The answer is: ∠N ≈ 33°
To find ∠N in ΔLMN, we can use the Law of Cosines which states that c² = a² + b² - 2abcos(C), where c is the side opposite angle C.
In this case, side LM (m) is opposite angle ∠N, side LN (n) is opposite angle ∠L, and side MN (x) is opposite the unknown angle.
So, we can write:
m² = n² + x² - 2nxcos(82°)
Substituting the given values:
x² = 35² + 59² - 2(35)(59)cos(82°)
Solving for x, we get:
x ≈ 64.27
Now, using the Law of Sines which states that a/sin(A) = b/sin(B) = c/sin(C), we can find ∠N:
sin(∠N)/35 = sin(82°)/64.27
sin(∠N) ≈ 0.5392
∠N ≈ sin⁻¹(0.857) ≈ 32.6344°
Therefore, ∠N ≈ 33° to the nearest degree.
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Victor drew trapezoid PQRS on a
coordinate plane. The coordinates of each
vertex are:
P(8,4) Q(10, 4) R(13,-1) S(8,-1)
ion
What is the length, in units, of side PS?
A. 2
B. 3
C. 4
D. 5
The coordinates of each vertex are: P(8,4) Q(10, 4) R(13,-1) S(8,-1) then the length of side PS is 0 units.
Side PS is the bottom base of trapezoid PQRS. To find its length, we need to calculate the horizontal distance between the x-coordinates of points P and S.
The x-coordinate of point P is 8, and the x-coordinate of point S is also 8. Therefore, the horizontal distance between these two points is 0. So, the length of side PS is 0 units.
The answer is (A) 2 is not correct because the length of side PS cannot be negative or less than zero, and the length of the other base of the trapezoid (QR) is 2 units, which is not equal to the length of side PS.
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The greenery landscaping company puts in an order for 2 pine trees and 5 hydrangea bushes for a neighborhood project. the order costs $150. they put in a second order for 3 pine trees and 4 hydrangea bushes that cost $144. 50.
what is the cost for one pine tree?
$: ?
The cost of one pine tree is $17.50.
To find the cost of one pine tree, we can use the information provided about the orders from the Greenery Landscaping Company. We have the following two equations:
1) 2P + 5H = $150 (2 pine trees and 5 hydrangea bushes)
2) 3P + 4H = $144.50 (3 pine trees and 4 hydrangea bushes)
Now, let's solve these equations using the substitution or elimination method. Here, we'll use the elimination method.
Step 1: Multiply the first equation by 3 and the second equation by 2 to make the coefficients of H the same:
1) 6P + 15H = $450
2) 6P + 8H = $289
Step 2: Subtract the second equation from the first equation:
(6P + 15H) - (6P + 8H) = $450 - $289
0P + 7H = $161
Step 3: Divide by 7 to find the cost of one hydrangea bush (H):
H = $161 / 7
H = $23
Step 4: Substitute the value of H back into one of the original equations to find the cost of one pine tree (P). We'll use the first equation:
2P + 5($23) = $150
2P + $115 = $150
Step 5: Subtract $115 from both sides of the equation:
2P = $35
Step 6: Divide by 2 to find the cost of one pine tree (P):
P = $35 / 2
P = $17.50
So, the cost of one pine tree is $17.50.
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The radius of a circle is 10 meters what is the area
Answer:
100π or ≈ 314.1592
Step-by-step explanation:
Area of a circle formula: A = πr²
A = π(10)²
A = π(100)
A = 100π ≈ 314.1592
Answer:
100pi
Step-by-step explanation:
pi*radius^2=area
pi*10^2=100pi
The point R(-2, 9) is rotated 90° clockwise around the origin. What are the coordinates of the resulting point, R′?
The coordinates of the resulting point, R′ are R' = (-9, -2)
What are the coordinates of the resulting point, R′?From the question, we have the following parameters that can be used in our computation:
The point R(-2, 9) is rotated 90° clockwise around the origin.
This means thar
R = (-2, 9)
Rotation rule = 90° clockwise around the origin.
The rotation rule of 90° clockwise around the origin is
(x,y) becomes (y,-x)
So, we have
R' = (y, -x)
Substitute the known values in the above equation, so, we have the following representation
R' = (-9, -2)
Hence, the coordinates of the resulting point, R′ are R' = (-9, -2)
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In the driest part of an Outback ranch, each cow needs about 40 acres for grazing. Use an equation to find how many cows can graze on 720 acres of land.
