The critical points of f(x,y) are:
Along the x-axis at (x,0) where [tex]sin(xy^{2/2}) = 0[/tex] and y = 0 or [tex]xy^{2/2[/tex] = nπ for some integer n.
Along the y-axis at (0,y) where sin([tex]xy^{2/2[/tex]) = 0 and x = 0 or [tex]xy^{2/2[/tex] = nπ for some integer n.
At (±[tex]\sqrt{(2n\pi /y)}[/tex]),y) where sin([tex]xy^{2/2[/tex]) = 0 and[tex]xy^{2/2[/tex] = nπ for some integer n.
To find the critical points of the function f(x,y) = 1 − cos([tex]xy^{2/2[/tex]), we need to find where the gradient vector is zero or undefined.
Let's start by finding the partial derivatives with respect to x and y:
fx(x,y) = [tex]y^{2/2}[/tex] sin([tex]xy^2/2[/tex])
fy(x,y) = xy sin([tex]xy^2/2[/tex])
Now, we need to find where both fx(x,y) and fy(x,y) are zero or undefined.
Setting fx(x,y) = 0 gives us either y = 0 or sin([tex]xy^{2/2[/tex]) = 0.
If y = 0, then fy(x,y) = 0 and we have a critical point at (x,0).
If sin([tex]xy^{2/2[/tex]) = 0, then either [tex]xy^{2/2[/tex] = nπ for some integer n, or x = 0.
If [tex]xy^{2/2[/tex] = nπ, then fy(x,y) = 0 and we have a critical point at (x,±[tex]\sqrt{(2n\pi /x)}[/tex]).
If x = 0, then fy(x,y) = 0 and we have critical points along the y-axis.
Setting fy(x,y) = 0 gives us either x = 0 or sin([tex]xy^{2/2[/tex]) = 0.
If x = 0, then fx(x,y) = 0 and we have critical points along the y-axis.
If sin([tex]xy^{2/2[/tex]) = 0, then either [tex]xy^{2/2[/tex] = nπ for some integer n, or y = 0.
If [tex]xy^{2/2[/tex] = nπ, then fx(x,y) = 0 and we have critical points at (±[tex]\sqrt{(2n\pi /y)}[/tex],y). If y = 0, then fx(x,y) = 0 and we have a critical point at (x,0).
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Question is attached. Show workings
Answer:
First question(unnumbered):
[tex]\mathrm{Choice \; (B) : \;\;\dfrac{1}{2}}[/tex]
Question 48
[tex]\mathrm{Choice \;(C)\;\; (x, y) = \left(-1, \dfrac{3}{2}\right)}[/tex]
Question 49
[tex]\mathrm{Choice \; (D)\;\;-1}[/tex]
Step-by-step explanation:
The imaginary variable i is [tex]\sqrt{-1}[/tex] and [tex]i^2 = -1[/tex]
For the unnumbered first question we have
the equation
[tex]\dfrac{1}{1-i}[/tex]
Multiply numerator and denominator by [tex]i + 1[/tex]
[tex]\rightarrow \dfrac{1\cdot \left(1+i\right)}{\left(1-i\right)\left(1+i\right)}\\\\\\= \dfrac{1 + i}{1 - i^2} \;\;\;\;\;\;\;since \;(a +b)(a-b) = a^2 - b^2\\\\\textrm{Since $i^2 = -1$, the denominator becomes:}\\\\1-i^2 = 1 -(-1) = 2\\\\\\[/tex]
Therefore ,
[tex]\dfrac{1}{1-i} = \dfrac{1 + i}{2}\\\\= \dfrac{1}{2} + \dfrac{i}{2}\\\\[/tex]
The real part is [tex]\dfrac{1}{2}[/tex] and the imaginary part is [tex]\dfrac{i}{2}[/tex]
Answer to first question is: ) [tex]\dfrac{1}{2}[/tex]
-------------------------------------------------------------------------
#48
Solve [tex](x, y)[/tex] for [tex]2y\:+\:xi\:=\:4+x-i[/tex] for [tex]$x,y\in\mathbb{R}$[/tex]
For the equation
[tex]2y\:+\:xi\:=\:4+x-i[/tex]
Move the the x and i terms from the right to left side:
[tex]2y + xi - x + i = 4\\\\2y - x +xi + i =4\\\\[/tex]
Since this works out to a real number, the imaginary part is 0
So
[tex]xi + i = 0\\\\xi = -i\\\\x = \dfrac{-i}{i}\\\\x = -1[/tex]
and the real part
[tex]2y - x = 4[/tex]
