^(7y^2+14√8y) = 0[/tex]
Dividing both sides by [tex]14e^(7y^2+14√8y)[/tex], we get:
y + √8 = 0
Thus, the critical point of g is (0, -√8).
(b) To find the value of D(a,b) from the Second Partials test, we need to compute the second-order partial derivatives of g with respect to x and y:
[tex]∂^2g/∂x^2 = 32x^2e^(-4x^2+7y^2+14√8y) - 8e^(-4x^2+7y^2+14√8y)[/tex]
[tex]∂^2g/∂y^2 = 98y^2e^(-4x^2+7y^2+14√8y) + 196√8ye^(-4x^2+7y^2+14√8y) + 686e^(-4x^2+7y^2+14√8y)[/tex]
[tex]∂^2g/∂x∂y = -112xye^(-4x^2+7y^2+14√8y) - 196√8xe^(-4x^2+7y^2+14√8y)[/tex]
At the critical point (0, -√8), we have:
[tex]∂^2g/∂x^2 = -8[/tex]
[tex]∂^2g/∂y^2 = 686[/tex]
[tex]∂^2g/∂x∂y = 0[/tex]
Therefore, D(0, -√8) =[tex](∂^2g/∂x^2)(∂^2g/∂y^2) - (∂^2g/∂x∂y)^2[/tex] = (-8)(686) - [tex](0)^2[/tex] = -5488.
(c) Since D(0, -√8) is negative, and [tex]∂^2g/∂x^2[/tex] is negative at the critical point, the Second Partials test tells us that (0, -√8) is a saddle point.
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