A scatter plot is a graphical representation of a set of data points that displays the relationship between two variables.
How to solve the question?
The plot uses a Cartesian coordinate system, with one variable plotted on the x-axis and the other on the y-axis. Each data point is represented by a dot, with its location on the plot determined by its values for the two variables. The pattern of the dots on the plot gives insight into the relationship between the two variables.
Scatter plots are useful for visualizing the strength and direction of a relationship between two variables. A positive relationship between the variables means that as one variable increases, the other variable also increases. In contrast, a negative relationship means that as one variable increases, the other variable decreases. A scatter plot can also help to identify any outliers or unusual data points.
Scatter plots can be used to analyze many types of data, including scientific data, financial data, and social data. They can be used to identify trends, correlations, and patterns that might not be immediately apparent from a simple numerical analysis of the data. Scatter plots are a valuable tool for data analysis and visualization, and are widely used in scientific research, data analysis, and data-driven decision making.
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In week 2, if 400 drinks are sold, to estimate the number of smoothies write the proportion _______. There would be _______ smoothies
Answer:
57 seven drinks sold each day 200 drinks a week
Step-by-step explanation:
what is the amount of work done, in foot-pounds, to lift a crate weighing 30 pounds a total of 3 feet?
The amount of work done, in foot-pounds, to lift a crate weighing 30 pounds a total of 3 feet is 90 foot-pounds.
To calculate the amount of work done, in foot-pounds, to lift a crate weighing 30 pounds a total of 3 feet, we need to use the formula:
Work = Force x Distance
In this case, the force is the weight of the crate, which is 30 pounds. The distance is the height the crate is lifted, which is 3 feet.
So, Work = 30 pounds x 3 feet
We can simplify this by converting pounds to foot-pounds, which is the unit used to measure work. To convert pounds to foot-pounds, we need to multiply the weight by the distance. So:
Work = 30 pounds x 3 feet = 90 foot-pounds
Therefore, the amount of work done, in foot-pounds, to lift a crate weighing 30 pounds a total of 3 feet is 90 foot-pounds.
To calculate the amount of work done, in foot-pounds, for lifting a crate weighing 30 pounds a total of 3 feet, you can follow these steps:
Step 1: Identify the weight of the crate (30 pounds) and the distance it is lifted (3 feet).
Step 2: Multiply the weight of the crate by the distance lifted.
Work (in foot-pounds) = Weight (in pounds) × Distance (in feet)
In this case:
Work = 30 pounds × 3 feet
Work = 90 foot-pounds
So, the amount of work done is 90 foot-pounds.
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The work done to lift a crate weighing 30 pounds a total of 3 feet is:
Work = 30 pounds x 3 feet x cos(0 degrees) = 90 foot-pounds
So the amount of work done to lift the crate is 90 foot-pounds.
The amount of work done, in foot-pounds, to lift a crate weighing 30 pounds a total of 3 feet is given by the formula:
Work = Force x Distance x cos(theta)
where Force is the weight of the crate, Distance is the vertical distance lifted, and theta is the angle between the direction of the force and the direction of motion (which is 0 degrees in this case since the force is acting vertically upwards).
Therefore, the work done to lift a crate weighing 30 pounds a total of 3 feet is:
Work = 30 pounds x 3 feet x cos(0 degrees) = 90 foot-pounds
So the amount of work done to lift the crate is 90 foot-pounds.
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Help. Please? Really need some help please?
Answer:
To solve this problem, we need to calculate the compound interest on the loan over the 9-year period. We can break this down into two parts: the first 5 years, during which the interest rate is 7%, and the remaining 4 years, during which the interest rate is 11%.
For the first 5 years, we can use the formula for compound interest:
A = P * (1 + r/n)^(n*t)
where A is the amount after t years, P is the principal (the initial amount borrowed), r is the annual interest rate as a decimal, n is the number of times the interest is compounded per year, and t is the time in years.
