We can use the method of integrating factors to solve the equation.
Y' + x = (x-1)y' + 1
How can we use the integrating factors method here?The integrating factor method is a technique used to solve linear first-order ordinary differential equations.
Find the integrating factor:
The integrating factor is given by:
I(x) = e^(∫x dx) = e^(x^2/2)
Multiply both sides of the equation by the integrating factor I(x)
e^(x^2/2)Y' + xe^(x^2/2) = (x-1)e^(x^2/2)y' + e^(x^2/2)
Let's simplify the left-hand side using the product rule of differentiation
(d/dx)(e^(x^2/2)Y) = e^(x^2/2)Y' + xe^(x^2/2)
Rewrite the equation using the simplified left-hand side
(d/dx)(e^(x^2/2)Y) = (x-1)e^(x^2/2)y' + e^(x^2/2)
Then, Integrate both sides with respect to x
∫(d/dx)(e^(x^2/2)Y) dx = ∫((x-1)e^(x^2/2)y' + e^(x^2/2)) dx
e^(x^2/2)Y = ∫((x-1)e^(x^2/2)y' + e^(x^2/2)) dx
Using integration by substitution and integration by parts, we can evaluate the integral on the right-hand side and simplify the equation:
e^(x^2/2)Y = e^(x^2/2) + C - e^(x^2/2)y + D
where C and D are constants of integration.
We Solve for Y:
Y = 1 - y + Ce^(-x^2/2) + De^(x^2/2)
Therefore, the solution of the given differential equation is:
Y = 1 - y + Ce^(-x^2/2) + De^(x^2/2)
where C and D are constants of integration.
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Three distinct, five-sided dice are rolled, and the numbers showing are recorded. How many different outcomes are possible
Answer:
Step-by-step explanation:
Each die can show one of the five possible numbers. So, the number of possible outcomes for rolling one die is 5.
Since there are three dice being rolled, the total number of possible outcomes is the product of the number of possible outcomes for each die. That is:
Number of possible outcomes = 5 x 5 x 5 = 125
Therefore, there are 125 different outcomes possible when rolling three distinct, five-sided dice.
Complete the table to show equipment measures
Mathematics is a field that involves primarily abstract reasoning and mental calculations, but there are still several types of equipment that mathematicians may use to aid in their work.
What are different equipments use in mathematics?
1.) Calculator: A calculator is an electronic device that performs arithmetic operations quickly and accurately. It is often used by mathematicians to check their calculations, solve equations, and perform more complex calculations.
2.) Protractor: A protractor is a tool used to measure and draw angles. It is commonly used in geometry and trigonometry to draw and measure angles of various shapes.
3.) Compass: A compass is a tool used to draw circles and arcs. It is commonly used in geometry to draw circles, bisect angles, and construct various geometric shapes.
4.) Ruler: A ruler is a straightedge tool used to measure lengths and draw straight lines. It is commonly used in geometry and algebra to draw graphs, construct figures, and measure distances.
5.) Graph paper: Graph paper is a type of paper that has a grid of small squares printed on it. It is often used in algebra and geometry to draw graphs, plot data, and make calculations.
The question doesn't contain any table and is incomplete, below is the complete question -
Fill in the table with equipment measurements.
Equipment Code: The required field for the primary key.
Standard for equipment: The kind of equipment. confirmed by the listing of equipment standards.
Use of Equipment: Describe how the piece of equipment is used in this field.
categorization: Select a categorization for this piece of equipment using the field provided. The Condition Assessment programme organises and categorises your data using this classification.
Equipment state: To characterise the equipment's condition, select one of the values from the list box: new, good, fair, or poor.
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Consider a political discussion group consisting of 10 Democrats, 9 Republicans, and 3 Independents. Suppose that
two group members are randomly selected, in succession, to attend a political convention. Find the probability of
selecting two Independents.
As a result, there is a 1/77 chance of choosing two Independents consecutively as probabilities in order to get the likelihood of choosing two Independents.
what is probability ?The chance of an event happening is gauged by probability. It is stated as a value between zero and 1, with 0 denoting that the occurrence is unlikely and 1 denoting that it is unavoidable. Typically, P stands for the likelihood of something happening A. (A). It is computed by dividing the number of positive outcomes but by total number of potential outcomes. For instance, if you flip a fair coin, the likelihood of receiving heads is 50% since there is only one positive result (heads) so out two possible possibilities (heads or tails).
given
The likelihood of choosing an Independent on the first draw is: There are 3 Independents out of a total of 10 Democrats, 9 Republicans, and 3 Independents.
