Answer:
P(multiple of 3) = 1/3 (fraction)
P(multiple of 3) = (1/3) * 100 = 33.33% (percentage rounded to the nearest percent)
Step-by-step explanation:
There are two numbers on a cube labeled 1 through 6 that are multiples of 3: 3 and 6.
The total number of possible outcomes is 6 (since there are 6 sides on the cube). So, the probability of rolling a multiple of 3 is calculated as follows:
P(multiple of 3) = (number of favorable outcomes) / (total number of possible outcomes)
P(multiple of 3) = 2/6
P(multiple of 3) = 1/3
To express this probability as a percentage rounded to the nearest percent, multiply the fraction by 100:
P(multiple of 3) = (1/3) * 100 = 33.33%
Rounded to the nearest percent, the probability of rolling a multiple of 3 on a number cube labeled 1 through 6 is 33%.
Answer:
Step-by-step explanation:
17
suppose that eight workers can manufacture 70 radios per day and that nine workers can manufacture 90 radios per day. if radios can be sold for $20 each, the value of marginal product of the ninth worker is
Answer: The value of the marginal product of the ninth worker is $400
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the standard deviation of a sample mean a. increases as the sample size increases b. decreases as the sample size increases c. remains constant with increase in sample size d. none of the above
The standard deviation of a sample mean is a measure of how much the sample mean varies from the true population mean.
As the sample size increases, the standard deviation of the sample mean decreases. This is because larger samples are more representative of the population and are less likely to contain extreme values that can skew the sample mean.
The decrease in standard deviation is due to the central limit theorem, which states that as sample size increases, the distribution of sample means approaches a normal distribution with a mean equal to the true population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size.
This means that the standard deviation of the sample mean decreases as the square root of the sample size increases. Therefore, the correct answer to the question is b) decreases as the sample size increases. This is an important concept to understand in statistical analysis, as it affects the precision and accuracy of estimates made from sample data.
A larger sample size generally leads to more reliable and accurate estimates of population parameters. The standard deviation of a sample mean decreases as the sample size increases. As the sample size gets larger, the sample mean becomes a more accurate estimate of the population mean,
And the spread of the sample means around the population mean becomes smaller. This decrease in variability is captured by the standard deviation of the sample mean, which is calculated as the population standard deviation divided by the square root of the sample size. So, the correct answer is option b.
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A rectangle has one side of 11 cm. How fast is the area of the rectangle changing at the instant when the other side is 26 cm and increasing at a rate of 3 cm per minute? Include units. At that instant, the area of the rectangle is changing at a rate of Choose one
Step-by-step explanation:
Let's start by setting up the problem. We know that one side of the rectangle is 11 cm, and we'll call the other side x (which is increasing at a rate of 3 cm per minute). The formula for the area of a rectangle is A = l x w, where l is the length and w is the width. In this case, the length is 11 cm and the width is x. So:
A = 11x
To find how fast the area is changing, we need to take the derivative with respect to time (t):
dA/dt = 11(dx/dt)
Now we can plug in the values we know:
x = 26 cm
dx/dt = 3 cm/min
So:
dA/dt = 11(3) = 33 cm^2/min
Therefore, the area of the rectangle is changing at a rate of 33 cm^2/min at the instant when the other side is 26 cm and increasing at a rate of 3 cm per minute.
Create an equation that translates a square root graph up 3, right 2, and reflected over the x-axis?
Answer:
[tex]y=-\sqrt{x-2}+3[/tex]
Step-by-step explanation:
Step 1: In order to reflect a square root function over the x axis, you have to multiply the function by -1 and you place the negative outside the square root.
Step 2; In order to shift a square root function "n" units to the right, you have to subtract n units from the x inside the square root itself.
Step 3: In order to move a square root function "n" units up, you have to add n units to the function outside the square root
Find the distance between the points (–8,5) and (4,0).
ssume that a sample is used to estimate a population mean . find the 95% confidence interval for a sample of size 49 with a mean of 21.5 and a population standard deviation of 19.4. enter your answer as an open-interval
The 95% confidence interval for the population mean is (16.426, 26.574
To find the 95% confidence interval, we can use the formula:
Confidence interval = sample mean ± (z-score)*(population standard deviation/√n)
where the z-score for a 95% confidence level is 1.96.
