Step-by-step explanation:
The SSS (Side-Side-Side) Congruence Postulate states that if three sides of a triangle are congruent to three sides of another triangle, then the triangles are congruent.
To create your own SSS Congruence Postulate, you could use the following statement:
"If the corresponding sides of two triangles are congruent, then the triangles are congruent."
This postulate would encompass the SSS Postulate, as well as two other postulates: the SAS (Side-Angle-Side) Postulate and the ASA (Angle-Side-Angle) Postulate.
Using this postulate, you could show that two triangles are congruent if, for example, their corresponding sides are all 5cm long. This would mean that the triangles have the same shape and size, even if they are oriented differently.
Hopes this helps
The function in the table is quadratic:
x f(x)
-1 1/3
0 1
1 3
2 9
The quadratic function that fits the given points is: f(x) = (4/3)x² - (10/3)x + 1
By using the simplification formula
f(x) = ax² + bx + c
where a, b, and c are constants
a(-1)² + b(-1) + c = 1/3
a(0)² + b(0) + c = 1
a(1)² + b(1) + c = 3
a(2)² + b(2) + c = 9
Simplifying each
a - b + c = 1/3
c = 1
a + b + c = 3
4a + 2b + c = 9
We can solve this using any method like substitution, elimination
a - b + c = 1/3
a + b + c = 3
2a + 2c = 9/3
Adding the first two equations 2a + 2c = 10/3
Subtracting the third equation
b = 5a/3 - 2
a - (5a/3 - 2) + 1 = 1/3
a = 4/3
Finally, we can substitute a = 4/3 and b = 5a/3 - 2, and c = 1 into the standard form of the quadratic equation to get: f(x) = (4/3)x² - (10/3)x + 1.
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Solve the problem. (EASY! 10 Points)
If the length of a rectangle is decreased by 6cm and the width is increased by 3cm, the result is a square, the area of which will be 27cm^2 smaller than the area of the rectangle. Find the area of the rectangle.
Let L be the original length of the rectangle and W be the original width of the rectangle. We know that:
(L - 6) = (W + 3) (1) (since the length is decreased by 6cm and the width is increased by 3cm, the result is a square)
The area of the rectangle is LW, and the area of the square is (L - 6)(W + 3). We also know that the area of the square is 27cm^2 smaller than the area of the rectangle. So we can write:
(L - 6)(W + 3) = LW - 27 (2)
Expanding the left side of equation (2), we get:
LW - 6W + 3L - 18 = LW - 27
Simplifying and rearranging, we get:
3L - 6W = 9
Dividing both sides by 3, we get:
L - 2W = 3 (3)
Now we have two equations with two unknowns, equations (1) and (3). We can solve this system of equations by substitution. Rearranging equation (1), we get:
L = W + 9
Substituting this into equation (3), we get:
(W + 9) - 2W = 3
Simplifying, we get:
W = 6
Substituting this value of W into equation (1), we get:
L - 6 = 9
So:
L = 15
Therefore, the area of the rectangle is:
A = LW = 15 x 6 = 90 cm^2.
Answer:
252
Step-by-step explanation:
lol I take rsm too and I just guessed and checked
According to National Collegiate Athletic Association (NCAA) data, the means and standard deviations of eligibility and retention rates (based on a 1,000-point scale) for the 2013–2014 academic year are presented, along with the fictional scores for two basketball teams, A and B. Assume that rates are normally distributed.
Normal Distribution Practice data
Question 9
1 Point
On which criterion (eligibility or retention) did Team A do better than Team B? Calculate appropriate statistics to answer this question.
Team A has bette
By calculating the z-score, we can conclude that, Team A has better eligibility rates than Team B.
What is z-score?
A z-score (also called a standard score) is a measure of how many standard deviations a given data point is away from the mean of its distribution. It is calculated by subtracting the mean of the distribution from the data point, and then dividing the difference by the standard deviation.
To determine this, we can compare the z-scores for each team's eligibility rates.
Let's assume that Team A's eligibility rate is 875 and Team B's eligibility rate is 825.