In this case, P = $1350, r = 0.07, n = 1 (compounded annually), and t = 5. Plugging in these values, we get: A = 1350 * (1 + 0.07/1)^(1*5) = $1872.73 (rounded to the nearest cent)
So after 5 years, Imaan owes $1872.73.
For the remaining 4 years, we can use the same formula with r = 0.11 and t = 4: A = 1872.73 * (1 + 0.11/1)^(1*4) = $2959.77 (rounded to the nearest cent)
So after 9 years, Imaan owes $2959.77. However, this is the amount she would owe if she paid back the loan in a single payment at the end of 9 years. If she pays back the loan in installments, she will need to pay interest on the outstanding balance each year. Without information about the payment schedule, we can't calculate the exact amount she will have to pay back.
The amount Imaan has to pay back, after 9 years at 5% and 11% per annum interest, obtained using the compound interest formula is about $2,616
What is a compound interest rate?A compound interest is an interest calculated using based on the interest earned on from previous periods of the investment or loan.
The amount, A, Imaan pays back after 9 years can be obtained using the compound interest formula as follows;
A = P·(1 + r)ⁿ
Where;
P = The principal amount borrowed = $1,350
r = The interest rate = 7% and 11%
n = The number of years = 5 years and (9 - 5) years
The value of the loan after the first 5 years is therefore;
A = $1350 × (1 + 5/100)⁵ ≈ $1722.98
The value of the loan after the next four years, at 11% per annum, interest rate is therefore;
A = $1,722.98 × (1 + 11/100)⁴ ≈ $2616
The value of the loan, and the amount Imaan has to pay back after the 9 years is about $2,616
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Click an item in the list or group of pictures at the bottom of the problem and, holding the button down, drag it into the correct position in the answer box. Release your mouse button when the item is place. If you change your mind, drag the item to the trashcan. Click the trashcan to clear all your answers.
Factor completely and then place the factors in the proper location on the grid.
81a2 + 36a + 4
When we factor the expression, we are going to obtain the result;
9a(9a + 4) + 4
How do you factor an expression?Factoring an expression involves breaking it down into simpler parts that can be multiplied together to obtain the original expression. The specific method used to factor an expression depends on the type of expression.
It's important to note that factoring can sometimes be a challenging process, and not all expressions can be factored using real numbers.
We can carry out this factorization by the use of the nesting method, Hence;
81a^2 + 36a + 4
(81a^2 + 36a ) + 4
9a(9a + 4) + 4
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trigonometry, please help.
The value of x is given as follows:
x = 9.40 cm.
What are the trigonometric ratios?The three trigonometric ratios are the sine, the cosine and the tangent, and they are defined as follows:
Sine of angle = length of opposite side to the angle divided by the length of the hypotenuse.Cosine of angle = length of adjacent side to the angle divided by the length of the hypotenuse.Tangent of angle = length of opposite side to the angle divided by the length of the adjacent side to the angle.The parameters for this problem are given as follows:
Hypotenuse of 10 cm.Side length of x cm adjacent to the angle of 20º.Applying the cosine, the value of x is obtained as follows:
cos(20º) = x/10
x = 10 x cosine of 20 degrees
x = 9.40 cm.
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Find the equation for the line of best fit.
1.
X235
7
11
16
24
Equation:
quation
r value:
y
5.8
4.55
2.05
-.45
-5.45
-11.7
-21.7
The equation of the line of best fit is given as follows:
y = -1.25x + 8.3.
How to find the equation of linear regression?To find the regression equation, which is also called called line of best fit or least squares regression equation, we need to insert the points (x,y) in the calculator.
The points for this problem are given as follows:
(2, 5.8), (3, 4.55), (5, 2.05), (7, -0.45), (11, -5.45), (16, -11.7), (24, -21.7).
Inserting these points into a calculator, the equation is given as follows:
y = -1.25x + 8.3.
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On a piece of paper, use a protractor and a ruler to construct two equilateral triangles: one with a side length of 4 inches and one with a side length of 5 inches.
Which statement is true about the two triangles?
Responses
The two triangles are the same size but not the same shape.
The two triangles are the same shape but not the same size.