P(Unrelated to the initial draw) = 3/22
P(Independent on First Draw and Second Draw) = 2/21
We must compound these probabilities in order to get the likelihood of choosing two Independents consecutively:
P(Choosing two Independents) = P(Independent on first draw) + P(Independent on second draw | Independent on first draw)
= (3/22) * (2/21)
= 6/462
= 1/77
As a result, there is a 1/77 chance of choosing two Independents consecutively as probabilities in order to get the likelihood of choosing two Independents.
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$12715= x2 - 5x + 65
Step-by-step explanation:
To solve for x, we need to isolate the variable on one side of the equation. We can start by subtracting 65 from both sides of the equation:
$12715 - 65 = x^2 - 5x
Simplifying:
$12650 = x^2 - 5x
Next, we can move all terms to one side of the equation:
x^2 - 5x - 12650 = 0
Now, we can use the quadratic formula to solve for x:
x = [-(-5) ± sqrt((-5)^2 - 4(1)(-12650))]/(2(1))
Simplifying:
x = [5 ± sqrt(63225)]/2
x = [5 ± 251]/2
So the solutions are:
x = (5 + 251)/2 = 128
x = (5 - 251)/2 = -123
Therefore, x can be either 128 or -123.
The ratio of the width to the length of a flower bed is 10:19. How long is the flower bed if it's width is 6feet
Answer:
The length of the flower bed is 11.4 feet.
Step-by-step explanation:
If the ratio of width to length of the flower bed is 10:19, this means that for every 10 units of width, there are 19 units of length.
Let's call the width of the flower bed "W" and the length "L". We know that W is 6 feet, so we can write:
W:L = 10:19
Substituting the value of W, we get:
6:L = 10:19
To solve for L, we can cross-multiply to get:
10L = 6 x 19
10L = 114
L = 11.4
Therefore, the length of the flower bed is 11.4 feet.
The latin dance club needs go raise more than 178.50 to buy costumes. The club already has 35.70. which inequality shows how much money each of the 7 club number needs to raise m if each person rasises the same amount
Answer:
its on quizzes
Step-by-step explanation:
5 ) A basketball league has a single- elimination tournament that begins with 32 teams playing in 16 games. After the first game, only 16 teams remain in the tournament to play 8 games. This continues until there is a winner. How many total games were played in the tournament?
To solve this problem, we can start by noticing that each round cuts the number of teams in half. Therefore, we can create a table to see how many teams are left after each round:
Round Number of Teams
1 32
2 16
3 8
4 4
5 2
6 1
We can see that there are 6 rounds in total, and that the number of games in each round is half the number of teams. So we can calculate the number of games for each round and add them up to get the total number of games:
Round Number of Teams Number of Games
1 32 16
2 16 8
3 8 4
4 4 2
5 2 1
6 1 0
Adding up the number of games for each round, we get:
16 + 8 + 4 + 2 + 1 + 0 = 31
Therefore, a total of 31 games were played in the tournament.
each shape is 1 whole. write a fraction in standard form, word form, and numerical-word form for the parts that are shaded
If each shape is 1 whole and the circle is divided in half and shaded on both sides, then the fraction of the shaded part would be [tex]\frac{1}{2}[/tex].
In standard form, the fraction [tex]\frac{1}{2}[/tex] is already in its simplest form, as both the numerator and denominator have no common factors other than 1. In word form, the fraction [tex]\frac{1}{2}[/tex] can be expressed as "one-half," which indicates that one of the two equal parts of the circle is shaded. In numerical-word form, the fraction [tex]\frac{1}{2}[/tex] can be written as "0.5" or "zero point five," which represents the decimal equivalent of the fraction. This form is useful when working with measurements and calculations that involve fractions.
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The complete question is:
Each shape is 1 whole. Write a fraction in standard form, word form, and numerical-word form for the parts that are shaded
HELP ASAP 25 PONITS PLEASE HELP
Convert 3% to a decimal by doing 3/100. You should get 0.03.
Now multiply 0.03 and 44.50, you will get 1.335.
Add 1.335 to 44.50 and you will get 45.835.
The new price of the item after it was marked up is $45.835
Here is a graph of the function g
Use the graph to find the following.