The mean eligibility rate for all teams is given as 870, with a standard deviation of 50. Therefore, we can calculate the z-scores for each team's eligibility rate as follows:
z-score for Team A's eligibility rate = (875 - 870) / 50 = 0.1
z-score for Team B's eligibility rate = (825 - 870) / 50 = -0.9
Since the z-score for Team A's eligibility rate is positive and greater than the z-score for Team B's eligibility rate, we can conclude that Team A has better eligibility rates than Team B.
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3(26+14)/(2x2) i need help please
Answer:
30
Step-by-step explanation:
its right
order of operations
Answer:
The answer is 30...
Step-by-step explanation:
Apply the rule BODMAS...
3(40)/(4)
120/4
30
pls help fast!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
The number of non-fiction novels among the following 100 books that are released is anticipated to be 36%.
What is Probability?Probability refers to the likelihood that an occurrence will occur.
In actual life, we frequently have to make predictions about the future. We might or might not be conscious of how an event will turn out.
When this occurs, we proclaim that there is a possibility that the event will occur. In conclusion, probability has a broad range of amazing uses in both business and this rapidly developing area of artificial intelligence.
Simply dividing the favorable number of possibilities by the total number of possible outcomes using the probability formula will yield the chance of an event.
According to our question-
Total number of books that arrived that day = 23 + 41 = 64.
Let E be the event of arriving at a non-fictional book.
The event of arriving at a non-fictional book is n(E) = 23.
Total sample space, n(S) = 64.
The probability of a non-fictional book to arrive is = n(E) / n(S)
= 23 / 64 = 0.359375
= 0.359375 × 100
= 35.9375 ≈ 36.
Hence, The number of non-fiction novels among the following 100 books that are released is anticipated to be 36%.
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An expression is shown.
x³ + 2x² - 7x + 3x² + x³ - X
Given that x does not = 0, which of the following is equivalent to the expression?
Select one:
A. 2x³ + 5x² - 8x
B. −x^12
C. x³ + x²- x
D. x²+ x -1
Answer:
A: [tex]2x^3+5x^2-8x[/tex]
Step-by-step explanation:
In order to get the answer to this, you have to combine like terms to simplify the answer.
Go through and organize the x's by the size of the exponents. (this step isn't necessary but it can help you visualize it if you are having trouble with that)
[tex]x^3+x^3+2x^2+3x^2-7x-x[/tex]
When the variables are raised to the same degree, the coefficients can be added together.
[tex]2x^3+5x^2-8x[/tex]
one fourth of a number is no less than -3
The solution to the problem is that the number "x" must be greater than or equal to -12.
What solution is provided for "1/4 of number not less than -3"?Let's define the variable as "x," representing the unknown number. According to the problem statement, one fourth of the number is no less than -3.
Mathematically, we can express this statement as:
x/4 ≥ -3
To solve for "x," we can start by multiplying both sides of the inequality by 4 to eliminate the fraction:
x ≥ -3*4
x ≥ -12
Therefore, the solution to the problem is that the number "x" must be greater than or equal to -12.
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Four hundred gallons of 89-octane gasoline is obtained by mixing 87-octane gasoline with 92-octane gasoline.
(a)
Write a system of equations in which one equation represents the total amount of final mixture required and the other represents the amounts of 87- and 92-octane gasoline in the final mixture. Let x and y represent the numbers of gallons of 87-octane and 92-octane gasolines, respectively.
amount of final mixture required
amounts of 87- and 92-octane gasolines in the final mixture
(b)
Use a graphing utility to graph the two equations in part (a) in the same viewing window. As the amount of 87-octane gasoline increases, how does the amount of 92-octane gasoline change?
There is not enough information given.
As the amount of 87-octane gasoline increases, the amount of 92-octane gasoline stays the same.
As the amount of 87-octane gasoline increases, the amount of 92-octane gasoline increases.
As the amount of 87-octane gasoline increases, the amount of 92-octane gasoline decreases.