It is impossible to construct equilateral triangles with these side lengths.
The two triangles are the same size and same shape.
The correct statement is "The two triangles are the same shape but not the same size."
How to determine Which statement is true about the two trianglesBoth triangles will have three congruent sides and three congruent angles, which means they will be equilateral and equiangular. However, since the side lengths are different, the triangles will be different in size. In other words, one triangle will be larger than the other.
To see why they cannot be the same size, imagine drawing a line from one vertex of each triangle to the opposite side. Since both triangles are equilateral, these lines will bisect the opposite side and form two congruent right triangles. The hypotenuse of the larger triangle (with side length 5) will be longer than the hypotenuse of the smaller triangle (with side length 4), which means the two triangles are not the same size.
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draw an appropriate tree diagram, and use the multiplication principle to calculate the probabilities of all the outcomes. hint [see example 3.]there is a 60% chance of rain today and a 60% chance of rain tomorrow. assume that the event that it rains today is independent of the event that it rains tomorrow. what is the probability that there will be no rain today and no rain tomorrow?
An appropriate tree diagram.
/ \
/ \
0.6 / \ 0.4
/ \
R NR
/ \ / \
/ \ / \
0.6 0.4 0.6 0.4
R NR R NR
Where R represents rain and NR represents no rain.
By using the multiplication principle, we can calculate the probabilities of all the outcomes by multiplying the probabilities along each path. The probability of there being no rain today and no rain tomorrow is the probability of following the path NR-NR, which has a probability of
P(NR-NR) = P(NR|NR) * P(NR) = 0.4 * 0.4 = 0.16
Hence, the probability that there will be no rain today and no rain tomorrow is 0.16 or 16%.
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(a) Show that a differentiable function f decreases most rapidly at x in the direction opposite the gradient vector, that is, in the direction of −∇f(x). Let θ be the angle between ∇f(x) and unit vector u. Then Du f = |∇f| cos(theta). Since the minimum value of cos(theta) is -1 occurring, for 0 ≤ θ < 2π, when θ = π , the minimum value of Du f is −|∇f|, occurring when the direction of u is the opposite of. The direction of ∇f (assuming ∇f is not zero).
(b) Use the result of part (a) to find the direction in which the function f(x, y) = x^(3)y − x^(2)y^(3) decreases fastest at the point (4, −4)
(a) As we have shown that the differentiable function f decreases most rapidly at x in the direction opposite the gradient vector, that is, in the direction of −∇f(x).
(b) The direction in which f(x, y) = x³y − x²y³ decreases fastest at (4, -4) is the direction of the unit vector u = <-3/5, 4/5>.
To show that a differentiable function f decreases most rapidly at x in the direction opposite the gradient vector, we first need to define the directional derivative. The directional derivative of f at x in the direction of a unit vector u is denoted by Du f and is given by the dot product of the gradient vector ∇f(x) and u.
Du f = ∇f(x)·u
Now, we want to find the direction in which Du f is minimized. Let θ be the angle between ∇f(x) and u. Using the dot product formula, we have:
Du f = |∇f(x)| cos(θ)
where |∇f(x)| is the magnitude of the gradient vector. Since cos(θ) is maximum when θ = 0 (i.e., when u points in the same direction as ∇f(x)) and minimum when θ = π (i.e., when u points in the opposite direction of ∇f(x)), we can conclude that the direction in which f decreases most rapidly at x is opposite the gradient vector −∇f(x).
Now, let's apply this result to the function f(x, y) = x³y − x²y³ and find the direction in which it decreases fastest at the point (4, −4).
First, we need to find the gradient vector of f:
∇f(x, y) = <3x²y-2xy³, x³-3x²y²>
Evaluating at (4, -4), we have:
∇f(4, -4) = <192, -256>
The magnitude of the gradient vector is |∇f(4, -4)| = √(192² + (-256)²) = 320.
To find the direction of fastest decrease, we need to consider the opposite of the gradient vector:
−∇f(4, -4) = <-192, 256>
To make this a unit vector, we divide by its magnitude:
u = <-192, 256>/320 = <-3/5, 4/5>.