If there is more than one answer, separate them with commas.
(a) All values at which g has a local minimum:___
(b) All local minimum values of g :___
(a) All values at which g has a local minimum: -2, 3
(b) All local minimum values of g: g(-2) ≈ -2.5, g(3) ≈ -1.5
What is the Graph?
The graph is a visual representation of the function g, which shows how the output of the function varies with the input. It plots the values of g(x) on the vertical axis against the values of x on the horizontal axis.
(a) All values at which g has a local minimum:
From the graph, it appears that there are two values at which g has a local minimum:
x = -2
x = 3
(b) All local minimum values of g:
From the graph, the local minimum value at x = -2 appears to be approximately -2.5, and the local minimum value at x = 3 appears to be approximately -1.5. So, the local minimum values of g are:
g(-2) ≈ -2.5
g(3) ≈ -1.5
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What is 7/10 changed into a decimal
Answer:
0.7
Step-by-step explanation:
[tex]\frac{7}{10}=\frac{70}{100}=0.70[/tex]
What are the multiplicative and additive inverses of -7?
a
multiplicative inverse 1/7; additive inverse -7
b
multiplicative inverse 1/7; additive inverse 7
c
multiplicative inverse -1/7; additive inverse -7
d
multiplicative inverse -1/7; additive inverse 7
pleaase help im have a 77 in her class smh
Can someone Please answer this?
Gets brainliest and 30 points!!!!!!
I think the correct answer is 326
A grassland ecosystem experiences an especially rainy year. During the same
year, the rabbit population in the grassland increases sharply. What is the
most likely reason for this increase?
OA. The ecosystem's carrying capacity is lower, so fewer rabbits have
immigrated and more have died.
B. The ecosystem's carrying capacity is higher, so fewer rabbits have
been born and more have emigrated.
OC. The ecosystem's carrying capacity is lower, so more rabbits have
emigrated and fewer have been born.
D. The ecosystem's carrying capacity is higher, so more rabbits have
been born and fewer have died.
Answer: D. The ecosystem's carrying capacity is higher, so more rabbits have been born and fewer have died.
Explanation: Carrying capacity refers to the maximum number of individuals that an ecosystem can support without causing significant harm to the environment. In this case, the grassland ecosystem experienced an especially rainy year, which likely led to favorable conditions for both the rabbits and their food sources.
With more rainfall, there would be an abundance of vegetation in the grassland ecosystem. This increase in food availability would provide ample resources for the rabbits to thrive and reproduce. As a result, more rabbits would be born during this year.
Additionally, the favorable conditions provided by the rainy year may have also reduced the mortality rate of the rabbits. With plentiful food and water, the rabbits would be less likely to face resource scarcity and increased competition, leading to fewer deaths.
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A fast-food restaurant has a cost of production C(x)=13x+132 and a revenue function R(x)=7x . When does the company start to turn a profit?
The company will start to turn a profit when x is greater than -22. In other words, the company will start to turn a profit when they sell more than 22 units of their product.
What is function?
In mathematics, a function is a relationship between two sets of elements, called the domain and the range, such that each element in the domain is associated with a unique element in the range.
To find the point at which the company starts to turn a profit, we need to find the value of x for which the revenue is greater than the cost of production.
The revenue function is R(x) = 7x, and the cost of production is C(x) = 13x + 132.
So, we need to solve the inequality:
R(x) > C(x)
7x > 13x + 132
Subtracting 13x from both sides, we get:
-6x > 132
Dividing both sides by -6 (and flipping the inequality because we're dividing by a negative number), we get:
x < -22
Therefore, the company will start to turn a profit when x is greater than -22. In other words, the company will start to turn a profit when they sell more than 22 units of their product.
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There is a card under 30% of the chairs at a booster meeting. Finn wants to find the probability that the first card he finds will be under the third chair that he checks. He generates 10 sets of random numbers from 1 to 10. The numbers 1-3 represent a chair with a card and 4-10 represent a chair without a card. Complete the table, and find the experimental probability of the event.