As the amount of 87-octane gasoline increases, the amount of 92-octane gasoline fluctuates.
(c)
How much (in gallons) of each type of gasoline is required to obtain the 400 gallons of 89-octane gasoline?
87-octane gal
92-octane gal
a) The total volume equals the sum of the volumes.
[tex]500 = x + y[/tex]
The total octane amount equals the sum of the octane amounts.
[tex]89(500) = 87x + 92y[/tex]
[tex]44500 = 87x + 92y[/tex]
b)
As x increases, y decreases.
c) Use substitution or elimination to solve the system of equations.
[tex]44500 = 87x + 92(500-x)[/tex]
[tex]44500 = 87x + 46000 - 92x[/tex]
[tex]5x = 1500[/tex]
[tex]x = 300[/tex]
[tex]y = 200[/tex]
The required volumes are 300 gallons of 87 gasoline and 200 gallons of 92 gasoline.
Jan’s pencil is 8.5 cm long Ted’s pencil is longer write a decimal that could represent the length of teds
pencil
Answer:
Without knowing the exact length of Ted's pencil, we cannot give an exact decimal representation of its length. However, we do know that Ted's pencil is longer than Jan's pencil, which is 8.5 cm long.
If we assume that Ted's pencil is one centimeter longer than Jan's pencil, then its length would be 9.5 cm. In decimal form, this would be written as 9.5.
If we assume that Ted's pencil is two centimeters longer than Jan's pencil, then its length would be 10.5 cm. In decimal form, this would be written as 10.5.
So, the decimal that could represent the length of Ted's pencil depends on how much longer it is than Jan's pencil.
1. In a cooking class 20% of
the students attend two
sessions. 60% of students
attend the first session of
cooking class. What is the
probability that a student that
attends the first session also
attends the second session?
The probability that a student who attends the first session also attends the second session is 1/3
Evaluating the probabilityWe can approach this problem by using conditional probability.
Let's use the notation "S1" to represent the event that a student attends the first session, and "S2" to represent the event that a student attends the second session.
Then we can use the formula for conditional probability:
P(S2 | S1) = P(S1 and S2) / P(S1)
We know that P(S1) = 0.6, since 60% of students attend the first session. We also know that P(S1 and S2) = 0.2, since 20% of students attend both sessions.
So we can plug in these values and solve for P(S2 | S1):
P(S2 | S1) = 0.2 / 0.6
P(S2 | S1) = 1/3
Therefore, the probability that a student who attends is 1/3 or approximately 0.333.
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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 days 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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The table of values forms a quadratic function f(x)
x f(x)
-1 24
0 30
1 32
2 30
3 24
4 14
5 0
What is the equation that represents f(x)?
Of(x) = -2x² + 4x + 30
Of(x) = 2x² - 4x-30
Of(x) = -x² + 2x + 15
Of(x)=x²-2x-15
Answer:
the answer is A) -2x² + 4x + 30
Step-by-step explanation:
To find the equation of the quadratic function f(x), we can use the standard form of a quadratic function: f(x) = ax^2 + bx + c, where a, b, and c are constants.
We can plug in the values of x and f(x) from the table to get three equations:
a(-1)^2 + b(-1) + c = 24
a(0)^2 + b(0) + c = 30
a(1)^2 + b(1) + c = 32
Simplifying each equation, we get:
a - b + c = 24
c = 30
a + b + c = 32
We can substitute c = 30 into the first and third equations to get:
a - b + 30 = 24
a + b + 30 = 32
Simplifying these equations, we get:
a - b = -6
a + b = 2
Adding these two equations, we get:
2a = -4
Dividing by 2, we get:
a = -2
Substituting a = -2 into one of the equations above, we get:
-2 - b = -6
Solving for b, we get:
b = 4
Therefore, the equation that represents f(x) is:
f(x) = -2x^2 + 4x + 30
So the answer is A) -2x² + 4x + 30
Fill in the missing values to make the equation true
Answer:
the missing values are 2, 8, and 3.
Write the quadratic equation whose roots are 5 and 2 , and whose leading coefficient is 4.