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What are the zeros of the function f (x) = (2x +6)(x - 4)?
Answer:
x = - 3 , x = 4
Step-by-step explanation:
to find the zeros let f(x) = 0 , that is
(2x + 6)(x - 4) = 0
equate each factor to zero and solve for x
2x + 6 = 0 ( subtract 6 from both sides )
2x = - 6 ( divide both sides by 2 )
x = - 3
x - 4 = 0 ( add 4 to both sides )
x = 4
the zeros are x = - 3 , x = 4
Roller Coaster Crew
Ray and Kelsey have summer internships at an engineering firm. As part of their internship, they get to assist in the planning of a brand new roller coaster. For this assignment, you help Ray and Kelsey as they tackle the math behind some simple curves in the coaster's track.
Part A
The first part of Ray and Kelsey's roller coaster is a curved pattern that can be represented by a polynomial function.
Ray and Kelsey are working to graph a third-degree polynomial function that represents the first pattern in the coaster plan. Ray says the third-degree polynomial has four intercepts. Kelsey argues the function can have as many as three zeros only. Is there a way for the both of them to be correct? Explain your answer.
Kelsey has a list of possible functions. Pick one of the g(x) functions below and then describe to Kelsey the key features of g(x), including the end behavior, y-intercept, and zeros.
g(x) = (x + 2)(x − 1)(x − 2)
g(x) = (x + 3)(x + 2)(x − 3)
g(x) = (x + 2)(x − 2)(x − 3)
g(x) = (x + 5)(x + 2)(x − 5)
g(x) = (x + 7)(x + 1)(x − 1)
Create a graph of the polynomial function you selected from Question 2.
Part B
The second part of the new coaster is a parabola.
Ray needs help creating the second part of the coaster. Create a unique parabola in the pattern f(x) = (x − a)(x − b). Describe the direction of the parabola and determine the y-intercept and zeros.
Create a graph of the polynomial function you created in Question 4.
Part C
Now that the curve pieces are determined, use those pieces as sections of a complete coaster. By hand or by using a drawing program, sketch a design of Ray and Kelsey's coaster that includes the shape of the g(x) and f(x) functions that you chose in the Parts A and B. You do not have to include the coordinate plane. You may arrange the functions in any order you choose, but label each section of the graph with the corresponding function for your instructor to view.
Part D
Create an ad campaign to promote Ray and Kelsey's roller coaster. It can be a 15-second advertisement for television or radio, an interview for a magazine or news report, or a song, poem, or slideshow presentation for a company. These are just examples; you are not limited to how you prepare your advertisement, so be creative. Make sure to include a script of what each of you will say if you are preparing an interview or a report. The purpose of this ad is to get everyone excited about the roller coaster.
Logan takes a trip to an amusement park and rides a Ferris wheel. The graph below shows the height, in feet above the ground, of his car over time, t, measured in minutes. Write an equation in terms of y, height in feet above the ground, and t, time in minutes, to represent the given context.
We can write the equation as:
y = 80 sin(2π/5 t) + 280
How can we calculate?
Finding the oscillation's period will help us get started. Since the height repeats every five minutes, it appears from the graph that the period is five minutes.
The distance between the oscillation's highest and lowest points, or its amplitude, can then be determined. Given that 80 feet separate the highest and lowest points on the graph, it appears that the amplitude is 80 feet.
We determine the vertical shift of the graph, which is the height of the Ferris wheel when t = 0. From the graph, it appears that the vertical shift is 280 feet, since that is the height of the car when t = 0.
Putting it all together, we can write the equation as:
y = 80 sin(2π/5 t) + 280
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An athlete is planning to run a certain distance in three days. • The first day, he is planning to run 1/3 • The second day, he wants to run 3/4 the distance. The distance left. • The last day, he plans to run the last 4 miles. Part A: Create an equation that represents the situation and can be used to solve for x, the total distance that the athlete is x s planning to run in the three days. Enter your equation in the first response box. Part B: Enter the total distance, in miles, that the athlete is planning to run in the three days in the second response box
The equation that represents the situation is 1/3x + 3/4(2/3x) + 4 = x, where x is the total distance. The athlete is planning to run a total distance of 24 miles.