The calculated value of the number of cows that can graze on 720 acres of land is 18
From the question, the statements that can be used in our computation are given as
The area of grazing needed by each cow is 40 acres
From the above statement, the equation to use is
Cows = Area of land/Unit rate of cows
By substituting the given values in the above equation, we have the following equation
Cows = 720/40
Evaluate
Cows = 18
Hence, the calculated number of cows is 18
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In a 2 digit number the tens digit is 5 less than the units digit. The number itself is 5 more tha 3 times the sum of its digits. What is the number
Answer:
Step-by-step explanation:
(2-5)3=-9. (-9)5 =-45
Pls help me find the exponent
Answer:
1.6 × 10^4
Step-by-step explanation:
As a general guideline, the research hypothesis should be stated as the:.
As a general guideline, the research hypothesis should be stated as the alternative hypothesis, which is the statement that researchers are trying to support or prove.
The research hypothesis is a statement that describes the expected relationship between variables or the expected difference between groups in a research study. It should be based on a clear and specific research question, and it should be testable using appropriate statistical methods.
In other words, the research hypothesis should be a clear and concise statement that proposes a relationship or difference between variables that can be tested through data analysis. It should also be framed in a way that allows for the rejection or acceptance of the hypothesis based on the results of the study.
The null hypothesis, on the other hand, is the statement that there is no significant relationship or difference between variables. It serves as the default assumption until evidence is provided to support the alternative hypothesis.
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A research hypothesis should be stated as the predicted outcome of the study. It's often a declarative sentence that shows the relationship between variables in a study. The research hypothesis is often contrasted with a null hypothesis, which claims no significant relationship between the study's variables.
Explanation:A Research HypothesisIn general, a research hypothesis should be stated as the predicted outcome of the study. A research hypothesis is usually written in a declarative sentence format and states the relationship between variables in the study. For example, if your research is about studying the impact of amount of study time on test scores, your hypothesis could be: 'Students who spend more time studying will have higher test scores.'
The research hypothesis is often contrasted with a null hypothesis, which states there will be no significant relationship between the study's variables. In our example, the null hypothesis would be: 'The amount of study time will not impact the test scores significantly.' Remember, a research hypothesis should always be testable through research methods.
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a city maintains a database of all traffic tickets that were issued over the past ten years. the tickets are divided into the following two categories. moving violations nonmoving violations the data recorded for each ticket include only the following information. the month and year in which the ticket was issued the category of the ticket which of the following questions cannot be answered using only the information in the database? responses have the total number of traffic tickets per year increased each year over the past ten years? have the total number of traffic tickets per year increased each year over the past ten years? in the past ten years, were nonmoving violations more likely to occur on a weekend than on a weekday? in the past ten years, were nonmoving violations more likely to occur on a weekend than on a weekday? in the past ten years, were there any months when moving violations occurred more often than nonmoving violations? in the past ten years, were there any months when moving violations occurred more often than nonmoving violations? in how many of the past ten years were there more than one million moving violations?
The statement about the chances of occurrence of nonmoving violations more on a weekend than on a weekday in past ten years is cannot be answered using only the information in the database. So, option(b) is provide the right answer.
Database: The database is a large collection of data and it is used to store and retrieve the accordingly. It has an organization of a certain amount of data that needs and it contains. We have a database of all traffic tickets that were issued over the past ten years in a city. The informations including are
the tickets are divided into two categories. 1) moving violations2) nonmoving violations
the month and year in which the ticket was issued the category of the ticket.We have to recongise the statement which is not answering by the above informations in database. First statement is can be answerable, because we have data of year in which tickets are issued. Similarly, statements third and fourth related to data of tickets issued in month and year in past ten years. So, both are answerable statements. Therefore, the second statement
is cannot be answerable here.
