The requried radius of the cylinder is 4 inches.
We are given the volume of the cylinder as 48π cubic inches and the height as 3 inches. So we can use the formula V = πr²h to solve for the radius.
48π = πr²(3)
16 = r²
r = ±4
Since the radius of a cylinder cannot be negative, we take the positive value of r, which is 4.
Therefore, the radius of the cylinder is 4 inches.
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dr. smith is instructing his graduate students to put together a sample for an upcoming research study of college students. the graduate students were asked to stand outside of the student union to solicit participants, finding 50 freshmen, 50 sophomores, 50 juniors, and 50 seniors. what sort of sampling method is being used?
When the strata is not proportional to the population, it is called disproportionate stratified sampling.
The sampling method used in this case is a stratified sampling method. What is Stratified Sampling? Stratified sampling is a type of probability sampling in which a population is divided into subgroups or strata based on a certain criterion. It is used to ensure that the sample drawn is representative of the population as a whole. When the strata is proportional to the population, it is called proportionate stratified sampling, and Here are the steps followed in this process: Divide the population into strata based on some criterion relevant to the research question. Identify the sample size required from each stratum based on its proportion to the population. Draw a random sample from each stratum to arrive at the final sample.
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What is the gradient of a line through the following pair of points (a,b),(b,a)
Answer:
(a-b)/(b-a)
Step-by-step explanation:
The gradient = difference of y values / difference of corresponding x value
= (a-b)/(b-a)
what is the horizontal asymptote of the function, r(x)=3x^2+9x/-8x^2+3x+10? write your answer in reduced fraction form
To find the horizontal asymptote of the given function, we need to compare the degrees of the numerator and denominator.
r(x) = (3x^2 + 9x) / (-8x^2 + 3x + 10)
As x becomes very large or very small, the higher degree terms in the numerator and denominator dominate the fraction. Since the degree of the denominator (-8x^2) is higher than the degree of the numerator (3x^2), the horizontal asymptote is a horizontal line at y = 0.
Therefore, the horizontal asymptote of the function r(x) is y = 0, which can be written as the reduced fraction 0/1.
A rectangular playing field is 80m by 100m, if Ali wants to make 3 rounds around this field, how many meters will he cover altogether?
(please explain well)
Answer:
Ali will cover 1,080m altogether.
Step-by-step explanation
If the field is 80m by 100m and there are 2 sides of each of those measurements to make a whole round around the field, then one round would be (80m x 2) + (100m x 2) = 360. Since Ali wants to do three laps around the field, then you multiply 360 x 3 to get 1080m total.
Please help! How do I find n in this equation?
Answer: - 6
Step-by-step explanation:
Since n is c in [tex]ax^{2}[/tex] + bx + c, that means that n is the y - intercept of the graph. The y - intercept in the graph is -6 which means that n = -6.
sharp discounts wholesale club has two service desks, one at each entrance of the store. customers arrive at each service desk at an average of one every six minutes. the service rate at each service desk is four minutes per customer. [2 points each part] a.how often (what percentage of time) is each service desk idle? b.what is the probability that both service clerks are busy? c.what is the probability that both service clerks are idle? d.how many customers, on average, are waiting in line in front of each service desk? e.how long does it take a customer to get serviced from the moment they join the queue (i.e., waiting plus service time)?
a. 44.4% of the time each service desk is idle.
b. The probability that both service clerks are busy is 0.22 or 22%.
c. The probability that both service clerks are idle is 11.1%.
d. On average, 0.67 customers are waiting in line in front of each service desk.
e. A customer takes, on average, 5.33 minutes to get serviced from the moment they join the queue (i.e., waiting plus service time).
a) The arrival rate is 1 customer every 6 minutes. Therefore, 10 customers arrive at each service desk every hour. The service rate is 1 customer every 4 minutes, which means that a service desk can handle 15 customers every hour.
The utilization rate of each service desk = arrival rate/service rate = (10/60) ÷ (15/60) = 2/3 Average number of customers in the system at any time (queue + service) = utilization/(1 - utilization) = (2/3)/(1 - 2/3) = 2
Therefore, the average number of customers in the queue is the average number of customers in the system minus the average number of customers in service:Average number of customers in the queue = 2 - 2 = 0
Therefore, each service desk is idle for 60% - 15.6% = 44.4% of the time.
b) The probability of a service desk being busy is equal to its utilization rate = 2/3.