If the roots of a quadratic equation are given, we can write the equation in factored form as [tex](x - r1)(x - r2) = 0[/tex] , where r1 and r2 are the roots.the quadratic equation with roots 5 and 2 and leading coefficient 4 is: [tex]4x^2 - 28x + 40 = 0.[/tex]
What is the quadratic equation?A quadratic equation can be written in the form:
[tex]ax^2 + bx + c = 0[/tex]
where a, b, and c are constants. Since the roots of the equation are 5 and 2, we can write:
[tex](x - 5)(x - 2) = 0[/tex]
Expanding this equation gives:
[tex]x^2 - 7x + 10 = 0[/tex]
To make the leading coefficient of this equation 4, we can multiply both sides by 4/1, which gives:
[tex]4x^2 - 28x + 40 = 0[/tex]
Therefore, the quadratic equation with roots 5 and 2 and leading coefficient 4 is: [tex]4x^2 - 28x + 40 = 0.[/tex]
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Question 3
Ordered: Erythromycin 0.4 g BID for infection
Available: Erythromycin susp. 500 mg/10 mL
Give:____________ml(s)
Answer:
Step-by-step explanation:
500/0.4 = 1250
The function g(x) is the height of a football x seconds after it is thrown in the air. The football reaches its maximum height of 28 feet in 6 seconds, and hits the ground at 12 seconds.
What is the practical domain for the function f(x)?
Type your answer in interval notation.
It can be expressed in interval notation as:[0, 12]
The practical domain for the function g(x) is [0,12], as this is the time during which the football is in the air, from the time it is thrown until it hits the ground.
The practical domain for the function g(x) would be the time interval during which the football is in the air, since it only makes sense to talk about the height of the football while it is airborne.
From the problem statement, we know that the football is thrown in the air at time x = 0, reaches its maximum height of 28 feet at time x = 6, and hits the ground at time x = 12. Therefore, the practical domain for the function g(x) is:
0 <= x <= 12
This means that the function g(x) is defined and meaningful for any value of x between 0 and 12, inclusive. Beyond this domain, the function does not have a practical interpretation because the football is either not yet thrown or has already hit the ground.
The function g(x) is the height of a football x seconds after it is thrown in the air. The football reaches its maximum height of 30 feet in 4 seconds, and hits the ground at 10 seconds.
What is the practical domain for the function f(x)
Write your answer in interval notation.
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A block of wood has the shape of a triangular prism. The bases are right triangles. Find its surface area
The formula used to calculate the surface area of a triangular prism is:
S = bh + 2ah
Where S = surface area, b = length or side of the triangle, and h = height of the triangle.
For right triangles, the height can be calculated as:
c = a^(2) \+ b^(2)
c = the hypotenuse
a and b = the two sides of the triangle
PLEASE HELP SOLVE ALL !!!!!
Step-by-step explanation:
6.
the figure can be considered as a combination of a 16×8 rectangle and a 10×8 right-angled triangle on the left side.
for both, perimeter and area, we need to find the length of the missing third side of the right-angled triangle, as this is a part of the long baseline of the overall figure.
as it is a right-angled triangle, we can use Pythagoras :
c² = a² + b²
"c" being the Hypotenuse (the side opposite of the 90° angle, in our case 10 ft). "a" and "b" being the legs (in our case 8 ft and unknown).
10² = 8² + (leg2)²
100 = 64 + (leg2)²
36 = (leg2)²
leg2 = 6 ft
that means the bottom baseline is 16 + 6 = 22 ft long.
a.
Perimeter = 10 + 16 + 8 + 22 = 56 ft
b.
Area is the sum of the area of the rectangle and the area of the triangle.
area rectangle = 16×8 = 128 ft²
area triangle (in a right-angled triangle the legs can be considered baseline and height) = 8×6/2 = 24 ft²
total Area = 128 + 24 = 152 ft²
7.
it was important that the bottom left angle of the quadrilateral was indicated as right angle (90°). otherwise this would not be solvable.
but so we know, it is actually a rectangle.
that means all corner angles are 90°.
therefore, the angle AMT = 90 - 20 = 70°.
the diagonals split these 90° angles into 2 parts that are equal in both corners of the diagonal, they are just up-down mirrored.
a.
so, the angle HTM = AMT = 70°.
b.
