Suppose you look by eye at a star near the edge of a dusty interstellar cloud. The star will look dimmer than it would if it were outside the cloud.
Interstellar clouds are vast, dense regions of dust and gas that exist between stars. These clouds contain tiny solid particles of dust, which scatter and absorb light as it passes through them.
When we observe a star that is located near the edge of an interstellar cloud, the light from that star has to pass through a greater amount of dust before it reaches our eyes or telescopes. The dust scatters and absorbs some of the light, causing the star to appear dimmer than it would if it were outside the cloud.
This effect is similar to how the sun appears dimmer when it is viewed through a thick layer of clouds or fog. In both cases, the amount of light that reaches us is reduced due to the presence of a barrier that scatters and absorbs some of the light.
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Given two points (x1, y1) and (x2, y2) in the cartesian plane, show that the slope
m of a line is of the form
m =y2 − y1÷x2 − x1
assuming that x2≠ x1
therefore, we have shown that: [tex]m= (y_{2} -y_{1} )/(x_{2}-x_{1})[/tex] assuming that x2 ≠ x1.
What is slope?Slope refers to the measure of steepness of a line or a curve. In mathematics, slope is usually denoted by the letter "m" and is defined as the ratio of the change in the y-coordinate to the change in the x-coordinate between two points on a line.
The formula for calculating the slope between two points (x1, y1) and (x2, y2) on a line is:
[tex]m= (y_{2} -y_{1} )/(x_{2}-x_{1})[/tex]
by the question.
To finds the slope of a line passing through two points (x1, y1) and (x2, y2), we use the slope formula:
[tex]m= (y_{2} -y_{1} )/(x_{2}-x_{1})[/tex]
This formula represents the change in y divided by the change in x between the two points.
Now, assuming that x2 ≠ x1, we can simplify the formula as follows:
[tex]m= (y_{2} -y_{1} )/(x_{2}-x_{1})*(1/1)[/tex]
Multiplying the numerator and denominator by 1, which in this case is (x2 - x1) / (x2 - x1), we get:
[tex]m= (y_{2} -y_{1} )/(x_{2}-x_{1})*(x_{2}-x_{1})/(x_{2}-x_{1})[/tex]
Simplifying the numerator, we have:
[tex]m= (y_{2} -y_{1} )/(x_{2}-x_{1})/[(x_{2}-x_{1})*1][/tex]
The term (x2 - x1) cancels out, leaving us with:
[tex]m=(y_{2}-y_{1} /1[/tex]
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15.with regard to p-charts, the general recommendation for the number of samples to be taken when estimating p is .
With regard to p-charts, the general recommendation for the number of samples to be taken when estimating p is to take at least 20-25 samples.
The number of samples needed when estimating p using p-charts depends on the desired level of accuracy and confidence.
A general recommendation is to take at least 20-25 samples to obtain a reasonably accurate estimate of p. This recommendation is based on the central limit theorem, which states that the distribution of sample proportions approaches a normal distribution as the sample size increases.
With a sample size of 20-25, the estimate of p is likely to be within a reasonable margin of error and have a sufficient level of confidence.
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select all conditions for a discrete probability distribution also referred to as a probability distribution
The two requirements for a discrete probability distribution are that the probabilities assigned to each possible outcome must be between 0 and 1 inclusive, and the sum of probabilities of all possible outcomes must be equal to 1.
There are two main requirements for a discrete probability distribution,
The probabilities assigned to each possible outcome must be between 0 and 1 inclusive. That is, the probability of each outcome must be a non-negative number, and the sum of probabilities of all possible outcomes must be equal to 1.
Each possible outcome must be mutually exclusive. That is, only one of the possible outcomes can occur on any given trial of the random experiment.
These two requirements ensure that the probabilities assigned to each outcome are valid and that the total probability space is complete, allowing us to make valid inferences and predictions about the outcomes of random events.
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I have solved the question in general, as the given question is incomplete.
The complete question is:
What are the two requirements for a discrete probability distribution?
multiple linear regression with coefficient standard deviation and mean to get new multiple regression
Multiple linear regression is a statistical technique used to model the relationship between a dependent variable and multiple independent variables.
In multiple linear regression,
The dependent variable is modeled as a linear function of several independent variables.