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Complete question:
a city maintains a database of all traffic tickets that were issued over the past ten years. the tickets are divided into the following two categories. moving violations nonmoving violations the data recorded for each ticket include only the following information. the month and year in which the ticket was issued the category of the ticket which of the following questions cannot be answered using only the information in the database? responses
a) have the total number of traffic tickets per year increased each year over the past ten years?
b) in the past ten years, were nonmoving violations more likely to occur on a weekend than on a weekday?
c) in the past ten years, were there any months when moving violations occurred more often than nonmoving violations?
d) in how many of the past ten years were there more than one million moving violations?
how many five-digit positive integers exist where the digits are non increasing from left to right? (for example, 87743 and 10000 fulfill the conditions. 78987 and 33429 do not.)
There are 715 five-digit positive integers where the digits are non-increasing from left to right.
Here, we have to find the number of five-digit positive integers where the digits are non-increasing from left to right, you can think of this as selecting five digits (from 0 to 9) with repetition allowed, while ensuring that the selected digits are arranged in a non-increasing order.
This is essentially a combinations with repetition problem.
For each digit, there are 10 choices (0 to 9). Since repetition is allowed, you can use a stars and bars approach, where you place 4 bars among 10 possible positions (one for each digit choice) to separate the digits into groups.
The number of ways to arrange 5 digits with repetition allowed is given by the formula:
Number of arrangements = (n + k - 1) choose k,
where n is the number of digits (10 choices) and k is the number of bars (4). Plugging in the values:
Number of arrangements = (10 + 4 - 1) choose 4 = 13 choose 4 = 715.
So, there are 715 five-digit positive integers where the digits are non-increasing from left to right.
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Three stages at a music festival are arranged in a triangle. The distances between the centers of each stage are given: Stage A and Stage B: 100 yards Stage B and Stage C: 135 yards Stage C and Stage A: 210 yards List the interior angle measures formed at the center of each stage in descending order (largest to smallest). â
Answer:
BCA
Step-by-step explanation:
The interior angle measures at stages A, B, and C are approximately 71.7 degrees, 133.4 degrees, and 175.0 degrees, respectively. Therefore, the angles in descending order are C, B, and A.
Let's label the stages A, B, and C. To find the interior angles at each stage, we can use the Law of Cosines:
cos(A) = (b^2 + c^2 - a^2) / 2bc
cos(B) = (c^2 + a^2 - b^2) / 2ca
cos(C) = (a^2 + b^2 - c^2) / 2ab
where a, b, and c are the distances between the centers of the stages.
Using the given distances, we have:
a = 100, b = 135, c = 210
cos(A) = (135^2 + 210^2 - 100^2) / (2*135*210) ≈ 0.309
cos(B) = (210^2 + 100^2 - 135^2) / (2*210*100) ≈ -0.692
cos(C) = (100^2 + 135^2 - 210^2) / (2*100*135) ≈ -0.905
To find the interior angles, we can take the inverse cosine (also known as arccosine) of each value:
A ≈ 71.7 degrees
B ≈ 133.4 degrees
C ≈ 175.0 degrees
So the interior angle measures at stages A, B, and C are approximately 71.7 degrees, 133.4 degrees, and 175.0 degrees, respectively. Therefore, the angles in descending order are C, B, and A.
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Which of the following quadratics would NOT have zeros that are 5 and -7?
(1) y= (x + 12-36
(3) y = x2 + 2x - 35
(2) y = (× + 5)(x - 7)
(4) y = 2(× + 7)(× - 5)
The quadratics that would not have 5 and -7 as zeros are (1) and (3)
Identifying the quadratics that would not have 5 and -7 as zeros?From the question, we have the following parameters that can be used in our computation:
Zeros: 5 and -7
This means that
x = 5 and x = -7
Rewrite as
x - 5 = 0 and x + 7 = 0
Multiply both equation
This gives
(x - 5)(x + 7) = 0
Rewrite as
f(x) = (x - 5)(x + 7)
When expanded, we have
f(x) = x² + 2x - 35
So, we have
(2) f(x) = x² + 2x - 35
(4) f(x) = 2(x + 7)(x - 5)
Hence, the quadratics that would not have 5 and -7 as zeros are (1) and (3)
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Assume that Jocoro Provincial Park occupies an area modeled by the domain D. This domain is bordered by the coordinate axes and the lines y = 60 and y = 180 - 2x (where units are in kilometers). Futhermore, assume that snow depth measurements (in meters) were taken over the park on April 15 and that the depth measurements are modeled by the function d(x, y) = -0.024x + 0,012y + 1.2. Estimate the volume of the snowpack in the park in cubic meters. (Give your answer as a whole or exact number.) V= ____billion m
The estimated volume of the snowpack in the park is approximately 1.94 billion cubic meters.