MEA is an isoceles triangle.
so, the angles AME and EAM are equal.
the angle AME = AMT = EAM = 70°.
the sum of all angles in a triangle is always 180°.
therefore,
the angle MEA = 180 - 70 - 70 = 40°
c.
both diagonals are equally long, and they intersect each other at their corresponding midpoints.
so, when AE = 15 cm, then AH = 2×15 = 30 cm.
TM = AH = 30 cm.
I don’t know how to find the side length and the word problems are confusing(don’t worry about the ones I did).
The side of the given square is 36W^2 + 12W +1 of is 6W +1 and 81W^2 -72W + 16 is (9W-4).
How to calculate the area of the square?The area is calculated by multiplying the length of a shape by its width.
and the unit of the square is a square unit.
Given Area of the square :
1) [tex]36W^{2}+12W+1[/tex]
Area of square = [tex]36W^{2}+12W+1[/tex]
[tex]side^{2}[/tex] = [tex]36W^{2} + 6W +6W +1\\6W (6W +1) +1 (6W +1)\\(6W +1)(6W+1)\\(6W +1 )^{2} \\side^{2} = (6W + 1)^{2} \\side = 6W + 1[/tex]
[tex]Area of the square = 81W^{2} - 72W + 16\\(side)^{2} = (9W-4)(9W-4)\\(side)^{2} = (9W-4)^{2} \\side = (9W-4)[/tex]
Therefore the side of the square is 6W+1 and 9x-4.
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A TV cable company has 4800 subscribers who are each paying $24 per month. It can get 120 more subscribers for each $0.50 decrease in the monthly fee. What rate will yield maximum revenue, and what will this revenue be?
The rate that yields maximum revenue is $9 per month, and the maximum revenue is $1,296,000.
To find the rate that yields maximum revenue, we need to find the price that maximizes the revenue. Let x be the number of $0.50 decreases in the monthly fee, and let y be the number of subscribers who sign up at the new rate. Then we have the following equations:
y = 4800 + 120x (number of subscribers)
p = 24 - 0.5x (price per month)
r = xy(p) = (4800 + 120x)(24 - 0.5x) (revenue)
To find the rate that yields maximum revenue, we need to take the derivative of the revenue function concerning x, set it equal to zero, and solve for x:
r' = 120(24 - x) - (4800 + 120x)(0.5) = 0
x = 60
Therefore, the optimal number of $0.50 decreases is 60, and the corresponding price per month is $24 - 0.5(60) = $9. The number of subscribers at this rate is 4800 + 120(60) = 12000.
Finally, the maximum revenue is given by r = xy(p) = (4800 + 120(60))(24 - 0.5(60)) = $1,296,000.
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What is the equation of the
circle with centre (2,-3) and
radius 5?
Answer:
Step-by-step explanation:
Solution:
[tex](x-2)^2+(y+3)^2=25[/tex]
Equation of general circle:
[tex](x-a)^2+(y-b)^2=r^2[/tex] where (a,b) is circle center and r is the radius.
HELP ASP!!!
Ramon is filling cups with juice. Each cup is shaped like a cylinder and has a diameter of 4.2 inches and a height of 7 inches. How much juice can Ramon pour into 6 cups? Round to the nearest hundredth and approximate using π = 3.14.
96.93 cubic inches
553.90 cubic inches
581.59 cubic inches
2,326.36 cubic inches
Answer:
581,58 in^3
Step-by-step explanation:
Given:
Cylinder shaped cups
d (diameter) = 4,2 in
r (radius) = 0,5 × 4,2 = 2,1 in
h (height) = 7 in
π = 3.14
.
First, let's find how much juice can he pour into 1 cup:
.
We need to find the base of the cylinder:
.