Regression coefficient that quantifies the strength and direction of the relationship between independent and dependent variable.
Coefficient standard deviation refers to the standard deviation of the estimated regression coefficients in multiple linear regression.
Provides a measure of the variability of the estimated coefficients and can be used to assess the precision of the estimates.
New multiple regression model,
Collect data on the dependent variable and multiple independent variables of interest.
Perform a multiple linear regression analysis, which involves fitting a linear equation to the data using the method of least squares.
Estimates the regression coefficients and their standard deviations.
Used to assess the significance of each independent variable and the overall fit of the model.
Therefore, multiple regression model consider adding or removing independent variables, transforming the data, or machine learning algorithms.
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The above question is incomplete , the complete question is:
What is multiple linear regression with coefficient standard deviation and mean to get new multiple regression ?
List all the factors of
36
We can see that 36 is a composite number.
We also know that [tex]36=1\times36[/tex], [tex]3\times15[/tex], or [tex]4\times9[/tex]. All of these numbers are prime numbers so they are all factors of 36.
Thus we see that the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.
The table below shows the number of painted pebbles of Claire and Laura. If Greg chooses a pebble at random from the box 75 times, replacing the pebble each time, how many times should he expect to choose a yellow pebble?
A) 11
B) 33
C) 32
D) 22
So, out of 75 pulls, we would anticipate choosing a yellow pebble 25 times.
what is probability ?The measurement and study of random events are the focus of the mathematic branch known as probability. It entails calculating the probability of an event happening, with a scale from 0 (impossible) to 1. (certain). The fundamental meaning of chance is: Number of favourable outcomes minus the total number of outcomes equals the probability of an occurrence.
given
There are a total of the following painted stones in the box:
Total painted stones equals Claire's painted stones plus Laura's painted stones.
Total number of decorated stones: 35 + 40 = 75
Number of yellow stones equals the sum of Claire's and Laura's yellow pebbles.
5 Plus 20 = 25 yellow pebbles total.
As a result, the likelihood of getting a yellow pebble on any given draw is:
Number of yellow pebbles / total painted pebbles represents the likelihood of painting a yellow pebble.
25/75 is the likelihood of getting a yellow pebble.
The likelihood of drawing a yellow pebble on each draw is the same because we are drawing with substitution.
In 75 pulls, there should be an average of:
Number of draws times the likelihood of getting a yellow pebble yields the expected number of yellow pebbles.
Number of yellow pebbles anticipated Equals 75 * (1/3)
Expected quantity of golden stones: 25
So, out of 75 pulls, we would anticipate choosing a yellow pebble 25 times.
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Use matrix inversion to solve the given system of linear equations. (You previously solved this system using row reduction.)
4x + y = −4
4x − 3y = −4
The solution using matrix inversion is x = -1 and y = 0.
To solve this system of equations using matrix inversion, we first need to write the system in matrix form:
[tex]\left[\begin{array}{cc}4&1\\4&-3\end{array}\right] \left[\begin{array}{c}x\\y\end{array}\right] = \left[\begin{array}{c}-4\\-4\end{array}\right][/tex]
Next, we need to invert the coefficient matrix on the left-hand side of the equation by finding its inverse. The inverse of a 2x2 matrix is given by:
[tex]\left[\begin{array}{cc}a&b\\c&d\end{array}\right]^{-1} =\frac{1}{(ad - bc)} \left[\begin{array}{cc}-d&b\\c&-a\end{array}\right][/tex]
Using this formula, we can find the inverse of the coefficient matrix [4 1; 4 -3]:
[tex]\left[\begin{array}{cc}4&1\\4&-3\end{array}\right]^{-1} =\frac{1}{(-12 - 4)} \left[\begin{array}{cc}3&1\\4&-4\end{array}\right]=\frac{-1}{16} \left[\begin{array}{cc}3&1\\4&-4\end{array}\right][/tex]
Now we can solve for [x y] by multiplying both sides of the equation by the inverse of the coefficient matrix:
[tex]\left[\begin{array}{cc}4&1\\4&-3\end{array}\right]^{-1}\left[\begin{array}{cc}4&1\\4&-3\end{array}\right] \left[\begin{array}{c}x\\y\end{array}\right] = \frac{-1}{16} \left[\begin{array}{cc}3&1\\4&-4\end{array}\right]\left[\begin{array}{c}-4\\-4\end{array}\right][/tex]
[tex]\left[\begin{array}{cc}1&0\\0&1\end{array}\right] \left[\begin{array}{c}x\\y\end{array}\right] = \frac{-1}{16} \left[\begin{array}{c}-12-4\\-16+16\end{array}\right] = \left[\begin{array}{c}-1\\0\end{array}\right][/tex]
Therefore, x = -1 and y = 0, which is the same solution we obtained using row reduction.