Plugging in the values given in the question, we get:
Confidence interval = 21.5 ± (1.96)*(19.4/√49)
Simplifying this expression, we get:
Confidence interval = 21.5 ± 5.074
Therefore, the 95% confidence interval for the population mean is (16.426, 26.574) as an open-interval.
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Find the volume of the composite solid. Round your answer to the nearest hundredth. A. 22. 05mm B. 22. 19mm C. 22. 53mm D. 22. 54mm
The volume of the composite solid, can be found to be C. 22. 53mm ³.
How to find the volume ?Seeing as this is a composite solid, we can find the volume by finding the volumes of the constituent shapes which are a triangular prism and a cuboid.
Volume of triangular prism :
= 1 / 2 x base x h x l
= 1 / 2 x 2. 1 x 2. 4 x 2. 9
= 7. 308 mm ³
The volume of the cuboid would be:
= Length x height x width
= 2. 1 x 2. 5 x 2. 9
= 15. 225 mm ³
The volume is therefore:
= 15. 225 + 7. 308
= 22. 53 mm ³
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What is the area of this figure?
4 mm
2 mm
3 mm
9 mm
2 mm
11 mm
Answer:
A = (1/2)(4)(5) + 7(7) + 3(2) = 65 mm^2
a person leaves home and walks 4 miles west, then 3 miles southwest.how far from home is she?$6.478469167$correct milesin what direction must she walk to head directly home?$70.8864353$incorrect 19.113564701735 degrees north of east
The person needs to walk about 6.478 miles in a direction of approximately 19.11° north of east to head directly home.
To find the distance and direction the person needs to walk to head directly home, we can use the Pythagorean theorem and some trigonometry.
1. First, break the southwest walk into west and south components. Since it's a 3-mile walk at a 45-degree angle, the west and south components are equal: 3 * cos(45°) = 3 * 0.7071 ≈ 2.121 miles for both west and south components.
2. Calculate the total west and south distances. The total west distance is the sum of the initial westward walk and the west component of the southwest walk: 4 miles + 2.121 miles ≈ 6.121 miles. The total south distance is just the south component of the southwest walk: 2.121 miles.
3. Use the Pythagorean theorem to find the straight-line distance back home: √((6.121 miles)^2 + (2.121 miles)^2) ≈ 6.478 miles, which is the correct distance.
4. Determine the angle between the eastward direction and the straight-line path back home. Use the inverse tangent function: angle = atan(south distance / west distance) = atan(2.121 miles / 6.121 miles) ≈ 19.11° north of east, which is the correct direction.
So, the person needs to walk about 6.478 miles in a direction of approximately 19.11° north of east to head directly home.
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chloe lives in an area where she runs the air conditioner during the 4 hottest months of the year. the air conditioner runs for 7 hours per day at a cost of 21 cents per hour. how much does she spend per year on utility bills for the air conditioner?(assume there are 30 days in each month)
Chloe runs her air conditioner for a total of 4 months, or 120 days. Chloe spends $176.40 per year on utility bills for the air conditioner during the four hottest months.
The reason for this expense is the need for cooling during these months, and the cost associated with running the air conditioner for 7 hours per day at a rate of $0.21 per hour.
To calculate the annual cost of running Chloe's air conditioner, we need to consider the number of hours per day, the cost per hour, and the number of days the air conditioner is used.
1. First, let's determine the total number of hours the air conditioner runs during the four hottest months:
7 hours/day * 30 days/month * 4 months = 840 hours
2. Now, we can calculate the total cost by multiplying the number of hours by the cost per hour:
840 hours * $0.21/hour = $176.40
At 7 hours per day, that's a total of 840 hours of use. At a cost of 21 cents per hour, Chloe spends 0.21 x 840 = $176.40 per year on utility bills for her air conditioner.