Let x be the total distance the athlete is planning to run in three days. Then, the distance he plans to run on the first day is 1/3 of x, the distance he plans to run on the second day is 3/4 of the distance left after the first day, which is (2/3)x.
Finally, the distance he plans to run on the third day is 4 miles. The equation that represents the situation is
1/3x + 3/4(2/3x) + 4 = x
To solve for x, we can simplify the equation obtained in Part A
1/3x + 1/2x + 4 = x
Multiplying both sides by 6 to get rid of the fractions, we get
2x + 3x + 24 = 6x
Simplifying and solving for x, we get
x = 24
Therefore, the athlete is planning to run a total distance of 24 miles in the three days.
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Pre-Caclus 75 points + [tex]Brainliest[/tex]
Question shown in picture.
Options:
[tex]\frac{x - 28}{x-46}[/tex]
[tex]\frac{x - 13}{x-28}[/tex]
[tex]\frac{x - 7}{x-13}[/tex]
[tex]\frac{x - 13}{x-46}[/tex]
Expectations:
Correct
Explantion
Reasonable
Must not:
Incorrect
No explanation
Spam
Gibberish
Question:
Answer:
Factoring the polynomials and then simplifying:
[tex] \frac{ {x}^{2} - 20x + 91}{ {x}^{2} - 35x + 196} \times \frac{ {x}^{2} - 74x + 1288}{ {x}^{2} - 92x + 2116 } [/tex]
[tex] \frac{(x - 7)(x - 13)}{(x - 7)(x - 28)} \times \frac{(x - 28)(x - 46)}{(x - 46)(x - 46)} [/tex]
[tex] \frac{x - 13}{x - 46} [/tex]
Answer:
x−46
x−13
Step-by-step explanation:
Explanation in picture :)
What is the y-intercept of the line with the equation y = one-half x minus 3? a. -3 c. -6 b. 6 d. One-half
If every 3 cm on a scale drawing is equal to 4 feet in real life, which lines on the drawing would be greater than 12 feet in real life? Select all that apply. A) 12 cm B) 6 cm C) 15 cm D) 10 cm
Answer: C) 15 cm, D) 10 cm
Step-by-step explanation:
Let's use the scale to convert the measurements on the drawing to real-life measurements in feet:
3 cm on the drawing = 4 feet in real life
To determine which lines on the drawing would be greater than 12 feet in real life, we need to find the lines on the drawing that correspond to more than 12 feet in real life.
For each line on the drawing, we can convert its length to feet using the scale:
A) 12 cm = (12/3) * 4 = 16 feet
B) 6 cm = (6/3) * 4 = 8 feet
C) 15 cm = (15/3) * 4 = 20 feet
D) 10 cm = (10/3) * 4 = 13.33 feet
Therefore, the lines that are greater than 12 feet in real life are C) 15 cm and D) 10 cm.
Answer:
C + D
Step-by-step explanation:
Pythagorean Theorem 100 points brainliest
A group of medical professionals predicts that the number of yearly occurrences of a disease will decrease from the year 2008 to the year 2016. They constructed a spreadsheet with formulas to display their predictions. How many occurrences of the disease do they predict in the year 2016?
One way to predict occurrences of a disease for a future year based on previous years' data is to use exponential modeling.
How can we predict occurrences of disease for year mathematically based on previous years data?Specifically, the formula for exponential growth or decay can be used to project the number of cases for a future time period based on the growth or decline rate observed in the past.
For example, let's say that we have data on the number of cases of a certain disease for the past 10 years. We can use this data to fit an exponential model that describes the growth or decline rate of the disease over time.
Once we have this model, we can use it to predict the number of cases for a future year. For instance, if the model predicts that the disease will grow at a rate of 5% per year, we can use this rate to estimate the number of cases for the next year by multiplying the current number of cases by 1.05. By doing this, we can forecast the occurrence of the disease for the next year based on the previous years' data.