Substitute for x = -1 in the above to get
[tex]2y -(-1) = 4\\\\2y + 1 = 4\\\\2y = 4-1\\\\2y = 3\\\\y = \dfrac{3}{2}\\\\[/tex]
[tex](x, y) = \left(-1, \dfrac{3}{2}\right)[/tex]
which is choice (C)
------------------------------------------------------------------------
#49
For [tex]$a,b\in\mathb{R}$[/tex]
[tex]a+ib\:=\:\left(2-i\right)^2 \\\\[/tex]
Expand [tex]x(2-i)^2[/tex]
[tex]= 2^2 - 2.2.i +i^2\\\\[/tex]
Since i² = -1 this works out to:
[tex]4 - 4i -1[/tex]
[tex]3 - 4i[/tex]
Therefore
[tex]a + ib = 3 - 4i\\\\[/tex]
Equation real part on left to the real part on the right :
[tex]a = 3[/tex]
Equation imaginary part on left to the imaginary part on the right :
[tex]ib = 4i[/tex]
[tex]b = -4[/tex]
[tex](a + b) = (3 +(-4))\\\\= 3 - 4 \\\\= -1\\\\[/tex]
This would be option (D)
The area of the South China Sea i 6. 963 × 106 km2. The area of the Artic Ocean i 1. 4056 × 107 km2. Find the total area. Write the final anwer in cientific notation with the correct number of ignificant digit
So the total area of the South China Sea and the Arctic Ocean is 2.10286 x 107 km2
To find the total area of the South China Sea and the Arctic Ocean, you would add their individual areas together.
The formula for finding the total area is:
Total Area = Area of South China Sea + Area of Arctic Ocean
By applying this formula we get:
Total Area = 6.963 × 106 km2 + 1.4056 × 107 km2
Total Area = 2.10286 x 107 km2
So the total area of the South China Sea and the Arctic Ocean is 2.10286 x 107 km2. The final answer is correct and has correct number of significant digits as per the given figures.
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Find Dy/Dx By Implicit Differentiation. 5x^2-xy-4y^2=0
The Implicit Differentiation of 5x^2-xy-4y^2=0 is Dy/Dx= (5x-y)/(-4y)
Using implicit differentiation, Dy/Dx can be found by taking the derivative of both sides with respect to x.
On the left side, the derivative with respect to x of 5x2 is 10x and the derivative with respect to x of -xy is -y.
On the right side, the derivative of 4y2 with respect to x is 0.
Therefore, the equation becomes 10x - y = 0.
Solving for y gives y = 10x.
Substituting this into the original equation, we get 5x2 - 10x2 - 4(10x)2 = 0, which simplifies to -5x2 - 40x2 = 0.
Finally, we can solve for Dy/Dx:
Dy/Dx= (5x-10x)/(-4(10x))
Dy/Dx= (5x-y)/(-4y)
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Happy Paws charges $18.00 plus $4.50 per hour to keep a dog during the day. Woof Watchers charges $12.00 plus $5.75 per hour. Complete the equation and solve it to find for how many hours the total cost of the services is equal. Use the variable h to represent the number of hours. The equation is . The total cost of the services will be equal at hours.
HELP ME PLEASE!
Answer:
This is just a basic mathematical function in which 18.00$ is the starting cost of Happy Paws service and they charge 4.50$ an hour, and we can represent this with [tex]18.00$ + 4.50h = C[/tex].
C is the total cost and h represents the number of hours that took place.
Now looking at Woof Watchers, the starting cost is 12.00$ and they charge 5.75$ an hour, we can also represent this with the equation [tex]12.00 +5.75h=C[/tex].