The experimental probability of finding a card under the third chair is 4/10 or 2/5, which is 0.4 or 40%
How to find the experimental probability of the event?The experimental probability of finding a card is 30%. The complete table is given below:
Trial Numbers Generated Chairs Checked
1 6,5,1 No Card, No Card, Card
2 8,1 No Card, Card
3 10,1 No Card, Card
4 7,8,4,1 No Card, No Card, No Card, Card
5 9,3 No Card, Card
6 5,4,2 No Card, No Card, Card
7 3 Card
8 9,5,8,2 No Card, No Card, No Card, Card
9 7,3 No Card, Card
10 9,6,8,4,7,3 No Card, No Card, No Card, No Card, No Card, Card
The probability that the first card he finds will be under the third chair that he checks is calculated by taking the total number of trials in which the third chair had a card (3) and dividing it by the total number of trials (10).
The table can be completed by generating 10 sets of random numbers from 1 to 10 and checking the first three numbers in each set. If the first number is between 1 and 3, then the first chair has a card. If the second number is between 1 and 3, then the second chair has a card. If the third number is between 1 and 3, then the third chair has a card. If none of the first three numbers are between 1 and 3, then there are no cards in the first three chairs
Out of the 10 trials, 4 have a card under the third chair. Therefore, the experimental probability of finding a card under the third chair is 4/10 or 2/5, which is 0.4 or 40%.
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I need help with this
Answer:
The solution for problem a is cross multiplication
5×500=2500 and 15×x=15× so 15×=2500
divide both sides by the coefficient 15. 15x ÷15=×
divide 2500 by 15=500
x=500/3= approximately 166.7
Solution for b:
This one's easy!!
10/20=x/500
cross multiply
10×500=5000
20×X=20x
now we have the equation. 20x=5000
stay with me. I hope this makes sense ^_^
divide both sides by the coefficient 20
20x÷20=x
5000÷20=250
x=250
a is true
i hope this helps ;)
2 + ( − 5 ) + 1 5 + ( − 3 )
Answer:
9
Step-by-step explanation:
[tex]2 + ( - 5) + 15 + ( - 3)[/tex]
[tex]2 - 5 + 15 - 3[/tex]
Add the negative and the positive numbers seperately:
[tex]17 - 8 = 9[/tex]
HELP ASAP
A composite figure is represented in the image.
A four-sided shape with the base side labeled as 21.3 yards. The height is labeled 12.8 yards. A portion of the top from the perpendicular side to a right vertex is labeled 6.4 yards. A portion of the top from the perpendicular side to a left vertex is labeled 14.9 yards.
What is the total area of the figure?
272.64 yd2
231.68 yd2
190.72 yd2
136.32 yd2
The total area of the figure is 272.64 yd². Hence the correct option is (a) 272.64 yd².
The shape described in the problem is a trapezoid. The formula to calculate the area of a trapezoid is:
[tex]Area = (base1 + base2) x height / 2[/tex]
We are given the following measurements:
Base1 = 21.3 yards
Height = 12.8 yards
A portion of the top from the perpendicular side to a right vertex is labeled 6.4 yards, which is a part of the base2.
A portion of the top from the perpendicular side to a left vertex is labeled 14.9 yards, which is the remaining part of the base2.
To calculate the missing part of base2, we subtract the known parts of base2 from the total length of the top, which is:
21.3 - 6.4 - 14.9 = 0
This means that the missing part of base2 is zero, and the trapezoid becomes a rectangle. Therefore, the area of the trapezoid is equal to the area of the rectangle, which is:
Area = base x height
Area = 21.3 x 12.8
Area = 272.64 square yards
Therefore, the total area of the figure is 272.64 yd². Hence the correct option is (a) 272.64 yd².
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the line L passes through the points (5,-5) (4,3)
The equation of the line L is y = -8x + 35.
What is the slope?
The slope of a line is a measure of its steepness. Mathematically, the slope is calculated as "rise over run" (change in y divided by change in x).
To find the equation of the line L that passes through the points (5,-5) and (4,3), we need to use the point-slope form of a linear equation:
y - y1 = m(x - x1)
where m is the slope of the line and (x1, y1) is one of the given points.
First, we need to find the slope of the line L:
m = (y2 - y1) / (x2 - x1)
m = (3 - (-5)) / (4 - 5)
m = 8 / (-1)
m = -8
Now we can use either point to write the equation of the line. Let's use (5, -5):
y - (-5) = -8(x - 5)
y + 5 = -8x + 40
y = -8x + 35
Therefore, the equation of the line L is y = -8x + 35.
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The diameter of a circle is 5in. Find the length of its radius.