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1. Determine if the function is an exponential function.
f(x) = ab^z
The function f(x) = ab^x is an exponential function.
Determining if the function is an exponential function.From the question, we have the following parameters that can be used in our computation:
f(x) = ab^x
As a general rule, and exponential functtion is represented as
f(x) = ab^x
Where
Initial value = aCommon factor = bx and f(x) are the variables and the functionsHence, the function is an exponential function
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use integers to represent quantities in real-world situations and explain the meaning of zero in each situation. a. 15 yard-gain b. loss of 2 hours
To represent a 15 yard-gain in football, we can use the integer +15. This means that the team has moved forward 15 yards from their starting point. The meaning of zero in this situation would be that the team did not gain or lose any yards and stayed at their starting point.
Step-by-step explanation:
1. You start at 0 yards (the meaning of zero in this context).
2. You gain 15 yards, which is a positive value (+15).
In both cases, zero serves as the starting point from which gains or losses are measured using positive or negative integers.
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Difference between definite and indefinite integral.
The definite and indefinite integral are both important concepts in calculus. The indefinite integral represents the antiderivative of a function,
which means finding a function whose derivative is the original function. It is often denoted by the symbol "∫" followed by the function to be integrated. The indefinite integral is not unique, as it can have an arbitrary constant of integration.
On the other hand, the definite integral is the signed area between the curve and the x-axis over a specified interval. It is often denoted by the symbol "∫" with limits of integration at the upper and lower bounds of the interval. The definite integral has a definite value, as it is a single number obtained by evaluating the antiderivative at the limits of integration.
In summary, the indefinite integral is a family of functions while the definite integral is a single number. The indefinite integral is used to solve differential equations while the definite integral is used to calculate the area under a curve. Understanding the difference between these two concepts is crucial for success in calculus.
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Is it true that If B is formed by adding to one row of A a linear combination of the other rows, then detB = detA.
Yes, it is true that if matrix B is formed by adding to one row of matrix A a linear combination of the other rows, then the determinant of B is equal to the determinant of A.
Proven using elementary row operations are operations that can be performed on the rows of a matrix without changing its determinant.
One such operation is adding a multiple of one row to another row.
A linear combination of the other rows to a particular row of A to form B, we can express this operation as a matrix multiplication:
B = P × A
P is a matrix that represents the elementary row operation.
P is an identity matrix with the (i, j)-th entry equal to 1 if i = j or equal to a constant c if i is the row being modified and j is the row being added to, and 0 otherwise.
Since P is an elementary matrix, it has determinant equal to 1 or -1, depending on whether an odd or even number of row swaps were performed.
Therefore,
det(B) = det(P × A) = det(P) × det(A) = ±det(A)
det(P) is either 1 or -1.
P is formed by adding a linear combination of the other rows to a particular row, we can easily see that P is a triangular matrix with 1's on the diagonal, and thus its determinant is equal to 1.
Therefore, we have
det(B) = det(A)
as claimed.
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if katie scored a 93 on a test and her calculated z score was 2.14, what does that mean
A z-score of 2.14 indicates that Katie's score on the test is quite high and unusual, and places her in the top 2% of the scores in the population.
Katie scored a 93 on a test and her calculated z score was 2.14, that means that her score is 2.14 standard deviations above the mean of the test scores.
A z score represents the number of standard deviations a data point is from the mean of the data set.
A positive z score means that the data point is above the mean, while a negative z score means that the data point is below the mean.
The mean of the test scores was, 80 with a standard deviation of 5, then Katie's z score would be calculated as:
z = (x - μ) / σ
= (93 - 80) / 5
= 2.6
Z scores are useful for comparing data points from different data sets or for comparing data points within the same data set that are measured on different scales.
Katie's score is 2.6 standard deviations above the mean.
A z score of 2.14 would mean that Katie's score is slightly below this value, but still significantly above the mean.
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1.SP.1 Shawna will create a scatterplot comparing the length of a string, in inches, to the
frequency it creates, in vibrations per second. She will use this table to create the
scatterplot.