Now we can figure out the answer the long way or the short way.
The short way is a lot more simple, so I will use that.
The short way is plugging these equations into a graphing calculator.
After that, the number of hours they both babysit to have equal total costs is 4.8 hours or 4 hours and 48 minutes, and the total cost for both services after 4.8 hours will be approximately 40$ or to be exact, 39.60$
Anyone who genius apin math? Help me, ty
Answer: B -3
Step-by-step explanation
-3/4 my teacher would say rise over run, run means x axes and rise is y axes to get 2 to 8 you need 6 if you times -3/4 by 2 you would get -6/8 6 is x +2 from (5,2) you would get 8 so (m,8) then -8 +5 is -3 creating (-3,8)
chapter 7.1.6 puzzle core connections algebra
The exponential function is y = 1200(1.04)^t and the investment amount after 16 years is $2247.58.
What is exponential function?
The formula for an exponential function is f(x) = a^x, where x is a variable and a is a constant that serves as the function's base and must be bigger than 0.
Jayden invests $1,200 in a savings account. Each year, the value increases by 4%.
Let t = time in years
Let y = amount of money invested in savings account
Start with $1200 and each year add 4% interest to the amount in the bank.
1st year: Interest = 1200 × (0.04) = $48
This is added to the previous amount -
= $1200 + $1200 × (0.04)
= $1200 (1+0.04)
= $1200 (1.04)
= $1248
2nd year: Previous amount + interest on the previous amount
= $1248 (1+0.06)
= $1248 (1.06)
= $1297.92
So, the exponential formula becomes -
y = 1200(1.04)^t
Now, to find the value of investment after 16 years, substitute t = 16.
y = 1200(1.04)^t
y = 1200(1.04)^16
y = 1200(1.87298125)
y =$2247.58
Therefore, the value of investment after 16 years is $2247.58.
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please help me answer this question!
17+44
[tex]10 + 51[/tex]
What is the solution
Answer :
17+44 = 61
10+51= 61
What is the greatest common factor of 10 and 4x?
The GCF of 10 and 4x is 2.
The greatest common factor (GCF) of two or more numbers is the largest number that divides evenly into each of them. Factoring a number means breaking it down into its prime factors. That is;
10 = 2 * 5 , we can see that 2 and 5 are the prime factors of 10.
4x = 4 * x, we can see that 4 and x are the prime factors of 4x.
Since 2 is the only common factor between 10 and 4x, we can conclude that 2 is the GCF of 10 and 4x.
It means that 2 is the greatest number that divides into both 10 and 4x with a remainder.
Therefore, the GCF of 10 and 4x is 2.
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Is the meaning of 9?
The number 9 is a numerical digit used in many numeral systems. It is the highest single-digit number and has no specific meaning in and of itself.
Write about meaning of 9.The meaning of 9 may vary depending on the context in which it is used. For example, in mathematics, 9 is a composite number, meaning it can be divided by numbers other than itself and 1. In numerology, the number 9 is considered to be a powerful, spiritual number and is often associated with universal love, selflessness, and completion. In other cultures, 9 may hold different meanings or symbolism.
If we check the single digit series we will find the 9 is the 10[tex]th[/tex] and last digit in single digit series.
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Complete question is: What is the meaning of 9?
What is the expression written as a single power?
(−4)8 · (−4) −5. (-4)⁰
O (-4)⁰
O-4
(-4)³
O-43
The expression (−4)⁸(−4)⁻⁵(-4)⁰ written as a single power is given by (- 4)³.
What is expression?In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.
Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax.
Given is the expression as -
(−4)⁸(−4)⁻⁵(-4)⁰
The given expression is -
(−4)⁸(−4)⁻⁵(-4)⁰
(- 4)⁸ ⁻ ⁵ ⁺ ⁰
(- 4)³
Therefore, the expression (−4)⁸(−4)⁻⁵(-4)⁰ written as a single power is given by (- 4)³.