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What is the gravitational force on a 35.0 kg object standing on the Earth’s surface?
Answer:341.97 N
Step-by-step explanation:Answer and Explanation: The gravitational force on the 35 kg mass at the surface of the Earth is 341.97 N while at the surface of the Moon the 35 kg mass feels only 57.56 N. The gravitational force on the surface of the Earth is 5.94 times greater than that of the surface of the moon.
The weight of a miniature Tootsie roll is normally distributed with a mean of 3.30 grams and standard deviation of .13 gram
We can estimate that the middle 95% of all miniature Tootsie rolls will fall within the range of 3.04 grams to 3.56 grams for standard deviation of 0.13 gram.
What is a normal distribution?A normal distribution is a symmetric, bell-shaped continuous probability distribution that is defined by its mean and standard deviation. The majority of the data in a normal distribution is located close to the mean, and the number of data points decreases as you deviate from the mean in either direction. Because many real-world events, like human height or test scores, have a tendency to follow a normal distribution, the normal distribution is frequently utilised in statistics. A helpful technique for determining the range of values within a normal distribution based on the mean and standard deviation is the empirical rule, commonly known as the 68-95-99.7 rule.
Given that, the mean of 3.30 grams and standard deviation of 0.13 gram.
Using the empirical formula the range that falls in 95% is associated to two standard deviations.
Mean + 2 standard deviations = 3.30 + 2(0.13) = 3.56 grams
Mean - 2 standard deviations = 3.30 - 2(0.13) = 3.04 grams
Hence, we can estimate that the middle 95% of all miniature Tootsie rolls will fall within the range of 3.04 grams to 3.56 grams.
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100 POINTS) a linear function passes through (3,10) and 6,8 what is the slope,m, of the function?
Check the explanation.
Step-by-step explanation:
As the graph of a linear function f passes through the point (-2,-10) and has a slope of 5/2.
As the slop-intercept form is given by:
where m is the slope and b is the y-intercept.
substituting the values (-2, -10) and m = 5/2 in the slop-intercept form to determine y-intercept.
And the equation of the line in the slope-intercept form will be:
putting b = -5 and slope = m = 5/2
Determining the zero of function.
As we know that the real zero of a function is the x‐intercept(s) of the graph of the function.
so let us determine the value of x (zero of function) by setting y = 0.
Therefore, the zeros of the function will be:
x = 2
Answer:
To find the slope, m, of a linear function that passes through two points, (x1, y1) and (x2, y2), we can use the following formula:
m = (y2 - y1) / (x2 - x1)
In this case, the two points are (3, 10) and (6, 8), so we have:
x1 = 3
y1 = 10
x2 = 6
y2 = 8
Substituting these values into the formula, we get:
m = (8 - 10) / (6 - 3) = -2 / 3
Therefore, the slope of the linear function is -2/3.
Use the gradient to find the directional derivative of the function at P in the direction of PQ. f(x, y) = 3x2 - y2 + 4, P(1, 5), 2(4,2)
The directional derivative of f(x,y) at point P(1,5) in the direction of PQ is -2√2.
Find the directional derivative of the function f(x,y) = 3x² - y² + 4 at point P(1,5) in the direction of PQ, where P(1,5) as well as Q(4,2), we need to first calculate the gradient of f(x,y) at point P.
The gradient of f(x,y) at P is:
∇f(x,y) = [∂f/∂x, ∂f/∂y] = [6x, -2y]
Evaluating this at point P(1,5), we get:
∇f(1,5) = [6(1), -2(5)] = [6, -10]
Now, we need to find the unit vector in the direction of PQ. This can be calculated as follows:
u = PQ/|PQ|
where PQ = Q - P = [4 - 1, 2 - 5] = [3, -3] and |PQ| = √(3² + (-3)²) = √18 = 3√2
So, u = PQ/|PQ| = [3/3√2, -3/3√2] = [1/√2, -1/√2]
The directional derivative of f(x,y) at P in the direction of PQ is then given by:
D_u f(P) = ∇f(P) · u
where · represents the dot product.