[tex]a(base) = \pi {r}^{2} = 3.14 \times( {2.1})^{2} = 13.8474[/tex]
.
Now, we can find the volume of one cup:
V = a (base) × h
[tex]v = 13.8474 \times 7 ≈96.93[/tex]
Multiply this number by 6 and we'll get the answer (since there's 6 cups):
96,93 × 6 = 581,58
CAN someone please help me with this question please?!!!! Worth 30 points
The total surface area of the given hemispherical scoop is: 30.41 cm²
How to find the surface area?The formula for the total surface area of a hemisphere is:
TSA = 3πr² square units
Where:
π is a constant whose value is equal to 3.14 approximately.
r is the radius of the hemisphere.
Since the steel is 0.2cm thick and the outside of the scoop has a radius of 2cm, then we can say that:
TSA = 3π(2 + 0.2)²
= 3π(2.2)²
= 30.41 cm²
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Perform the indicated operations,the simplify
Answer:
[tex] \frac{13y ^{2} - 59y - 90 }{(y + 10)(y - 10)} [/tex]
What is the Surface Area of the Triangular Prism below?
Surface area is calculated as 48 + 120 = 168 square units (area of triangular faces + area of rectangular faces).
A triangular prism is what?A polyhedron with two triangular sides and three rectangles sides is referred to as a triangular prism. It is a three-dimensional shape with two base faces, three side faces, and connections between them at the edges.
Given :
We must calculate the area of each face of the triangular prism and put them together to determine its surface area.
The areas of the triangular faces are equal, so we may calculate one of their areas and multiply it by two:
One triangular face's area is equal to (1/2) the sum of its base and height, or (1/2) 6 x 8 x 6, or 24 square units.
Both triangular faces' surface area is 2 x 24 or 48 square units.
Finding the area of the rectangular faces is now necessary:
One rectangular face's area is given by length x breadth (10 x 6) = 60 square units.
120 square units are the area of both rectangular faces or 2 by 60.
Hence, the triangular prism's total surface area is:
Surface area is calculated as 48 + 120 = 168 square units (area of triangular faces + area of rectangular faces).
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Determine the simple interest. The rate is an annual rate. Assume 360 days in a year. p=$586.21, r=6.3%, t=83 days
so we're assuming there are 360 days in a year, so 83 days is really just 83/360 of a year, so
[tex]~~~~~~ \textit{Simple Interest Earned} \\\\ I = Prt\qquad \begin{cases} I=\textit{interest earned}\\ P=\textit{original amount deposited}\dotfill & \$586.21\\ r=rate\to 6.3\%\to \frac{6.3}{100}\dotfill &0.063\\ t=years\dotfill &\frac{83}{360} \end{cases} \\\\\\ I = (586.21)(0.063)(\frac{83}{360}) \implies I \approx 8.51[/tex]
can someone help me please
A car was valued at $44,000 in the year 1992. The value depreciated to $15,000 by the year 2006.
A) What was the annual rate of change between 1992 and 2006?
r=---------------Round the rate of decrease to 4 decimal places.
B) What is the correct answer to part A written in percentage form?
r=---------------%
C) Assume that the car value continues to drop by the same percentage. What will the value be in the year 2009 ?
value = $ -----------------Round to the nearest 50 dollars.
(A) the annual rate of change between 1992 and 2006 was 0.0804
(B) r = 0.0804 * 100% = 8.04%
(C) value in 2009 = $11,650
What is the rate of change?
The rate of change is a mathematical concept that measures how much one quantity changes with respect to a change in another quantity. It is the ratio of the change in the output value of a function to the change in the input value of the function. It describes how fast or slow a variable is changing over time or distance.
A) The initial value is $44,000 and the final value is $15,000. The time elapsed is 2006 - 1992 = 14 years.
Using the formula for an annual rate of change (r):
final value = initial value * [tex](1 - r)^t[/tex]
where t is the number of years and r is the annual rate of change expressed as a decimal.