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A building company ued 24 worker to prepare a building ite. The ite meaured 30 acre and the work wa completed in 10 day. The company i aked to prepare another ite meaured 45 acre. Thi work ha to be completed in 15 day. Calculate the leat amount of worker the company hould employ for thi work
6 Need to increase the number of workers.
Now, According to the question:
A building company used 24 worker to prepare a building site.
The site measured 30 acre and the work was completed in 10 day.
The company is asked to prepare another site measured 45 acre.
This work has to be completed in 15 days.
To find the least amount of worker the company should employs for this work.
=> 30 ÷ 10 = 3 Daily worked of 24 workers.
=>3 ÷ 24 = 1/8 Daily worked of 1 worker.
=> 45 ÷ 12 = 15/4 The amount of work that the project needs to complete each day
=> 15/4 ÷ 1/8 = 30 Number of workers needed.
=> 30 - 24 = 6 Need to increase the number of workers.
Hence, 6 Need to increase the number of workers.
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Find the volume of a pyramid with a square base, where the area of the base is 4. 6\text{ cm}^24. 6 cm 2 and the height of the pyramid is 2. 3\text{ cm}2. 3 cm. Round your answer to the nearest tenth of a cubic centimeter.
The volume of the pyramid is 3.5 cubic centimeters.
The volume of a pyramid can be calculated using the formula:
V = (1/3) * Base area * Height
Where Base area is the area of the base of the pyramid, and Height is the distance from the base to the apex of the pyramid.
Given that the area of the base of the pyramid is 4.6 cm^2 and the height of the pyramid is 2.3 cm, we can substitute these values into the formula and solve for the volume:
V = (1/3) * 4.6 cm^2 * 2.3 cm = 3.53 cm^3
So the volume of the pyramid is 3.53 cubic centimeters. Rounded to the nearest tenth of a cubic centimeter, the volume is 3.5 cm^3.
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Three college students are planning to share a 3-bedroom, batth apartment near the campus. The rent is
$1,475 per month. The three bedrooms measure 12 by 10, 10 by 112 and 12 by 126 im. How
could they family divide the cast of the ent? Explain your reasoning
The required rent is divided by the family as $456, $456, and $574.56.
What is simplification?The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.
here,
Total rent = $ 1475
The total area of the place = 12× 10 + 10×11.2 + 12 × 12.6 = 383.2 units
Now,
Rent per square units = 1475/ 383.2 = $3.8
Now,
rent of bedroom with measure 12 by 10 = 12 × 10 × 3.8 = $456
rent of bedroom with measure 10 by 11.2 = 10 × 11.2 × 3.8 = $456
rent of bedroom with measure 12 by 12.6 = 12 × 12.6 × 3.8 = $574.56
Thus, the required rent is divided by the family as $456, $456, and $574.56.
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Hello! I have an inverse function question I don’t understand. Please help!!!
How do you solve complex fractions?
To solve complex fractions there are two methods -
1 - Divide the numerator by the denominator by multiplying the numerator by the reciprocal of the denominator and then simplify further if necessary.
2 - Multiply the numerator and denominator of the overall complex fractions and then simplify further if necessary.
What is a complex fraction?A rational expression with a fraction in the numerator, denominator, or both is referred to as a complex fraction.
There are two ways to solve a complex fraction -
method 1 : Take an example - (1/3-1/4)/(1/8+1/2)
step1; If necessary, combine the denominator and numerator into a single fraction.
Add the numerator -
=1/3-1/4
=(4-3)/12
=1/12
Add the denominator -
=1/8+1/2
=(1+4)/8
=5/8
Combine back to form a complex fraction -
=(1/12)/(5/8)
Step 2: Multiply the numerator by the denominator's reciprocal, then divide the result by the denominator.
Step 3: Simplify the rational expression if necessary.
=(1/12)×(8/5)
=8/60
=2/15
Method 2: Take an example - [(2x+1)/(x^2-25)]/[(4x^2-1)/(x-5)]
Step 1 - Divide the LCD of the smaller fractions by the numerator and denominator of the total complex fractions.