Se utilizan 3/5 de la capacidad de un camión para transportar sacos de papa. Un tercio de los sacos de papas son de papas chilotas y el resto es papas pukará. El resto del cargamento se divide en partes iguales entre lechugas y zapallos
¿Qué producto ocupa más capacidad?
answer?
i been struggling crazy......................
Answer:
50
Step-by-step explanation:
25+25 whcih mdans this must be added to get 50 becuase 25 + 35 is 50
please help me quick
Answer:
Step-by-step explanation:
4. If 99.7 percent of the data is between 30 and 90, then please find the mean and the standard of deviation.
b. a:
a. u:
5. If 68 percent of the data is between 42 and 58, then please find the mean and the standard of deviation.
b. a:
a. H
6. If 95 percent of the data is between 34 and 64, then please find the mean and the standard of deviation.
b. a:
a. u
Need help with this too
The mean is 49 and the standard deviation is 5.33.
What is mean?
The Arithmetic Mean is the average of the numbers: a calculated "central" value of a set of numbers.
We can use the empirical rule to solve these problems. According to the empirical rule:
About 68% of the data falls within one standard deviation of the mean.
About 95% of the data falls within two standard deviations of the mean.
About 99.7% of the data falls within three standard deviations of the mean.
a) To find the mean (u) and standard deviation (sigma) for each problem, we can use the following formulas:
u = (a + b) / 2
sigma = (b - a) / 6
where a and b are the lower and upper bounds for the data, respectively.
b) For each problem, we can use the given information to set up an equation and solve for either the mean or standard deviation.
If 68 percent of the data is between 42 and 58:
a) H: We are not given enough information to solve for the mean or standard deviation.
b) To find one of the missing values, we can use the formulas above and the given information. Since we are given that 68% of the data falls within one standard deviation of the mean, we know that:
(58 - u) / sigma = 1
Simplifying this equation, we get:
58 - u = sigma
We can also use the fact that the range of the data is 16 (from 42 to 58):
b - a = 16
58 - u - (42 - u) = 16
16 = 16u/16
Solving for the mean, we get:
u = (42 + 58) / 2
u = 50
Using the equation 58 - u = sigma, we can solve for the standard deviation:
sigma = 58 - u
sigma = 8
Therefore, the mean is 50 and the standard deviation is 8.
If 95 percent of the data is between 34 and 64:
a) u: We can find the mean using the formula:
u = (a + b) / 2
u = (34 + 64) / 2
u = 49
b) To find the standard deviation, we can use the formula:
sigma = (b - a) / 6
sigma = (64 - 34) / 6
sigma = 5.33
Therefore, the mean is 49 and the standard deviation is 5.33.
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The shape below is formed from a rectangle measuring 13 cm by 16 cm from which a rectangle of length 7.3 cm has been removed. Work out the perimeter of the shape 13 cm 5.7 cm 16 cm 7.3 cm 6.2 cm
Thus, the perimeter of the left shape when length of rectangle ABCD is reduced to 7.3 cm is found as: P = 43.4 cm.
Explain about the features of rectangle:Each vertex of a rectangle has a 90° internal angle.The total of a rectangle's internal angles is 360°.The diagonals cut each other in half.The diagonals are all the same length.A rectangle is also referred to as a parallelogram since its sides are parallel.While all parallelograms are rectangles, not all rectangles are parallelograms.Given data:
Length of rectangle ABCD 'l' = 16 cmWidth of rectangle ABCD 'w' = 13 cmRemoved length AE = 7.3 cmPerimeter = sum of all sides:
The shaded portion in the figure shows the left over rectangle:
Perimeter of EFCD = EF + FC + CD + DE
ED = FC = 16 - 7.3 = 8.7
P = 8.7 + 13 + 8.7 + 13
P = 43.4 cm.
Thus, the perimeter of the left shape when length of rectangle ABCD is reduced to 7.3 cm is found as: P = 43.4 cm.
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complete question:
The shape below is formed from a rectangle measuring 13 cm by 16 cm from which a rectangle of length 7.3 cm has been removed. Work out the perimeter of the left shape.
What was the equation of the graph below before it was shifted to the right 1
unit?
GD)-(x-1.5)-(x-1.5)
OA. G(x) = (x)3
OB. G(x) = (x-0.5)³-(x-0.5)
C. G(x) = (x-2)3-(x-2)
D. G(x) = (x-.5)³
The equation of the graph before it was shifted to the right one unit is given as follows:
B. G(x) = (x - 0.5)³ - (x - 0.5).
What is a translation?A translation happens when either a figure or a function is moved horizontally or vertically on the coordinate plane.