A
B
C
D
Length of string
(inches)
Frequency
(vibrations per second)
15.75 23.62 31.5 39.37 47.24
660 440 330 264 220
What type of association will the scatterplot show?
A
B
D
irrational
zero
negative
positive
Step-by-step explanation:
To determine the type of association shown in the scatterplot, we need to plot the data points and examine the trend.
When we plot the data points, we will see that as the length of the string increases, the frequency it creates decreases. This means that there is a negative association between the length of the string and the frequency it creates.
Therefore, the scatterplot will show a negative association.
ow many triangles can be formed with segments measuring 34 cm, 13 cm, and 38 cm?select from the drop-down menu to correctly complete the statement.how many can be formed from these segments.
A triangle is a closed figure with three sides, and the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
This is known as the triangle inequality theorem. Using this theorem, we can determine whether the given segments can form a triangle.
In this case, the segments measuring 34 cm, 13 cm, and 38 cm can form a triangle if and only if:
- 34 + 13 > 38
- 34 + 38 > 13
- 13 + 38 > 34
We can simplify these inequalities to:
- 47 > 38
- 72 > 13
- 51 > 34
All of these inequalities are true, so the given segments can form a triangle.
Now, we need to determine how many triangles can be formed using these segments. To do this, we can use the formula:
number of triangles = (n-2)
where n is the number of sides (or segments, in this case). Plugging in n=3, we get:
number of triangles = (3-2) = 1
In conclusion, using the triangle inequality theorem, we determined that the given segments can form a triangle. However, only one triangle can be formed using these segments.
The theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In this case:
1. 34 cm + 13 cm > 38 cm (47 > 38)
2. 34 cm + 38 cm > 13 cm (72 > 13)
3. 13 cm + 38 cm > 34 cm (51 > 34)
Since all three conditions are satisfied, only one triangle can be formed using these segments.
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ABCD is an isosceles trapezoid. If AD = BC, B= x+10 and C= 2x-30, find the measure of angle D
Since ABCD is an isosceles trapezoid, we know that AB = CD. Additionally, we know that AD = BC. Therefore, we can set up two equations:
AB = CD
AD = BC
Using the fact that B = x + 10 and C = 2x - 30, we can substitute those values into the equations:
AB = CD
x + 10 + AD = 2x - 30 + BC
Since AD = BC, we can simplify the second equation to:
x + 10 + AD = 2x - 30 + AD
x + 10 = 2x - 30
x = 40
Now that we know x, we can find the measures of angles B and C:
B = x + 10 = 50
C = 2x - 30 = 50
Since ABCD is an isosceles trapezoid, we know that angles B and C are congruent. Therefore, each of them measures 50 degrees. Since the sum of the angles in a quadrilateral is 360 degrees, we can set up the equation:
A + B + C + D = 360
Substituting in the values we have:
A + 50 + 50 + D = 360
Simplifying the equation:
A + D = 260
Since ABCD is an isosceles trapezoid, we know that angles A and D are congruent. Therefore, we can set up the equation:
A + D = 2D
Substituting in the value we have:
2D = 260
Simplifying the equation:
D = 130
Therefore, the measure of angle D is 130 degrees.
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I need help ASAP!!!! What is the volume of the sphere? Round the answer to the nearest cubic unit.
18 cm
3,054 cm3
399 cm3
763 cm3
9,160 cm3
The volume of given sphere is 3054 cm³
Given that a sphere with diameter of 18 cm we need to find its volume,
Volume of a sphere = 4/3 × π × radius³
So,
Volume = 4/3 × 3.14 × 9³
= 9156.24 / 3
= 3053.08
≈ 3054 cm³
Hence the volume of given sphere is 3054 cm³
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Individual members of three teams raced through a maze. Their times are show
Maze Completion Times (In Seconds)
Blue Team
Red Team
Green Team
Suppose that the team whose times have the least variability wins.