Therefore, the probability of both service clerks being busy is:P(both busy) = (2/3)² = 4/9 = 0.44
c) The probability of a service desk being idle is equal to 1 - utilization rate = 1 - 2/3 = 1/3.
Therefore, the probability of both service clerks being idle is:P(both idle) = (1/3)² = 1/9 = 0.11 or 11%.
d) We can use Little's Law to calculate the average number of customers in the queue:
Average number of customers in the queue = arrival rate x average waiting time in queue Average waiting time in queue = average number of customers in queue / arrival rate = 0 / (10/60) = 0 Average number of customers in the queue = 0 Therefore, on average, 0.67 customers are waiting in line in front of each service desk.
e) The average time a customer spends in the system (waiting + service) is equal to the average number of customers in the system divided by the arrival rate:
Average time in system = average number of customers in system / arrival rate Average number of customers in system = utilization/(1 - utilization) = (2/3)/(1 - 2/3) = 2 Average time in system = 2 / (10/60) = 12/10 = 1.2 minutes = 72 seconds The service time is 4 minutes. Therefore, the average waiting time is:
Average waiting time = average time in system - service time = 1.2 - 4 = -2.8 minutes We can't have negative waiting time, so the actual waiting time is 0.
Therefore, a customer takes, on average, 5.33 minutes (4 minutes of service + 1.33 minutes of waiting) to get serviced from the moment they join the queue.
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5.2 If cos 12° = m, write down the following in terms of m: 5.2.1 sin168⁰
A gardener wants to divide a square piece of lawn in half diagonally. what is the length of the diagonal side of the square is 8 fr. leave your answer in simplest radical form.
16√8
2√8
8√2
4
The answer is option C, 8√2.
The length of the diagonal side of the square is 8 fr. Hence the gardener wants to divide a square piece of lawn in half diagonally, which means to find the length of a diagonal bisector of a square. Let's solve the problem: Length of the diagonal side of the square = 8 fr.
Let's apply the Pythagorean theorem, which states that the square of the length of the hypotenuse (the longest side of the triangle) is equal to the sum of the squares of the other two sides. Therefore, the formula to find the length of the diagonal (d) of the square with side a is given by d = a√2.
Now we can find the length of the diagonal of the square using the formula given above: d = a√2d = 8√2Thus, the length of the diagonal side of the square is 8 fr. Hence the length of the diagonal of the square is 8√2 fr. Therefore, the answer is option C, 8√2.
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Enter the missing numbers in the boxes to complete the table of equivalent ratios of lengths to widths.
? 18
8 ?
10 30
12 ?
To complete the table of equivalent ratios of lengths to widths, we need to find the missing values that make the ratios in each row equal to each other. One way to do this is to use cross-multiplication.
First, we can compare the first and second rows to find the value that makes them equivalent:
x/18 = 8/y
Multiplying both sides by 18y, we get:
xy = 144
So the missing value in the first row is 144/18 = 8, and the missing value in the second row is 144/8 = 18.
Next, we can compare the third and fourth rows:
10/30 = 12/z
Multiplying both sides by 30z, we get:
10z = 360
Dividing both sides by 10, we get:
z = 36
So the missing value in the third row is 36, and the missing value in the fourth row is 12/10 * 36 = 43.2.
Therefore, the completed table is:
8 18
8 144/10 = 14.4
10 30
12 36
2x-3>0
help me please
Answer:
x > 3/2
Step-by-step explanation:
2x -3 > 0
2x > 3
x > 3/2
Answer:
To solve the inequality 2x-3>0, we can use algebraic manipulation to isolate x on one side of the inequality.
First, we can add 3 to both sides of the inequality:
2x - 3 + 3 > 0 + 3
Simplifying the left side and the right side, we get:
2x > 3
Next, we can divide both sides of the inequality by 2:
2x/2 > 3/2
Simplifying the left side and the right side, we get:
x > 3/2
Therefore, the solution to the inequality 2x-3>0 is x > 3/2.
i need help fast lol.
The coordinates of the vertices of the translated shape are (-4, 7), (-1, 7), (-4, 5), and (1, 5).
What is translation and transformation?Transforming an object from one place to another without altering its size or shape is known as translation. Contrarily, the term "transformation" refers to a wider range of procedures that can alter the size, shape, or orientation of an item. Rotation, reflection, and scaling are some more forms of transformation in addition to translation.
The given vector is (-5, 3).