Substituting the values we obtained earlier, wehave:
D_u f(P) = [6, -10] · [1/√2, -1/√2]
D_u f(P) = (6/√2) + (-10/√2)
D_u f(P) = -2√2
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What is the period of f(x)=secx?
Enter your answer in the box.
period of f(x)=secx:
Therefore , the solution of the given problem of function comes out to be f(x) = sec(x) has a period of 2. 2 is the answer.
What is function?The midterm test questions will cover all of the topics, including actual as well as fictitious locations and arithmetic variable design. a diagram showing the relationships between different elements that cooperate to create the same result. A service is composed of numerous distinctive components that cooperate to create distinctive results for each input. Every mailbox has a particular spot that might be used as a haven.
Here,
=> F(x) = sec(x) has a 2 phase.
Because the secant function is periodic, its values recur after a predetermined amount of time.
This interval's length is equal to the secant function's duration.
The formula for the secant function is
=> sec(x) = 1/cos.(x).
The cosine function repeats its values every 2 units of x, which is known as its period.
Consequently, the secant function has a period of 2 as well.
Therefore, f(x) = sec(x) has a period of 2. 2 is the answer.
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Calculate the area of the shaded segments in the following diagrams. (a) 12 cm 40° (b) 58° 16 cm
(a) 12 cm 40° : Area of shaded segments = 301.44 sq. cm.
(b) 58° 16 cm : Area of shaded segments = 777.04 sq. cm.
Explain about the sector of circle?Two radii that meet at the center to form a sector define a circle. The sector is the portion of the circle created by these two radii. Knowing a circle's central angle calculation and radius measurement are both crucial for solving circle-related difficulties.
Area of sector of circle = Ф/360 * πr²
π = 3.14
r is the radius
Ф is the angle subtended.
(a) 12 cm 40°
Area of shaded segments = 40/60 * 3.14* 12²
Area of shaded segments = 40/60 * 452.16
Area of shaded segments = 301.44 sq. cm.
(b) 58° 16 cm
Area of shaded segments = 58/60 * 3.14* 16²
Area of shaded segments = 58/60 * 803.84
Area of shaded segments = 777.04 sq. cm.
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The diagram for the question is attached.
let x be a normal random variable with a mean of 5 and a standard deviation of 10. find p(-20 < x < 15).
The probability of -20 < x < 15, where x be a normal random variable with a mean of 5 and a standard deviation of 10 is approximately 0.8351.
To solve this problem, we need to find the area under the normal distribution curve between -20 and 15, with a mean of 5 and a standard deviation of 10.
We can standardize the normal distribution by subtracting the mean and dividing by the standard deviation, which gives us the standard normal distribution with a mean of 0 and a standard deviation of 1.
So, we can first calculate the z-scores for -20 and 15:
z1 = (-20 - 5) / 10 = -2.5
z2 = (15 - 5) / 10 = 1
Next, we use a standard normal distribution table or calculator to find the probabilities associated with these z-scores:
P(z < -2.5) = 0.0062
P(z < 1) = 0.8413
To find the probability of -20 < x < 15, we subtract the probability associated with the lower z-score from the probability associated with the higher z-score:
P(-20 < x < 15) = P(-2.5 < z < 1) = P(z < 1) - P(z < -2.5)
P(-20 < x < 15) = 0.8413 - 0.0062 = 0.8351
Therefore, the probability of -20 < x < 15 is approximately 0.8351.
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which of the following correctly relates the measures of the diameter (d) and radius (r) of a circle
The equation which correctly relates the measure of diameter and radius of a circle is (c) r = d/2.
The Diameter (d) of a circle is defined as the distance across the circle through its center. The radius (r) of a circle is defined as the distance from the center of the circle to any point on the circle.
We know that the radius of the circle is half of diameter, because it extends from the center to the edge of the circle, while the diameter extends all the way across the circle.