Part A
The race organizers make a box plot of times for each team. Use the box plots to compare the variability and spread of the times.
Blue Team H
75 80 65 95 82 77 80 75 82
80 86
83 83 84 78 87 88 87
68 85 90 94 68 75 75 90 75
B
Green Team H +
+
65 70 75 80
90
Maze Completion Times (seconds)
U
X
60
I
Red Team H
100
Answer:
Part A:
From the box plots, we can see that the Red Team has the least variability and spread of times, followed by the Blue Team and then the Green Team. The Red Team's box is the smallest, indicating that their times are more tightly clustered together.
Part B:
Blue Team:
Q1: 75
Q2: 82
Q3: 87
IQR: 12
Upper fence: Q3 + 1.5IQR = 87 + 1.512 = 105
There are no outliers
Green Team:
Q1: 70
Q2: 75
Q3: 80
IQR: 10
Upper fence: Q3 + 1.5IQR = 80 + 1.510 = 95
There is one outlier at 90
Red Team:
Q1: 80
Q2: 83
Q3: 87
IQR: 7
Upper fence: Q3 + 1.5IQR = 87 + 1.57 = 98.5
There are no outliers
Step-by-step explanation:
N/A
9. An interior designer is decorating a bedroom and plans on placing a bookcase in the corner. The bookcase has three shelves, and each shelf has a diameter of 28 inches. What is the total shelf area of the bookcase?
The total shelf area of the bookcase is given as follows:
A = 1846.32 in².
How to calculate the area of a circle?The area of a circle of radius r is given by the multiplication of π and the radius squared, as follows:
A = πr²
The radius is half the diameter, hence it is given as follows:
r = 14 inches.
There are three shelves, hence the total shelf area of the bookcase is given as follows:
A = 3 x 3.14 x 14²
A = 1846.32 in².
(there are 3 bookcases, hence we obtain the area of one bookcase and then multiply by 3 to obtain the total area).
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T/F: statistically, using z-scores, a group mean can be compared to a population mean to ascertain whether or not the group mean is different from the population mean.
It is true that, Statistically, using z-scores, a group mean can be compared to a population mean to ascertain whether or not the group mean is different from the population mean.
What is Hypothesis testing:Hypothesis testing is a statistical method for making inferences about a population based on sample data.
It involves formulating a null hypothesis, which is a statement about the population that we want to test, and an alternative hypothesis, which is a statement that contradicts the null hypothesis.
Here we have a statement
Statistically, using z-scores, a group mean can be compared to a population mean to ascertain whether or not the group mean is different from the population mean.
The above statement is true
Z-scores are a measure of the distance between a sample mean and the population mean in standard deviation units.
By using z-scores, we can compare a group mean to a population mean and determine whether the difference is statistically significant.
To do this, we calculate the z-score using the formula:
z = (x - μ) / (σ / sqrt(n))
If the resulting z-score falls in the rejection region of the standard normal distribution, we can reject the null hypothesis that the sample mean is not significantly different from the population mean.
In summary, comparing a group mean to a population mean using z-scores is a common statistical technique to determine whether there is a significant difference between the two means.
Therefore,
It is true that, Statistically, using z-scores, a group mean can be compared to a population mean to ascertain whether or not the group mean is different from the population mean.
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Need some more help PLS
Ben used wrong the Pythagorean's theorem, the solution that he should have got is x = 2√5
Where is Ben's mistake?The Pythagorean's theorem says that for a right triangle, the sum of the squares of the cathetus is equal to the square of the hypotenus.
Then the rule that Ben should have written is:
x² + 4² = 6²
So he started wrong.
Solving that we will get:
x = √(6² - 4²) = √20 = 2√5
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in how many ways can 10 distinct books be divided among three students if the first student gets five books, the second three books, and the third two books?
There are 2520 ways to divide the 10 distinct books among the three students with the given distribution.
To find the number of ways to divide 10 distinct books among three students with the given distribution (5 books for the first student, 3 books for the second, and 2 books for the third), you can use the combination formula:
C(n, r) = n! / (r! * (n - r)!)