To translate the graph add the vector to the coordinates as follows:
(1, 4) + (-5, 3) = (-4, 7)
(4, 4) + (-5, 3) = (-1, 7)
(1, 2) + (-5, 3) = (-4, 5)
(6, 2) + (-5, 3) = (1, 5)
Hence, the coordinates of the vertices of the translated shape are (-4, 7), (-1, 7), (-4, 5), and (1, 5).
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The equation 9x^2 - 6xy + y ^2 + 6x - 5y + 12 = 0 represents which of these shapes:
The equation 9x^2 - 6xy + y ^2 + 6x - 5y + 12 = 0 represents shape option (B) Ellipse.
The given equation is 9x^2 - 6xy + y^2 + 6x – 5y + 12 = 0.
To determine which shape the equation represents, we can look at the coefficients of x^2, y^2, and xy.
If the coefficients of x^2 and y^2 are equal but have opposite signs, and the coefficient of xy is 0, then the equation represents an ellipse.
If the coefficient of xy is zero and the coefficients of x^2 and y^2 are equal, then the equation represents a circle.
If the coefficient of x^2 or y^2 is zero, then the equation represents a parabola.
If the coefficients of x^2 and y^2 have the same sign, but the coefficient of xy is nonzero, then the equation represents a hyperbola.
Comparing the given equation to these conditions, we can see that 9x^2 - 6xy + y^2 + 6x – 5y + 12 = 0 represents an ellipse, because the coefficients of x^2 and y^2 are equal (9 and 1, respectively) and have opposite signs, and the coefficient of xy is -6.
Therefore, the answer is (B) Ellipse.
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The given question is incomplete, the complete question is:
The equation 9x^2 - 6xy + y^2 + 6x – 5y + 12 = 0 represents which of these shapes: A)Hyperbola B) Ellipse C) Parabola D) Circle
how to draw veen diagram in class 7 of two sets
To draw a Venn diagram of two sets in class 7, draw two overlapping circles inside a rectangle representing the universal set, label each circle with the name of its corresponding set, and identify the elements that belong to each set and the overlapping region.
Steps to draw venn diagram :
These are the steps to draw a Venn diagram for two sets, follow these steps.
1. Draw two overlapping circles, one for each set. Label them A and B.
2. Write the elements that are common to both sets in the overlapping region of the circles.
3. Write the elements that belong only to set A in the non-overlapping part of circle A.
4. Write the elements that belong only to set B in the non-overlapping part of circle B.
For example, let's say we have two sets:
A = {1, 2, 3, 4, 5}
B = {3, 4, 5, 6, 7}
To draw a Venn diagram for these sets, we would follow these steps:
1. Draw two overlapping circles, one for A and one for B.
2. Write the elements that are common to both sets in the overlapping region of the circles. In this case, the common elements are 3, 4, and 5.
3. Write the elements that belong only to set A in the non-overlapping part of circle A. In this case, the elements that belong only to set A are 1 and 2.
4. Write the elements that belong only to set B in the non-overlapping part of circle B. In this case, the elements that belong only to set B are 6 and 7.
5. The completed Venn diagram for these sets should look like two overlapping circles with the common elements in the overlap region and the unique elements in the non-overlapping regions, as shown below:
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Dipesh bought 3 notebooks and 2 pens for Rs. 80. His friend Ramesh said that price of each notebook could be
Rs. 25. Then three notebooks would cost Rs.75, the two pens would cost Rs. 5 and each pen could be for Rs. 2.50.
Another friend Amar felt that Rs. 2.50 for one pen was too little. It should be at least Rs. 16. Then the price of
each notebook would also be Rs.16.Dipesh bought 3 notebooks and 2 pens for Rs. 80. His friend Ramesh said that price of each notebook could be
Rs. 25. Then three notebooks would cost Rs.75, the two pens would cost Rs. 5 and each pen could be for Rs. 2.50.
Another friend Amar felt that Rs. 2.50 for one pen was too little. It should be at least Rs. 16. Then the price of
each notebook would also be Rs.16.
The total price if they buy the identical style of 12 pens and 15 notebooks.
What is unitary method?"A method to find a single unit value from a multiple unit value and to find a multiple unit value from a single unit value."
We always count the unit or amount value first and then calculate the more or less amount value.
For this reason, this procedure is called a unified procedure.
Many set values are found by multiplying the set value by the number of sets.
A set value is obtained by dividing many set values by the number of sets.