To estimate the volume of the snowpack in the park, we need to calculate a double integral of the function d(x, y) over the domain D:
V = ∬_D d(x, y) dA
We can evaluate this integral by using iterated integrals. First, we integrate with respect to x over the interval [0, 30] (since the line y = 180 - 2x intersects the x-axis at x = 90):
V = ∫_0^30 (∫_0^(180-2x) (-0.024x + 0.012y + 1.2) dy) dx
Simplifying the inner integral:
V = ∫_0^30 [-0.024xy + 0.006y^2 + 1.2y]_0^(180-2x) dx
V = ∫_0^30 (-0.432x^2 + 21.6x - 5832) dx
Simplifying the integral:
V = [-0.144x^3 + 10.8x^2 - 5832x]_0^30
V = -0.144(30)^3 + 10.8(30)^2 - 5832(30)
V = 1.942592 billion cubic meters (rounded to nine decimal places)
Therefore, the estimated volume of the snowpack in the park is approximately 1.94 billion cubic meters.
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which function is shown in the graph below
Answer: y=log 3 x
Step-by-step explanation:
5. a square has a vertex at (-15,-9) and is dilated at the origin with a scale factor of 1/3
what is the new coordinate of the of the vertex?
(3,5)
(5,3)
(-3,-5)
(-5,-3)
The new coordinate of the vertex after dilation is (-5, -3).
To find the new coordinate, you'll need to use the given scale factor of 1/3 and apply it to the original vertex coordinates (-15, -9). Here's a step-by-step explanation:
1. Identify the original vertex coordinates: (-15, -9).
2. Identify the scale factor for dilation: 1/3.
3. Apply the scale factor to the x-coordinate: (-15) * (1/3) = -5.
4. Apply the scale factor to the y-coordinate: (-9) * (1/3) = -3.
5. The new coordinates after dilation are (-5, -3).
By following these steps, you can find the new coordinates of the vertex after dilation at the origin using the given scale factor.
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A bag contains seven tiles labeled A B C D E F and G wich
One tile will be randomly picked.
What is the probability of picking a letter that is not a vowel
please i need answer to this Asap. Only question A, B, C, E . With solution Asap
Answer:
Solve for these and you'll find it! :D (Surface Area for all)
The answer is:
A: 6(12)^2 (it's a cube)
B: 2(6.4 x 15.2 + 15.2 x 10.5 + 6.4 x 10.5) (it's a rectangular prism)
C: 6(3x - 1)^2 (it's a cube)
D: 2(pi)(radius)(height) + 2(pi)(radius^2) (it's a cylinder)
A machine has an initial cost of $40,000 and operating costs of $3500 each year. Its salvage value decreases by $4,000 each year. The machine is now 4 years old.
Assuming an effective annual interest rate of 12%, what is the cost of owning and operating the machine for one more year.
The cost of owning and operating the machine for one more year is approximately $11,160.71.
To calculate the cost of owning and operating the machine for one more year, we need to consider both the operating costs and the decrease in salvage value.
The operating costs for one more year will be $3,500.
To calculate the decrease in salvage value, we need to know the salvage value of the machine after 4 years of use. If the salvage value decreased by $4,000 each year, then after 4 years the salvage value will have decreased by $16,000. Therefore, the salvage value of the machine after 4 years is:
Salvage value after 4 years = Initial salvage value - Total decrease in salvage value
Salvage value after 4 years = $40,000 - $16,000
Salvage value after 4 years = $24,000
To calculate the cost of owning and operating the machine for one more year, we need to consider the difference between the salvage value at the end of the additional year and the salvage value after 4 years. Assuming a straight-line depreciation model, the salvage value of the machine after one more year will be:
Salvage value after one more year = Salvage value after 4 years - (4 x $4,000)
Salvage value after one more year = $24,000 - $16,000
Salvage value after one more year = $8,000
To calculate the cost of owning and operating the machine for one more year, we need to calculate the present value of the difference between the salvage values, plus the operating costs for one more year. Assuming an effective annual interest rate of 12%, the present value can be calculated using the formula:
PV = FV / (1 + r[tex])^n[/tex]
where PV is the present value, FV is the future value, r is the effective annual interest rate, and n is the number of years.