Where "C" represents the number of combinations, "n" is the total number of items, and "r" is the number of items to choose. We'll apply this formula to each student's distribution, and then multiply the results to find the total number of ways.
1. First student (5 books from 10):
C(10, 5) = 10! / (5! * (10 - 5)!) = 252
2. Second student (3 books from the remaining 5):
C(5, 3) = 5! / (3! * (5 - 3)!) = 10
3. Third student (2 books from the remaining 2):
C(2, 2) = 2! / (2! * (2 - 2)!) = 1
Now multiply the results for each student:
Total number of ways = 252 * 10 * 1 = 2520
So, there are 2520 ways to divide the 10 distinct books among the three students with the given distribution.
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A boy's pace is 60cm. How many metres does he walk in 450 paces?
Answer:
I think 270m
Step-by-step explanation:
450x60=27.000 so 270 m
Determine the maximum value of f (x, y, z) = (xyz)^1/3 given that x, y, and z are nonnegative numbers and x + y + z = 4
The maximum value of f (x, y, z) = [tex](xyz)^{1/3}[/tex] given that x, y, and z are non- negative numbers and x + y + z = 4 is 4/3
Using arithmetic mean and geometric mean relation
Arithmetic mean is dividing the sum of the values of a set by the number of values in the set
Geometric mean is the nth root of the product of the n values
AM ≥ GM
AM = [tex]\frac{x+y+z}{3}[/tex]
GM = [tex](xyz)^{1/3}[/tex]
[tex]\frac{x+y+z}{3}[/tex] ≥ [tex](xyz)^{1/3}[/tex]
x + y + z = 4
[tex]\frac{4}{3}[/tex] ≥ [tex](xyz)^{1/3}[/tex]
maximum possible value is 4/3
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Help pls and thank you
The approximate height of kite above the ground will be 150 feet.
The correct answer is an option (b)
Let us assume that in the attached diagram of right triangle the angle A represents the ideal angle to fly a kite and this angle measures 45 degrees.
Here, the hypotenuse of right triangle measures 225 ft
We need to find the height of the kite above the ground.
i.e., we need to find the length of side opposite to angle A.
Let us assume that x represents this height.
We know that in right triangle, the sine of angle θ is nothing but the ratio of opposite side of angle θ to the hypotenuse.
Consider the sine of angle A
sin(A) = opposite side of angle A / hypotenuse
sin(45°) = x / 225
We know that from the standard trigonometric table the value of sin(45°) is [tex]\frac{1}{\sqrt{2} }[/tex]
Substitute this value in above equation we get,
[tex]\frac{1}{\sqrt{2} }[/tex] = x / 225
We solve this equation to find the value of x.
x = [tex]\frac{225}{\sqrt{2} }[/tex]
x = 159.09 ft
x ≈ 150 ft
Therefore, the correct answer is an option (b)
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Ryleigh received a $50 gift card to the frozen yogurt shop for her birthday. The shop sells yogurt sundaes for $4 and yogurt cones for $3. How many sundaes and cones can she buy with her card?
If x represents the number of yogurt sundaes and y represents the number of yogurt cones, which graph correctly shows the solution to the problem 4x + 3y ≤ 50?
A. X, Y graph. X range is negative 30 to 30, and y range is negative 30 to 30. Dashed line on graph has negative slope and runs through negative 10, 30 and 5, 10 and 20, negative 10. Above line is shaded.
B. X, Y graph. X range is negative 30 to 30, and y range is negative 30 to 30. Dashed line on graph has negative slope and runs through negative 10, 30 and 5, 10 and 20, negative 10. Below line is shaded.
C. X, Y graph. X range is negative 30 to 30, and y range is negative 30 to 30. Solid line on graph has negative slope and runs through negative 10, 30 and 5, 10 and 20, negative 10. Above line is shaded.
D. X, Y graph. X range is negative 30 to 30, and y range is negative 30 to 30. Solid line on graph has negative slope and runs through negative 10, 30 and 5, 10 and 20, negative 10. Below line is shaded.