Hence total cost
= Rs ( 300 + 120 )
= Rs 420
The cost of 15 notebooks
= Rs ( 15 × 20 )
= Rs 300
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COMPLETE QUESTION-Dipesh bought 3 notebooks and 2 pens for Rs. 80. His friend Ramesh said that price of each notebook could be
Rs. 25. Then three notebooks would cost Rs.75, the two pens would cost Rs. 5 and each pen could be for Rs. 2.50.
Find the values of the variables and the lengths of the sides of this kite. Hint: solve for x first.
Please help!,,,,,
Answer:
x=7 y=14
Step-by-step explanation:
2x +4=x +112x-x=11-4
x=7
y-4=x+3
y-4=7+3
y=7+3+4
y=14
A recipe requires 1/5 of sugar for every 1/3 cup of milk. How much milk is needed for each cup of sugar
For each cup of sugar, we need 5/3 cups of milk.
Define fractionA fraction is used to express a portion of a whole or a ratio of two values. It is represented by a numerator and a denominator separated by a horizontal line, with the numerator above and the denominator below.
To find : how much milk is needed for each cup of sugar
we need to use the given ratio of sugar to milk and invert it:
1/5 cup of sugar per 1/3 cup of milk
To find how much milk is needed for each cup of sugar, we need to divide the amount of milk by the amount of sugar:
(1/3 cup of milk) ÷ (1/5 cup of sugar) = (1/3) × (5/1) = 5/3 cup of milk per cup of sugar
Therefore, for each cup of sugar, we need 5/3 cups of milk.
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-z + 7 = 6 solve for z
Answer:
z = 1
Step-by-step explanation:
-z + 7 = 6
-7 -7
-1
So, basically add -7 on both side (or the kcf method) which cancels out on on the right side but makes -1 on then notice how the -z/-1 = a positive because a negative divided by a negative is a positive so its positive one.
How to check?
just replace -z with -1 and put -1 + 7 = 6 which makes this answer correct.
The following table shows the number of miles a hiker walked on a trail each day for 6 days.
1
2
3
4
5
6
Day
Number of Miles
8
5
7
2
2
9
8
What was the mean number of miles the hiker walked for the 6 days?
A
3. 5
B
4. 5
С
6. 5
D
7. 5
E
8
The mean number of miles the hiker walked for the 6 days is 5.5. We can find it in the following manner.
To find the mean number of miles the hiker walked for the 6 days, we need to calculate the sum of the number of miles and then divide it by 6 (the number of days).
Sum of miles = 8 + 5 + 7 + 2 + 2 + 9 = 33
Mean = Sum of miles / Number of days = 33 / 6 = 5.5
Therefore, the mean number of miles the hiker walked for the 6 days is 5.5.
The closest option to this value is option C: 6.5. However, it is not the correct answer.
In mathematics, the mean is the average of a set of data, which is calculated by adding all the numbers together and then dividing the result by the total number of numbers. For instance, the mean for the collection of values 8, 9, 5, 6, 7 is 7, since 8 + 9 + 5 + 6 + 7 = 35, and 35/5 = 7.
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4 Assignment
K
Question 4, 2.4.20
Part 3 of 8
Fill in the Venn diagram with the appropriate numbers based on the following information.
n(A)=33
n(B)=36
n(B n C) = 14
n(An C) = 9
n(AnBn C) = 5
n(U)= 70
n(C) = 24
n(An B) = 17
A Venn diagram is a graphical representation of sets. Remember that n(A) is the number of
elements in set A.
Region I contains 12 elements.
Region Il contains 12 elements.
Region III contains elements.
I
=
II
HW Score: 73.15%, 6.58 of S
Points: 0 of 1
VII
VI
V
с
VIII
III
IV
B
U
We can fill in the Venn diagram since the number of elements in Region VIII is negative, it means there is an error in the given information or in the calculations.
How to fill in the Venn diagram?Based on the given information, we can fill in the Venn diagram as follows:
n(A) = 33, n(B) = 36, n(C) = 24, n(B ∩ C) = 14, n(A ∩ C) = 9, n(A ∩ B ∩ C) = 5
n(A ∩ B) = n(A) + n(B) - n(A ∩ B) = 33 + 36 - 17 = 52
Now we can fill in the Venn diagram:
Region I: n(A ∩ B ∩ C) = 5
Region II: n(A ∩ B) - n(A ∩ B ∩ C) = 52 - 5 = 47
Region III: n(B ∩ C) - n(A ∩ B ∩ C) = 14 - 5 = 9
Region IV: n(B) - n(B ∩ C) - n(A ∩ B) = 36 - 14 - 17 = 5
Region V: n(A) - n(A ∩ C) - n(A ∩ B) = 33 - 9 - 17 = 7
Region VI: n(C) - n(A ∩ C) - n(B ∩ C) = 24 - 9 - 14 = 1
Region VII: n(U) - n(A) - n(B) - n(C) + n(A ∩ B) + n(A ∩ C) + n(B ∩ C) + n(A ∩ B ∩ C) = 70 - 33 - 36 - 24 + 52 + 9 + 14 + 5 = 57
Region VIII: n(U) - (n(I) + n(II) + n(III) + n(IV) + n(V) + n(VI) + n(VII)) = 70 - (5 + 47 + 9 + 5 + 7 + 1 + 57) = -4
Since the number of elements in Region VIII is negative, it means there is an error in the given information or in the calculations.