The future value of the salvage value difference plus the operating costs for one more year is:
FV = Salvage value after one more year - Salvage value after 4 years + Operating costs for one more year
FV = $8,000 - $24,000 + $3,500
FV = -$12,500
(Note that the negative value indicates a cost.)
Plugging in the values, we get:
PV = -$12,500 / (1 + 0.12[tex])^1[/tex]
PV = -$11,160.71
Therefore, the cost of owning and operating the machine for one more year is approximately $11,160.71.
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Create a table of values for the function y=x² - 4x + 12 for the domain {-2,-1,0,1}. What is the value of f¹(12)?
Answer:
f¹(12) = 20
Step-by-step explanation:
f(-2) = 24
f(-1) = 17
f(0) = 12
f(1) = 9
f¹(x) = 2x - 4
f¹(12) = 20
Which measurement is closest to the circumference of circumference of the parachute in feet
The closest measurement to the circumference of the parachute in feet is 78.54 feet. The calculation was done by multiplying the radius by 2π (the constant pi, approximately equal to 3.14), which gives the circumference of a circle is 78.54 feet.
The circumference of the parachute can be calculated using the formula
C = 2πr
where r is the radius of the parachute.
Given that the radius of the parachute is 12.5 feet, we can substitute this value in the formula and calculate the circumference
C = 2πr
C = 2π(12.5)
C ≈ 78.54 feet
Therefore, the circumference of the parachute is closest to 78.54 feet.
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--The given question is incomplete, the complete question is given
" Which measurement is closest to the circumference of circumference of the parachute in feet if radius is 12.5. "--
Building a campfire you start by stacking kindling wood to form a pentagonal pyramid that is 27 centimeters tall. the base area is 965 square centimeters. what is the volume of the campfire pyramid
The volume of the campfire pyramid is approximately 8,175 cubic centimeters. This is found by using the formula for the volume of a pyramid and plugging in the given values for the height and base area of the pentagonal pyramid.
The formula for the volume of a pyramid is
V = (1/3) * base_area * height
where V is the volume, base_area is the area of the base, and height is the height of the pyramid.
In this case, the height of the pyramid is given as 27 cm, and the base area is given as 965 square cm. To find the volume, we can plug these values into the formula
V = (1/3) * 965 cm² * 27 cm
V = 8,175 cm³
Therefore, the volume of the campfire pyramid is 8,175 cubic centimeters.
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The substitution u = 3x transforms the integral 31 de into
The substitution u = 3x transforms the integral 31 de into: ∫(3/31)du
This is because the substitution u = 3x implies that du/dx = 3, which means dx = (1/3)du. Substituting this expression for dx in the original integral and using the fact that e is a constant, we have:
∫e^(3x) dx = ∫e^(u) (1/3)du = (1/3)∫e^(u)du = (1/3)e^u + C
where C is the constant of integration. So the final answer in terms of x is:
(1/3)e^(3x) + C
which is equivalent to the original integral. This is an example of how the technique of substitution can be used to simplify an integral and make it easier to solve. It is also a common step in many integral transforms.
When performing an integral with substitution, you transform the original integral into a new one with a different variable. In your case, the substitution is given as u = 3x.
To apply substitution, first find the derivative of the substitution equation with respect to x, which is du/dx = 3. Then, solve for dx: dx = du/3.
Now, substitute u = 3x into the original integral and replace dx with du/3. The transformed integral will have the new variable u and a constant factor 1/3.
Without knowing the specific function you're trying to integrate (it seems like "31 de" might be a typo), I cannot provide the exact transformed integral.
However, I hope this explanation of substitution and transforming integrals is helpful. Please feel free to provide more information or clarify your question if you need further assistance!
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Which equation best represents the line of best fit for the scatterplot?a) y = 2.5x + 25 b) y = −2.5x + 20 c) y = −0.05x + 20 d) y = −0.005x + 22.5
The equation of the line representing that best fit the given scatterplot is given by option d. y = -0.005x + 22.5.
Consider the two points from the attached scatterplot.