So, we can express the relationship between d and r as:
⇒ d = 2r
To solve for r, we can divide both sides by 2:
We get,
⇒ r = d/2
Therefore, The correct equation is Option (c) r = d/2.
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The given question is incomplete, the complete question is
Which of the following correctly relates the measures of the diameter (d) and radius (r) of a circle?
(a) d = r/2
(b) r = 2d
(c) r = d/2
(d) d = 2/r
If a data set has an even number of observations, the median a. cannot be determined b. is the average value of the two middle items c. must be equal to the mean d. is the average value of the two middle items when all items are arranged in ascending order
If a data set has an even number of observations, the median is the middle value of the two middle values when all values are placed in ascending order. Thus, option d is correct.
The median is the central value in a set of groups of data. The data should be organized and ordered from smallest value to biggest value. To find the middle value, we should divide the number of observations by two.
To determine the median, we must determine the Mean. Then, we need to calculate out how many observations there are in the data set. If there are an odd number of observations, the median cannot be calculated as a regular calculation.
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A survey of 64 informed voters revealed the following information.
45 believe that Elvis is still alive.
49 believe that they have been abducted by rebels.
42 believe both of these things.
a.) Create a Venn diagram to model the information.
b.) How many believe neither of these things?
c.) How many believe Elvis is still alive but do not believe that they have been abducted by rebels?
3 individuals believe Elvis is still alive but do not believe that they have been abducted by rebels.
What is Venn diagram?A Venn diagram is a graphical representation of set theory that illustrates the relationships between sets. It consists of a rectangle or a circle representing the universal set and circles or ovals representing the subsets.
According to question:a) Here's a Venn diagram to model the information:
Label the left circle "Elvis is still alive" and the right circle "Abducted by rebels." The overlapping region of the circles represents the individuals who believe both.
b) To find out how many believe neither of these things, we need to subtract the number of individuals in the overlapping region from the total number of individuals surveyed.
Total surveyed = 64
Number who believe both = 42
Therefore, the number who believe neither is:
64 - 42 = 22
So, 22 individuals believe neither of these things.
c) To find out how many believe Elvis is still alive but do not believe that they have been abducted by rebels, we need to subtract the number of individuals in the overlapping region from the number who believe Elvis is still alive.
Number who believe Elvis is still alive = 45
Number who believe both = 42
Therefore, the number who believe Elvis is still alive but do not believe that they have been abducted by rebels is:
45 - 42 = 3
So, 3 individuals believe Elvis is still alive but do not believe that they have been abducted by rebels.
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Please answer Full question
(1) 4y-7z is a binomial.
(2) 8-xy² is a binomial.
(3) ab-a-b can be written as ab - (a + b) which is a binomial.
(4) z²-3z+8 is a trinomial.
What are monomials, binomials and trinomials?In algebra, monomials, binomials, and trinomials are expressions that contain one, two, and three terms, respectively.
A monomial is an algebraic expression with only one term. A monomial can be a number, a variable, or a product of numbers and variables.
A binomial is an algebraic expression with two terms that are connected by a plus or minus sign. For example, 2x + 3y and 4a - 5b are both binomials.
A trinomial is an algebraic expression with three terms that are connected by plus or minus signs.
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Classify into monomials, binomials and trinomials.
(1) 4y-7z
(1) 8-xy²
(v) ab-a-b
(ix) z2-3z+8
Construct an example of a function that satisfies the following conditions:
a) Its domain and range are both all real numbers except 5.
b) Its domain is all positive numbers greater than 1, including 1.
c) Its domain is all positive numbers greater than 1, but not including 1.
Answer:
f(x) = (x^2 - 25) / (x - 5)
Step-by-step explanation:
Note that this function is undefined at x=5, which satisfies condition (a). Also, the function is defined for all other real numbers, which satisfies the domain and range requirement of (a).
For condition (b), note that the function is defined for all positive numbers greater than 1, including 1, since the denominator (x-5) will be positive for these values of x.
For condition (c), note that the function is undefined at x=1, since the denominator (x-5) will be negative for x slightly less than 1. Therefore, the function is defined for all positive numbers greater than 1, but not including 1.
Uri paid a landscaping company to mow his lawn. The company charged $74 for the service plus
5% tax. After tax, Uri also included a 10% tip with his payment. How much did he pay in all?