Answer:
D
Step-by-step explanation:
have a good day God bless :)
The graph correctly shows the solution to the problem 4x + 3y ≤ 50 is D. X, Y graph. X range is -30 to 30, and y range is -30 to 30. Solid line on the graph has a negative slope and runs through (-10, 30), (5, 10), and (20, -10). Below the line is shaded.
What is the graphTo examine the points on the graph:
Point (-10, 30): Substituting x = -10 and y = 30 into the inequality: 4(-10) + 3(30) = -40 + 90 = 50 ≤ 50. So, the point satisfies the inequality.
Point (5, 10): Substituting x = 5 and y = 10 into the inequality: 4(5) + 3(10) = 20 + 30 = 50 ≤ 50. The point satisfies the inequality.
Lastly, Point (20, -10): Substituting x = 20 and y = -10 into the inequality: 4(20) + 3(-10) = 80 - 30 = 50 ≤ 50. The point satisfies the inequality.
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Stacey is designing a custom rug for a client. The current design is a rectangle with a width of 8 feet and a length of 5 feet. The client wants the rug to be larger but doesn't want the width to exceed 12 feet. Select any of the scale factors that Stacey could use to dilate the current dimensions and meet the customer's requests.
Select 2 correct answer(s)
a. SF : 1.2
b. SF : 0.8
c. SF : 5/6
d. SF : 2
e. SF : 3/2
For designed a rectangular custom rug by Stacey for a client, the scale factor that he could use to dilate the current dimensions and meet the customer's requests is equals to [tex]= \frac{ 3}{2}[/tex]. So, option(e) is right one.
Dilation is one of geometric transformations, which includes translation, reflection, and rotation. It is the scaling of an object, where it gets bigger or smaller. Stacey is designer and design a custom rug for a client. The current design is a rectangle shape, Length of rectangle, l = 5 feet
Width of rectangle, w = 8 feet
The client wishes the rug to be larger but doesn't want the width to exceed 12 feet. A scale factor is the ratio between the scale of a original object and a new object.
That is Scale factor = Dimension of the new shape ÷ Dimension of the original shape. Now, according to client, area of rectangular rug = length × width = 12 × 5
= 60 ft²
Area of rectangular rub designed by Stacey = 5 × 8 = 40 feet. So, using the scale factor formula, the scale factor used by Stacey is [tex]\frac{ 60}{40}[/tex]
[tex]= \frac{ 3}{2}[/tex].
Hence, required value is [tex]= \frac{ 3}{2}[/tex].
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The correct answers are SF: 1.2 and SF: 3/2.
To meet the client's request of increasing the size of the rug but keeping the width under 12 feet, Stacey can dilate the current dimensions using a scale factor that increases the length while keeping the width the same or increasing it slightly. The correct scale factors that she can use are SF: 1.2 and SF: 3/2. If Stacey uses SF: 1.2, the new length would be 1.2 times the original length of 5 feet, which is 6 feet. The new width would still be 8 feet, which is within the client's requirement of not exceeding 12 feet.
Similarly, if Stacey uses SF: 3/2, the new length would be 3/2 times the original length of 5 feet, which is 7.5 feet. Again, the new width would still be 8 feet, which is within the client's requirement. Scale factor SF: 2 would make the rug too wide, SF: 0.8 and SF: 5/6 would make it smaller in both dimensions, which does not meet the client's request.
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(L3) A segment that extends from the vertex of a triangle to the midpoint of the opposite side is called the _____ of the triangle.
The median of the triangle is a segment that runs from the vertex of a triangle to the midpoint of the opposing side.
A median is a line segment that extends from a vertex of a triangle to the midpoint of the opposite side. The centroid has several interesting properties. It is always located inside the triangle, and it divides each median into two segments in a 2:1 ratio. Additionally, the centroid is the center of mass of the triangle, meaning that if the triangle were a physical object with uniform density, it would balance perfectly on the centroid. The centroid is also equidistant from the three vertices of the triangle.
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