Substituting the given values, we get:
$15,000 = $44,000 * (1 - r)¹⁴
Solving for r, we get:
r = 0.0804
So, the annual rate of change between 1992 and 2006 was 0.0804 or approximately 0.0804.
B) To express the rate of change in percentage form, we need to multiply by 100 and add a percent sign:
r = 0.0804 * 100% = 8.04%
C) Assuming the car value continues to drop by the same percentage, we can use the same formula as before to find the value in the year 2009. The time elapsed from 2006 to 2009 is 3 years.
Substituting the known values, we get:
value in 2009 = $15,000 * (1 - 0.0804)³
value in 2009 = $11,628.40
Rounding to the nearest $50, we get:
value in 2009 = $11,650
Hence, (A) the annual rate of change between 1992 and 2006 was 0.0804
(B) r = 0.0804 * 100% = 8.04%
(C) value in 2009 = $11,650
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What is the rectangular equivalence to the parametric equations?
x(θ)=3cosθ+2,y(θ)=2sinθ−1 , where 0≤θ<2π .
Drag a term into each box to correctly complete the rectangular equation.
the rectangular equation that is equivalent to the given parametric equations is: [tex]4(x-2)^2 + 9(y+1)^2 = 36[/tex]
The given parametric equations describe a curve in the xy-plane traced by a particle that moves along a circular path centered at (2,-1) with radius 3. To find the rectangular equation, we can use the following trigonometric identity:
[tex]cos^2[/tex]θ + [tex]sin^2[/tex]θ = 1
Multiplying both sides by [tex]3^2[/tex], we get:
9[tex]cos^2[/tex]θ + [tex]9sin^2[/tex]θ = 9
Rearranging and using the fact that cosθ = (x-2)/3 and sinθ = (y+1)/2, we get:
[tex]9((x-2)/3)^2 + 9((y+1)/2)^2 = 9[/tex]
Simplifying, we get:
[tex](x-2)^2/3^2 + (y+1)^2/2^2 = 1[/tex]
Multiplying both sides by 36, we get:
[tex]4(x-2)^2 + 9(y+1)^2 = 36[/tex]
Therefore, the rectangular equation that is equivalent to the given parametric equations is:
[tex]4(x-2)^2 + 9(y+1)^2 = 36[/tex]
This equation represents an ellipse centered at (2,-1) with semi-axes of length 3 and 2 along the x-axis and y-axis, respectively. The parameter θ varies from 0 to 2π, which means the particle completes one full revolution around the ellipse.
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help asap
let n>=2. Calculate the sum of the nth roots of the unity.
The sum of the nth roots of unity is zero.
What are Complex Numbers?A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, which is defined as the square root of -1. Complex numbers are used in mathematics and science to represent quantities that have both a real and imaginary part.
The nth roots of unity are given by:
[tex]\omega ^0, \omega^1, \omega^2, ..., \omega^(n-1)[/tex]
where [tex]\omega = e^{(2\pi i/n)[/tex]is a complex number and i is the imaginary unit.
The sum of these roots is:
[tex]\omega^0 + \omega^1 + \omega^2 + ... + \omega^{(n-1)[/tex]
To simplify this expression, we can use the formula for the sum of a geometric series:
[tex]a + ar + ar^2 + ... + ar^(n-1) = a(1 - r^n)/(1 - r)[/tex]
Let a = 1 and [tex]r = ω.[/tex] Then the sum of the nth roots of unity is:
[tex]\omega^0 + \omega^1 + \omega^2 + ... + \omega^(n-1) = (1 - \omega^n)/(1 - \omega)[/tex]
Substituting [tex]ω = e^(2πi/n)[/tex], we get:
[tex]\omega^n = (e^(2\pi i/n))^n = e^2 \pi i = 1[/tex]
Therefore, the sum of the nth roots of unity is:[tex]\omega^0 + \omega^1 + \omega^2 + ... + \omega^(n-1) = (1 - 1)/(1 - \omega) = 0/(1 - \omega) = 0[/tex]
Hence, the sum of the nth roots of unity is zero.
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