Answer:
$85.47
Step-by-step explanation:
Before tax and before tip: $74
Tax is 5%.
5% of $74 = 0.05 × $74 = $3.70
Cost including tax (but not tip): $74 + $3.70 = $77.70
After tax but before tip: $77.70
Tip is 10%.
10% of $77.70 = 0.1 × $77.70 = $7.77
Total price after tax & tip:
$77.70 + $7.77 = $85.47
SELECT THE CORRECT ANSWER:
Use the power of a product property to answer the question.
Which expression equals (7Y)^1/3
?
Answer:
7^1/3 * y^1/3
Step-by-step explanation:
(7y) ^ 1/3
We know that (ab) ^c = a^c * b^c
7^1/3 * y^1/3
Find the value of r so the line that passes through the pair of points has the given slope. (3, 5), (-3, r), m = 3/4
Answer:
the value of r that makes the slope of the line passing through (3, 5) and (-3, r) equal to 3/4 is r = 1/2
Step-by-step explanation:
We can use the formula for the slope of a line passing through two points, which is:
m = (y2 - y1)/(x2 - x1)
where (x1, y1) and (x2, y2) are the coordinates of the two points.
In this case, we have two points (3, 5) and (-3, r), and we know that the slope m is 3/4. So we can write:
3/4 = (r - 5)/(-3 - 3)
Simplifying this equation, we get:
3/4 = (r - 5)/(-6)
Multiplying both sides by -6, we get:
-9/2 = r - 5
Adding 5 to both sides, we get:
r = -9/2 + 5
Simplifying, we get:
r = 1/2
y is directly proportional to x ,if x=20 when y=160,then what is the value of x When y=3.2
Answer:
when y=3.2, the value of x is 0.4.
Step-by-step explanation:
If y is directly proportional to x, it means that y = kx, where k is the constant of proportionality. To find the value of k, we can use the given information that when x=20, y=160:
y = kx
160 = k(20)
k = 8
Now that we have the value of k, we can use it to find the value of x when y=3.2:
y = kx
3.2 = 8x
x = 0.4
Therefore, when y=3.2, the value of x is 0.4.
Karina is making a quilt and she has determined she needs 420 square inches of green fabric and 688 square
inches of burgundy. How many square yards of each material will she need? Round your answers up to the
nearest quarter yard.
The green fabric:
square yards
The burgundy fabric:
How many total yards of fabric will she have to buy?
square yards
square yards
1. The total yards of each fabric that Karina will buy to make a quilt is as follows:
a) Green Fabric = 12 square yards
b) Burgundy Fabric = 19 square yards
2. The total yards of fabric she will buy is 31 square yards.
How are the total determined?The total yards of fabric can be determined by unit conversion using division operation.
Given that 36 inches = 1 yard, the square inches of fabric are converted to square yards by dividing the total by 36.
The total number of green fabric Karina requires = 420 square inches
= 12 square yards (420/36)
The total number of burgundy fabric Karina requires = 688 square inches
= 19 square yards (688/36)
The total number of fabric (green and burgundy) = 1,108 square inches (420 + 688)
36 inches = 1 yard
1,108 inches = 30.78 square yards (1,108/36)
= 31 square yards or (12 + 19)
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Quinn had a 300-centimeter ribbon that he will cut into smaller pieces for decorations. Which ribbon lengths could he cut from the original ribbon?
Solve the problem, show your work, and submit.
You will have 2 answers.
A. 1.8 meters and 110 centimeters
B. 200 centimeters and 2 meters
C. 0.63 meters and 230 centimeters
D. 17 centimeters and 2.9 meters
The answer is letter A and letter F.
The question is asking if any of these choices are less than 300 centimeters. To find this out, first, you have to convert the meters to centimeters, to make it easier. (1 centimeter is equal to 0.01 meters) Multiply the meters by 100. For example 1.8m x 100 = 180cm. Do the rest for the other meters and then add them to the centimeters. Example: 180cm + 110cm = 290cm, which is less than 300cm.
Nike Air Forces were on sale for 12% off at Footlocker. If the Air Forces cost $98.00, what will be the sale price?