(x^2-25)=(x+5)(x-5)
The denominator of the denominator’s fraction has the following factor -
(x-5)
Using the highest exponent and combining all the other factors, we arrive at the LCD shown below for all the little fractions -
(x+5)(x-5)
By multiplying the LCD by the numerator and denominator -
=[{(2x+1)/(x+5)(x-5)}×(x+5)(x-5)]/[{(2x+1)(2x-1)/(x-5)}×(x+5)(x-5)]
=(2x+1)/(2x+1)(2x-1)(x+5)
Step 2 - Simplify the rational expression if necessary.
=(2x+1)/(2x+1)(2x-1)(x+5)
=1/(2x-1)(x+5)
Therefore, there are two methods to solve complex fractions.
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What is the table of 13 to 15?
Therefore , the solution of the given problem of table comes out to be 15 to 20 table are given as follows : 14,28,42,56,70,84,98,112,126 and 140
Describe a table.A statistical list is a collection of numbers that shows the outcomes of a computation with numerous inputs. Data is made up of descriptions and numbers and is arranged in rows and columns. The table and indeed the pattern of 10 are easily recalled. In numerical order, the numbers are 2, 4, 6, 8, 10, 12, 14, 16, 18, then 20. Value mapping is used to show how multiple data pieces are related to one another. Any science class can benefit from including the values table.
Here,
Given :
=> 13,26,39,52,65,78,91,104,117 and 130
=> 14,28,42,56,70,84,98,112,126 and 140
=> 15,30,45,60,75,90,105,120,135 and 150
Therefore , the solution of the given problem of tables comes out to be 13 to 15 table are given as follows : 14,28,42,56,70,84,98,112,126 and 140
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Place the numbers in order from least to greatest.
= -1050 = 25 = 80 = -800
Answer:
-1050 = -800 = 25 = 80
Step-by-step explanation:
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
1) look at the your numbers and find the negatives. they always are the SMALLEST.
2) as you can see on the line above, there's the middle and from there negative numbers go to the left and positive numbers go to the right.
3) the bigger negative numbers are the smaller they are, super confusing ik
Find the missing values in the radio table. Then write the equivalent ratio
3. 6
4. 24
A ratio remains the same if both antecedent and consequent are multiplied or divided by the same non-zero number.
How to find the missing number?Step by step explanation:
i.e., a/b = pa/pb = q a/q b ,p ,q ≠0
From the given table we can see that:
For the second column
Since, 6*0.5=3
So, multiply 10 by same number
10*0.5=5
For third column:
since 24=10*5
So, multiply 6 by same number
5*6=30
Hence, the missing values are 30,5
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The total price of a shirt and a cap is $12. If the price of the shirt was doubled and the price of the cap was three times its original price, the total price of a shirt and a cap would be $26. The following system of equations models this scenario: x + y = 12 2x + 3y = 26 What is the price of each shirt and cap? The price of a shirt is $7, and the price of a cap is $5. The price of a shirt is $8, and the price of a cap is $4. The price of a shirt is $9, and the price of a cap is $3. The price of a shirt is $10, and the price of a cap is $2.
Answer:
x = price of shirt = $10
y = price of cap = $2
Step-by-step explanation:
x = price of shirt
y = price of cap
(1) x+y = 12
(2) 2x+3y = 26
x = 12-y substitute this into equation 2
2(12-y) + 3y = 26
24 - 2y + 3y = 26
y = 26-24 = 2 substitute this into equation 1
x+2 = 12
x = 12-2 = 10
Answer:
x = price of shirt = $10y = price of cap = $2
i need this like right now.
Answer:
y = 13
Step-by-step explanation:
given (7, 33 ) is a solution of the top equation, then
x = 7, y = 33 satisfies the equation , that is
33 = 7a + 5 ( subtract 5 from both sides )
28 = 7a ( divide both sides by 4 )
4 = a
then
y = 4x + 5 ← top equation
substitute y = 4x + 5 into the other equation
9x + 2(4x + 5) = 44
9x + 8x + 10 = 44
17x + 10 = 44 ( subtract 10 from both sides )
17x = 34 ( divide both sides by 17 )
x = 2
substitute x = 2 into the top equation for value of y
y = 4x + 5 = 4(2) + 5 = 8 + 5 = 13
How do I change my UCS icon from 3D to 2D in Autocad?