Let the coordinates of the two point be ( x₁ , y₁) = ( 1500 , 12.5 )
And other point be ( x₂ , y₂) = ( 2000 , 10 )
Slope of the line 'm' = ( y₂ - y₁ ) / ( x₂ - x₁ )
= ( 10 - 12.5) / ( 2000 - 1500 )
= -2.5 / 500
= -0.005
From the attached scatterplot we have,
y-intercept 'c' where x = 0 is equals to 22.5.
The equation best which represents the line of best fit for the scatterplot is equals to,
y = mx + c
Substitute the value we have,
y = -0.005x + 22.5
Therefore, the equation of the line representing scatterplot is equals to option d. y = -0.005x + 22.5.
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The above question is incomplete, the complete question is:
Which equation best represents the line of best fit for the scatterplot?
a) y = 2.5x + 25 b) y = −2.5x + 20 c) y = −0.05x + 20 d) y = −0.005x + 22.5
Attached scatterplot.
In a round of mini-golf , Clare records the number of strokes it takes to hit the ball into the hole of each green. She said that, if she redistributed the strokes on different greens, she could tell that her average number of strokes per hole is 3.
If Clare's average number of strokes per hole is 3, it means that the total number of strokes she took in the round divided by the number of holes she played is equal to 3.
Let's say Clare played n holes in total and took a total of s strokes in the round. Then we can write:
s/n = 3
Multiplying both sides by n, we get:
s = 3n
This means that Clare took 3 strokes per hole on average, and a total of 3n strokes in the round. If she were to redistribute the strokes on different greens, the total number of strokes would still be 3n, and her average number of strokes per hole would remain 3.
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Which best describes the scale factor for each dilation?
Dilation 1 has a scale factor
.
Dilation 2 has a scale factor
.
Dilation 3 has a scale factor
.
The statement "Dilation 2 has a scale factor 2" (option b).
Dilation is a transformation that changes the size of an object but not its shape. It is a type of transformation that enlarges or reduces an object by a certain factor called the scale factor. The scale factor is a ratio of the size of the dilated image to the size of the original image.
Now, let's talk about the scale factor for each dilation you asked about.
For dilation 1, the scale factor is 1. This means that the size of the dilated image is the same as the size of the original image. In other words, there is no change in size. This type of dilation is often referred to as the identity transformation because it doesn't change the shape or size of the original object.
For dilation 2, the scale factor is 2. This means that the size of the dilated image is twice as large as the size of the original image. In mathematical terms, if the original object has a length of 'x', then the length of the dilated image will be '2x'. Similarly, if the original object has a width of 'y', then the width of the dilated image will be '2y'.
For dilation 3, the scale factor is 3. This means that the size of the dilated image is three times as large as the size of the original image. In mathematical terms, if the original object has a length of 'x', then the length of the dilated image will be '3x'. Similarly, if the original object has a width of 'y', then the width of the dilated image will be '3y'.
Hence the correct option is (b).
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Answer:
Dilation 1 has a scale factor between 0 and 1
Dilation 2 has a scale factor greater than 1
Dilation 3 has a scale factor equal to 1
Step-by-step explanation:
please help ASAP (can give brainliest)
Answer:
x = 6
Step-by-step explanation:
In order for it to be a parallelogram, the 2 lines must be equal.
2x=3x-6
2x - 3x = 3x - 3x -6
-1x = -6
x = 6/1
x = 6
What is the diameter if the circumference is 11.27
The diameter of approximately 3.58.
What is the circumference of a circle?The circumference of a circle is the distance around its edge or perimeter. The diameter of a circle is the distance across it, passing through the center. These two measurements are related by the mathematical constant pi (π), which is the ratio of the circumference of any circle to its diameter.
The formula to find the diameter of a circle from its circumference is:
diameter = circumference/pi
So, if you know the circumference of a circle, you can simply divide it by pi to find the diameter. In the case of the given circumference of 11.27, dividing it by pi gives us the diameter of approximately 3.58.
It's important to note that the diameter of a circle is twice the length of its radius, which is the distance from the center of the circle to its edge. So, if you know the diameter of a circle, you can find its radius by dividing the diameter by 2:
radius = diameter / 2
In this case, the radius of the circle would be approximately 1.79 (since 3.58 / 2 = 1.79).