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skinner produce buys fresh boston lettuce daily. daily demand is normally distributed with a mean of 100 units and standard deviation of 15 units. at the beginning of the day skinner orders 140 units of lettuce. what is the probability that skinner will have at least 20 units left over by the end of the day?
For a normally distribution of daily demand of fresh boston lettuce produce by skinner. the probability that skinner will have at least 20 units left over by the end of the day is equals the one.
Skinner produce buys fresh boston lettuce daily. There is daily demand is normally distributed,
Mean = 100 units
Standard deviations = 15 units
At beginning of the day skinner orders 140 units of lettuce. We have to determine the probability that skinner will have at least 20 units left over by the end of the day, P( X ≥ 20). Using the Z-Score for normal distribution is written as
[tex]z = \frac{ X - \mu}{\sigma} [/tex]
where, z --> z-score
X --> excepted value
--> standard deviations
--> population mean
Subsritute the known values in above formula, [tex]z = \frac{ 20 - 100}{15} [/tex]
= - 5.33
Now, probability value, P ( X < 20)
[tex]= P( \frac{X - \mu}{\sigma} < \frac{ 20 - 100}{15} )[/tex] = P( z < - 5.33 )
= 0.000
So, P( X < 20) = P( z< -5.33) = 0 . Also, required probability is P( X≥ 20) = 1 - 0
= 1
Hence, required value is one.
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Assume that
log 4 ~ 0.6021, log 5 ~ 0.6990, and log 6 – 0.7782
Use the properties of logarithms to evaluate
the expression. Do not use your calculator.
(Round your answer to 4 decimal places. If your
answer includes a negative, enter it like: -2)
log 16 = [a]
Answer:
Step-by-step explanation:
?
50 POINTS ASAP Triangle NMO is drawn with vertices N(−5, 2), M(−2, 1), O(−3 , 3). Determine the image vertices of N′M′O′ if the preimage is reflected over x = −2.
N′(5, −2), M′(2, 1), O′(3, 3)
N′(−2, 2), M′(1, 1), O′(0, 3)
N′(1, 2), M′(−2, 1), O′(−1, 3)
N′(−5, −2), M′(−2, −1), O′(−3, −3)
Answer:
To reflect a point over a vertical line x = c, where c is a constant, we can use the formula (2c - x, y).
Given the line of reflection x = -2, and the original points N(-5, 2), M(-2, 1), O(-3, 3), we can apply the formula as follows:
For N(-5, 2):
N' = (2(-2) - (-5), 2) = (1, 2)
For M(-2, 1):
M' = (2(-2) - (-2), 1) = (2, 1)
For O(-3, 3):
O' = (2(-2) - (-3), 3) = (1, 3)
So, the correct image vertices of N'M'O' after reflecting over x = -2 are N'(1, 2), M'(2, 1), O'(1, 3), which corresponds to the option:
N′(1, 2), M′(2, 1), O′(1, 3)
Please double-check to make sure if its right ;D
What are 2 questions that would require an interpretation of the table?
1. What is the relationship between education level and income in this population? 2. How does the popularity of different music genres vary across different age groups?
What do you mean by an interpretation?An interpretation is a way of understanding or explaining something. Interpretation often involves making connections between different elements or pieces of information to create a coherent understanding. It can also involve applying theories, frameworks, or methods to analyze or explain a particular subject. Ultimately, interpretations are subjective and may vary based on individual experiences and perspectives, making it important to consider multiple interpretations when analyzing a particular subject.
Here is two example questions:
1. What is the relationship between education level and income in this population?
Interpretation: To answer this question, you would need to examine a table that includes columns for education level and income, such as a cross-tabulation or a summary statistics table. You could look for patterns or trends in the data, such as higher incomes being associated with higher levels of education. You might also look for any outliers or anomalies in the data that could be affecting the overall relationship between education level and income.