Answer:
86.24
Step-by-step explanation:
If you divide 12 ÷ 98 you will get 11.76
Now with that information you will subtract 11.76 to 98 and will get a total of
86.24
Hope this helped :)
the smaller leg of a right triangle is 14cm smaller than the larger leg the hypotenuse is 2cm larger than the larger leg find each side of the triangle
Answer:
Step-by-step explanation:
Let's use the Pythagorean theorem to solve this problem.
Let x be the length of the larger leg of the triangle. Then, the length of the smaller leg is x - 14 cm. The length of the hypotenuse is x + 2 cm.
Using the Pythagorean theorem, we have:
(x - 14)^2 + x^2 = (x + 2)^2
Simplifying and solving for x:
x^2 - 28x + 196 + x^2 = x^2 + 4x + 4
2x^2 - 32x + 192 = 0
Dividing both sides by 2:
x^2 - 16x + 96 = 0
Factorizing:
(x - 8)(x - 12) = 0
Therefore, x = 8 or x = 12.
If x = 8, then the smaller leg is 8 - 14 = -6 cm, which is not possible.
If x = 12, then the smaller leg is 12 - 14 = -2 cm, which is also not possible.
Therefore, there are no real solutions to this problem.
Line AB contains point A(1, 2) and point B (−2, −1). Find the coordinates of A′ and B′ after a dilation with a scale factor of 5 with a center point of dilation at the origin
The coordinates of A' and B' after a dilation with a scale factor of 5 and a center point of dilation at the origin are A'(5, 10) and B'(-10, -5), respectively.
How to find dilated coordinate of A and B?To find the coordinates of the points A' and B' after a dilation with a scale factor of 5 and a center point of dilation at the origin, we can use the following formula:
[tex]$$(x', y') = (5(x - 0), 5(y - 0)) = (5x, 5y)$$[/tex]
where (x, y) are the original coordinates of the point, and (x', y') are the new coordinates after the dilation.
For point A(1, 2), the new coordinates A' are:
[tex]$$(x_A', y_A') = (5(1), 5(2)) = (5, 10)$$[/tex]
Therefore, the coordinates of point A' are (5, 10).
For point B(-2, -1), the new coordinates B' are:
[tex]$$(x_B', y_B') = (5(-2), 5(-1)) = (-10, -5)$$[/tex]
Therefore, the coordinates of point B' are (-10, -5).
Therefore, the coordinates of A' and B' after a dilation with a scale factor of 5 and a center point of dilation at the origin are A'(5, 10) and B'(-10, -5), respectively.
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find the radius of a circle whose area is 28½cm²
Answer: 3 cm
Step-by-step explanation:
The formula for the area of a circle is A = πr², where A is the area and r is the radius. We are given that the area of the circle is 28½ cm².
So, 28½ = πr²
We need to solve for r. Dividing both sides by π, we get:
r² = 28½/π
r² = 9
Taking the square root of both sides, we get:
r = 3√1 = 3 cm
Therefore, the radius of the circle is 3 cm.
let v be the set of all positive real numbers. determine whether v is a vector space with the following operations. x y
V is not a vector space, since at least one of the vector space axioms fails. Specifically, the axiom of closure under addition fails, since for any positive real numbers x and y, their sum xy may not be positive.
V is not a vector space, since some of the vector space axioms fail with the given operations.
Closure under addition fails For example, taking x=2 and y=3, we have xy = 6, but xy is not a positive real number, so xy is not in V.
Distributivity of scalar multiplication over vector addition fails For example, taking x=2, y=3, and c=-1, we have c(x+y) = cxy = -6, but cx + cy = -2 -3 = -5, which is not in V.
Other vector space axioms, such as associativity of addition, existence of additive identity and inverse, and compatibility of scalar multiplication with field multiplication, are still satisfied with the given operations.
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_____The given question is incomplete, the complete question is given below:
Let V be the set of all positive real numbers. Determine whether V is a vector space with the following operations. x + y = xy Addition cx = x Scalar multiplication If it is, then verify each vector space axiom; if it is not, then state all vector space axioms that fail. STEP 1: Check each of the 10 axioms. (1) U + v is in V. This axiom holds. This axiom fails. (2) U + V = V + u This axiom holds. This axiom fails. (3) U+ (v + w) = (u + v) + w This axiom holds. This axiom fails. STEP 2: Use your results from Step 1 to decide whether V is a vector space. O V is a vector space. O Vis not a vector space.