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Solve 2^{3-x} + 2^{x+1} =17
Log x + log (x+3) = 1
To solve the equation [tex]2^{3-x} + 2^{x+1} = 17[/tex] and the equation log x + log (x+3) = 1, We get the solution to the system of equations is x = 2. To check if x = 2 satisfies the first equation [tex](2^{3-x} + 2^{x+1} = 17).[/tex]
Solve the first equation, [tex]2^{3-x} + 2^{x+1} = 17.[/tex] Rewrite the equation as[tex]2^{3-x} = 17 - 2^{x+1}.[/tex] Take the logarithm of both sides (base 2): [tex]3-x = log2(17 - 2^{x+1})[/tex]. Rewrite the equation as [tex]x = 3 - log2(17 - 2^{x+1}).[/tex]
Solve the second equation, log x + log (x+3) = 1. Combine the logarithms: log(x(x+3)) = 1. Remove the logarithm by taking the exponent of both sides: x(x+3) = 10. The equation: x^2 + 3x = 10. Rearrange to form a quadratic equation:[tex]x^2 + 3x - 10 = 0.[/tex]
Factor the quadratic equation: (x+5)(x-2) = 0. Set each factor to zero: x+5 = 0 or x-2 = 0. Solve for x: x = -5 or x = 2. The solutions. Check if x = -5 satisfies the first equation [tex](2^{3-x} + 2^{x+1} = 17).[/tex]([tex]2^{3-x} + 2^{x+1} = 17)[/tex]. If it does not, discard it.
The solution to the system of equations is x = 2.
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(1 point) Use the Integral Test to determine whether the infinite series is convergent. 00 n2 n=12 (n3 + 3) Fill in the corresponding integrand and the value of the improper integral. Enter inf for oo, -inf for -00, and DNE if the limit does not exist. Compare with dx = 00 By the Integral Test, 722 the infinite series n 12 (73+3) A. converges B. diverges
To use the Integral Test, we need to find an integral that is comparable to the series. We can do this by using a basic comparison test and comparing it to the p-series with p=2.
n^2 / (n^3 + 3) < n^2 / n^3 = 1/n
The series 1/n is a divergent p-series with p=1, so we can conclude that the original series is also divergent.
To find the corresponding integral, we can integrate the function 1/n^2:
∫(n=1 to ∞) 1/n^2 dn = [-1/n] (n=1 to ∞) = 1/1 - 0 = 1
Since the improper integral converges to 1, we can conclude that the infinite series is divergent by the Integral Test.
Hi there! To use the Integral Test to determine whether the given infinite series is convergent, first rewrite the series as a function:
f(x) = x^2 / (x^3 + 3)
Next, we need to check that the function is continuous, positive, and decreasing on the interval [1, ∞). This function satisfies these conditions.
Now, we will calculate the improper integral:
∫(from 1 to ∞) (x^2 / (x^3 + 3)) dx
Let's use substitution: u = x^3 + 3, so du = 3x^2 dx, and x^2 dx = (1/3)du.
Now, the integral becomes:
(1/3) ∫(from 1 to ∞) (1/u) du
This integral is the same as the integral of 1/u from 1 to ∞, which is a well-known improper integral that diverges (ln(u) evaluated from 1 to ∞ results in ∞).
Therefore, by the Integral Test, the infinite series ∑(from n=1 to ∞) (n^2 / (n^3 + 3)) diverges. So the correct answer is B. Diverges.
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If st - sv, m∠sut - w + °10, and m∠suv =3w, what is m∠sut
If st - sv, m∠sut - w + °10, and m∠suv =3w, the measure of angle SUT is 52.5°.
Given the information provided, we can set up the following equations:
1) m∠SUT + m∠SUV = 180° (since they are supplementary angles)
2) m∠SUT = w + 10°
3) m∠SUV = 3w
Now we can substitute equations (2) and (3) into equation (1):
(w + 10°) + (3w) = 180°
Combining like terms, we get:
4w + 10° = 180°
Now, subtract 10° from both sides:
4w = 170°
Finally, divide both sides by 4:
w = 42.5°
Now we can find m∠SUT by substituting the value of w back into equation (2):
m∠SUT = 42.5° + 10°
m∠SUT = 52.5°
So, the measure of angle SUT is 52.5°.
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