The four translation rules for functions are defined as follows:
Translation left a units: f(x + a).Translation right a units: f(x - a).Translation up a units: f(x) + a.Translation down a units: f(x) - a.The function in this problem is defined as follows:
G(x) = (x - 1.5)³ - (x - 1.5).
To remove the effect of a shift right one unit, we must shift the graph left one unit, hence the original rule is given as follows:
G(x) = (x - 1.5 + 1)³ - (x - 1.5 + 1).
G(x) = (x - 0.5)³ - (x - 0.5).
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Use the 2020 Federal Tax Rates for Individuals to calculate the estimated taxes for a taxable income of $63,700.
Tax rate
10%
12%
22%
24 %
32%
35%
37%
2020 Federal Tax Rates for Individuals
Taxable Income Bracket
$0 to $9875
$9876 to $40,125
$40,126 to $85,525
$85,526 to $163,300
$163,301 to $207,350
$207,351 to $518,400
$518,401 or more
The estimated taxes for a taxable income of $63,700 using the 2020 Federal Tax Rates for Individuals is $9,804.00.
How the estimated taxes are determined:The estimated taxes for an individual with a taxable income of $63,700 for the fiscal year uses the progressive tax (graduated) system of the Federal Tax Rates for Individuals, as computed below.
Taxable Income for 2020 = $63,700
2020 Federal Tax Rates for Individuals:Tax rate Taxable Income Bracket Tax Payable
10% $0 to $9875 $987.50 ($9,875 x 10%)
12% $9876 to $40,125 $3,630.00 ($30,250 x 12%)
22% $40,126 to $85,525 $5,186.50 ($23,575 x 22%)
Estimated taxes $9,804.00
Thus, an individual whose taxable income is $63,700 in 2020 is expected to pay taxes amounting to $9,804.00.
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Ella invested $4,800 in an account paying an interest rate of 2 1 8 2 8 1 % compounded quarterly. Santiago invested $4,800 in an account paying an interest rate of 2 3 8 2 8 3 % compounded daily. After 18 years, how much more money would Santiago have in his account than Ella, to the nearest dollar?
Answer:331
Step-by-step explanation:
7360.3097-7029.415≈330.8947
Santiago could have $81 more than Ella after 18 years.
To calculate the amount of money each person can have after 18 years, we are able to use the formulation for compound interest:
[tex]A = P(1 + r/n)^{(nt)[/tex]
Wherein:
A = the final amountP = the principalr = the interest price n = the number of times the interest is compounded in step with yeart = the number of yearsFor Ella's investment:
P = $4,800
r = 2.28125% = 0.0228125
n = 4
t = 18
[tex]A = 4800(1 + 0.0228125/4)^{(4*18)[/tex]
A = $8,481.48
For Santiago's investment:
P = $4,800
r = 2.28333% = 0.0228333
n = 365 (compounded daily)
t = 18
A = 4800(1 + 0.0228333/365)^(365*18)
A = $8,562.02
The distinction of their final amounts is:
$8,562.02 - $8,481.48 = $80.54
Therefore, Santiago could have $81 more than Ella after 18 years.
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Square
G
(x²-8)
meters
x meters
*.
...
What expression represents the total area of the
shaded figure. Simplify your expression.
The total area of the shaded portion is (x² - 16)(x²- 1) square meters.
What is a square?Square is a regular quadrilateral, which has all the four sides of equal length and all four angles are also equal. The angles of the square are at right-angle or equal to 90-degrees. Also, the diagonals of the square are equal and bisect each other at 90 degrees.
Equation:The area of the outer square is (x² - 8)², and the area of the inner square is x². Therefore, the area of the shaded region can be found by subtracting the area of the inner square from the area of the outer square:
Area of shaded region = Area of outer square - Area of inner square
= (x² - 8)² - x²
Expanding the first term using the formula for the square of a binomial, we get:
Area of shaded region = (x⁴ - 16x² + 64) - x²
= x⁴ - 17x² + 64
Therefore, the expression that represents the total area of the shaded figure is x⁴ - 17x² + 64, which can be simplified by factoring:
x⁴ - 17x² + 64 = (x² - 16)(x² - 1)
So the simplified expression for the total area of the shaded figure is (x² - 16)(x²- 1).
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