In AutoCAD, you can change the UCS icon from 3D to 2D by following these steps:
Type "UCSICON" in the command line and press Enter.In the UCS Icon dialog box, select "2D Wireframe" from the drop-down menu.Click OK to close the dialog box and apply the change.What is Autocad?AutoCAD is a commercial computer-aided design and drafting software application.
Alternatively, you can also change the UCS icon to 2D by typing "UCSICON" in the command line, then pressing the F6 key. This will toggle the UCS icon between 2D and 3D.
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Please Help Me!!!!!!!!!!
Thank you !!!!!!!!!!!!11
The proportional equation that is on the graph can be written as:
y = 0.83*x
How to find the graphed proportional equation?Remember that a general proportional equation is written as:
y = k*x
Where k is the constant of proportionality.
To find the value of the constant of proportionality, we need to identify one point that lies on its graph. For example, we could use the point (0.6, 0.5)
Where x = 0.6
y = 0.5
Replacing that in the general proportional equation we will get:
0.5 = k*0.6
0.5/0.6 = k
0.83 = k
Then the proportional equation is:
y = 0.83*x
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Can the sides of a triangle have lengths 3 6 and 7?
Answer:
No
Step-by-step explanation:
3+6 is 9 which is greater than 7 so no it can not be a triangle
Answer:
For three lengths to form a triangle, the sum of two of the sides has to be more than the length of the third line. 3,6,8 or 3,6,7 will form triangles.
Step-by-step explanation:
The following rule describes the relationship between x and y.
Rule: Multiply x by 4 to get y.
Complete the table for the given rule.
X Y
0 __
1 __
2 __
................ ................................
select all expressions that are equivalent to -3/4( 32 - 4f)
24 - 3f
3f - 24
-24 + 3f
8 - f
3(-8 +f)
Answer:
-24 + 3f
3f - 24.
3(-8 + f)
Step-by-step explanation:
-3/4( 32 - 4f)
= -3/4 * 32 - 3/4 * -4f
= -3*8 + 3f
= -24 + 3f.
= 3f - 24
= 3(-8 + f)
may i get some help with this
The value of x is that will ensure that quadrilateral VWXY is parallelogram is 0.25.
What is Quadrilateral?•A polygon called a quadrilateral has four sides, four angles, and four vertices. Parallelograms, Squares, Rectangles, and Rhombuses are the different types of quadrilaterals. Parallelograms also include squares, rectangles, and rhombi.
• All quadrilaterals have a few characteristics. Parallelograms characteristics are each of them has four vertices, each side has four sides, 360 degrees is the sum of all interior angles and two diagonals are present.
•There are regular and irregular quadrilaterals. An ordinary quadrilateral must have four equal sides, four equal angles, and diagonals that cross one another. Only the square can fulfill each of these requirements of a quadrilateral.
As we know that the opposite angles of a parallelogram are equal
hence 10x + 27 = 2x + 29
8x = 2
x = 0.25
Hence, the angles are 29.5
Let the value of angle v and x be a
Now using angle sum property of a parallelogram
29.5 +29.5 + a + a = 360
2a = 301
a = 150.5
hence angle V is 150.5
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Help on b please! Thanks
Answer: he could've gotten a 60
Step-by-step explanation:
if you follow the line up from 50 on paper 1, it fits onto a 60 for paper 2
What is the answer ?
Answer:
25x+125< 500 15 fewer seats
Step-by-step explanation
this is number 1
linda saves 25 dollar evry month in her saving account she took out 100 dollars
The number of months that it would take for Linda to replenish the amount she took out is 4 months.
How to find the number of months ?The number of months it would take Linda to make back the $ 100 that she took from her savings account will depend on the amount that she saves every month.
This amount is $ 25 per month.
The number of months it would take for Linda to replenish the $ 25 would therefore be:
= Amount taken out of account / Amount saved per month
= 100 / 25
= 4 months
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Rest of the qyestion is:
How many months will it take Linda to replenish the amount she took out of her savings account ?