2. How does the popularity of different music genres vary across different age groups?
Interpretation: To answer this question, you would need to examine a table that includes rows for different age groups and columns for different music genres, such as a frequency table or a bar chart. You could look for patterns or trends in the data, such as certain music genres being more popular among younger age groups or older age groups. You might also look for any outliers or anomalies in the data that could be affecting the overall relationship between age and music genre popularity.
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Solve the system of linear equations using any method
Answer:
(-1,-1)
Step-by-step explanation:
Rearange the equation
5y=6x+1 + 4y=-6x-10
9y = -9
y = -1
The only option with y as -1 is the second one
You can plug in -1 as x into any of the equations
5y=6x+1
-5 = -6+1
Hope this helps
Mary, Jane, Tom, Andy saved for 6 weeks like this:
-
M: 2, 4, 8, 16,
J: 10, 12, 14, 16,
T: 7, 13, 19, 25,
A:
3,6,9..
Work out how much each person saved so that you can put their names in
order of how much they saved, from smallest to largest amount.
Enter your code as a four-lettered "word"
write an equation that describes the line: a line has a slope of 0 and through the point (3,-4)
Answer:
Step-by-step explanation:
slope (m) =0
point =(x, y)
=(3, -4)
equation:
(y-y1) =m(x-x1)
{y-(-4)} =0
y+4 =0
y =-4
Explanation:
as we know, the equation to find the slope of a line is (y-y1) =m(x-x1). here, it is given that the slope of the line (m) is 0 and it passes through points 3 and -4. therefore, we can conclude that equation (y) would be 0 by calculating.
Find the area of the shaded regions:
Please help!!!
The areas of the composite figures are listed below:
Case 6: A = 134.126 ft²
Case 7: A = 212.5 cm²
Case 8: A = 957.305 m²
Case 9: A = 316.643 in²
How to determine the area of a composite figure
In this problem we find four cases of composite figure, that is, figures that are the result of adding and subtracting figures, whose area formulas are introduced below:
Semicircle
A = 0.5π · r²
Circle
A = π · r²
Triangle
A = 0.5 · b · h
Rectangle
A = b · h
Where:
A - Arear - Radiusb - Baseh - HeightNow we proceed to determine the area of each composite figure:
Case 6
A = (25 ft)² - π · (12.5 ft)²
A = 134.126 ft²
Case 7
A = (25 cm) · (14 cm) - 0.5 · (25 cm) · (11 cm)
A = 212.5 cm²
Case 8
A = 0.5π · [(21.6 m)² + (9 m)²] + 0.5 · (21.6 m) · (9 m)
A = 957.305 m²
Case 9
A = 0.5 · (20 in) · √[(25 in)² - (20 in)²] + (17 in) · √[(25 in)² - (20 in)²] - 0.5π · 0.25 · [(25 in)² - (20 in)²]
A = 316.643 in²
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Prove the following identity:
(cos 2x + cos 4x)/(cos 2x - cos 4x) = (cot 3x/tan x)
We have shown that the left-hand side of the equation is equal to the right-hand side of the equation, and the trigonometric identity is proved.
We have,
(cos 2x + cos 4x)/(cos 2x - cos 4x)
We can use the identity cos 2A = 1 - 2sin² A to rewrite cos 4x as:
cos 4x = 1 - 2sin² 2x
Then, substituting this into the expression, we get:
(cos 2x + 1 - 2sin² 2x)/(cos 2x - 1 + 2sin² 2x)
Using the identity sin² A = 1 - cos^2 A, we can rewrite sin² 2x as:
sin² 2x = 1 - cos² 2x
Substituting this into the expression, we get:
(cos 2x + 1 - 2(1 - cos² 2x))/(cos 2x - 1 + 2(1 - cos² 2x))
Simplifying the numerator and denominator separately, we get:
(2cos² 2x + 1)/(2cos² 2x - 1)
Using the identity cot A = 1/tan A, we can rewrite tan 3x as:
tan 3x = sin 3x/cos 3x
We can then use the identity sin 3A = 3sin A - 4sin^3 A and
cos 3A = 4cos³ A - 3cos A to rewrite sin 3x and cos 3x as:
sin 3x = 3sin x - 4sin³ x
cos 3x = 4cos³ x - 3cos x
Substituting these expressions into the right-hand side of the equation, we get:
cot 3x/tan x = (cos x/sin x)/(3sin x - 4sin³ x)/(cos x)
Simplifying by multiplying the numerator and denominator by the reciprocal of the fraction in the denominator, we get:
cot 3x/tan x = (cos x/sin x) * (cos x)/(3sin x - 4sin^3 x)
Multiplying the two fractions together, we get:
cot 3x/tan x = cos^2 x/(sin x(3 - 4sin^2 x))
Using the identity 1 - cos^2 A = sin^2 A, we can rewrite cos^2 x as:
cos^2 x = 1 - sin^2 x
Substituting this into the expression, we get:
cot 3x/tan x = (1 - sin^2 x)/(sin x(3 - 4sin^2 x))
Simplifying the numerator and denominator separately, we get:
cot 3x/tan x = (1/sin x) x (1 + sin x)/(3 - 4sin^2 x)
Using the identity 1 + sin A = 1/cos A, we can rewrite the expression as:
cot 3x/tan x = (1/sin x) x (1/cos x)/(3 - 4sin^2 x)
Finally, using the identity cot A = cos A/sin A, we can rewrite the expression as:
cot 3x/tan x = cos x/(sin x(3 - 4sin^2 x))
Therefore,
We have shown that the left-hand side of the equation is equal to the right-hand side of the equation, and the trigonometric identity is proved.
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Find the volume of the solid formed by rotating the space bounded by y = 2x², y=0, and x = 2 around the line y = 8
The volume of the solid is (256π/15) - (128π/3) cubic units, which is formed by rotating the space bounded by y = 2x², y=0, and x = 2 around the line y = 8.
To find the volume of the solid formed by rotating the space bounded by y = 2x², y=0, and x = 2 around the line y = 8, we can use the disk method.
First, we need to find the bounds of integration. Since x = 2 is the right boundary, we can integrate from 0 to 2.
The distance between the line y=8 and the curve y=2x² is 8 - 2x². Therefore, the radius of the disk at a given x-value is 8 - 2x².
The area of each disk is given by π(radius)². Therefore, the volume of the solid is given by the integral of π(radius)² dx from 0 to 2;
V = ∫₀² π(8 - 2x²)² dx
This integral can be evaluated by expanding the square and using the power rule of integration. After simplification, we get;
V = (256π/15) - (128π/3)
Therefore, the volume of the solid is (256π/15) - (128π/3) cubic units.
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Destiny sells newspapers at a
newsstand outside of her grocery
store. She normally charges $2.35 for
a newspaper. Today, Destiny has
lowered the price by $0.50. So far, she
has sold 8 newspapers at the new
price. How much money has Destiny
made so far from selling newspapers
at the new price?
Answer:
$14.80
Step-by-step explanation:
First you need to subtract .50 from 2.35, which is 1.85.
Next you need to multiply 1.85 times 8.
1.85 x 8 = 14.80
Therefore, she has made $14.80 since the new price.
Can somebody help, please?
When calculating the composite shape's area, the value of x that is needed is: x = 3.
What is rectangle?The internal angles of a rectangle, which has four sides, are all exactly 90 degrees. At each corner or vertex, the two sides come together at a straight angle. The rectangle differs from a square because its two opposite sides are of equal length.
What is area of rectangle?The area a two-dimensional form occupies in a plane is known as its area. It has a square unit of measurement. As a result, the rectangle's area is the space enclosed by its exterior lines. It is the same as the length times the width calculation.
The following is the rectangle's area formula:
Area = length*width
Thus;
Area of rectangle = 21x
Area of rectangle = 16 * 3x = 48x
Since the total area is 207 cm², then we have;
21x + 48x = 207
69x = 207
x = 207/69
x = 3
The value of x=3 is the answer which is required in question.
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