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Anastasia was trying to decide which investment plan would be best over 10 years. Bank A was offering 8.5% simple interest on her money using the formula I = P r t. Bank B was offering 8% compounded annually using the formula A = P (1 + r) Superscript t. Which bank is a better investment if she has $2,000 to invest for 10 years?
Answer:
8$ is the answer so chose that one
Calculate the length of minor arc Ab with the angle 72 with a circumference of 90
Answer:
18
Step-by-step explanation:
here,
circumference of the circle = 90
[tex]S = r\theta[/tex]
in above equation, S is the length of arch, r is the radius and theta is the angle formed by arc at the center,
As you can see, in a circle of given radius,
[tex]S \propto \theta[/tex]
therefore,
[tex]\dfrac{S_1}{S_2} = \dfrac{\theta_1}{\theta_2}[/tex]
we have the arc length for angle 360 (circumference of the circle).
let S1 be the length of minor arc, S2 be the arc formed by full circle (360 degree),
[tex]\dfrac{S_1}{90} = \dfrac{72}{360}[/tex]
solving the above equation, we get
[tex]S_1 = 18[/tex]
What is the value of x?
12 units
15 units
20 units
24 units
Answer:The value of x is 15 units.
Step-by-step explanation:
In the given triangle RSQ and ΔRST it is given that
∠RTS ≅ ∠SRQ ≅ 90°
∠RSQ is common in two triangles Δ RSQ and Δ RST.
Therefore two angles are equal so both the triangles are common.
From this property we opposite sides of the corresponding angles will be in the same ratio.
SR/SQ = ST/SR
x/(9 + 16) = 9/x
By cross multiplication on each side of the equation
x² = 25×9 = 225
x = √225 = 15
Therefore measurement of x is 15 units.
He made a baby quilt that was 3ft long and it’s area was 64 square feet. What is its perimeter?
The perimeter of the baby quilt is approximately 48.66 feet.
A quilt that is 3 feet long cannot have a surface area of 64 square feet, hence it appears that the information provided is incorrect. When calculating the area of a quilt, the length and breadth are multiplied. If the quilt is just 3 feet long, for example, the width would need to be 64/3 = 21.3 feet, which is not a practical size for a baby quilt.
Let's assume the information provided is false and try to determine the perimeter of a baby quilt whose surface area is 64 square feet.
To do this, we must first determine the quilt's proportions. Given that a rectangle's area is determined by multiplying its length by its width, we may determine the width by dividing the area by the length.
width = area / length, or 64 / 3 = 21.33 feet (rounded to two decimal places)
The baby quilt is 3 feet by 21.33 feet.
We add the lengths of the four sides to find the perimeter.
The perimeter is equal to 2 (length + width)/2 (3 + 21.33) = 48.66 ft (rounded to two decimal places)
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need help with this question
Answer:
Growth
3.8%
Step-by-step explanation:
Look at the number raised to a power.
In this case it is 1.038
If the number raised to a power is greater than 1, it is growth.
If that number is less than 1 it is decay.
Here, it is growth since 1.038 > 1.
Also, 1.038 = 1 + 0.038 = 1 + 3.8%
The rate of growth is 3.8%.
Operations With Functions
The given expression is [tex](f+g)(x)[/tex] so the value of this expression is [tex]= x^3 + 1/2 x^2 + 3/4 x - 5[/tex]
Is an expressive a mathematical response?A solution or the quantity of the variables is shown by an equation since it has a "equal to" sign. There isn't an equal sign inside the instance of an express, hence there is no clear answer and it cannot demonstrate the value of something like the associated variable.
Where in mathematics is an expression?An expression in mathematics is a grouping of numbers, variables, and mathematical operations including arithmetic, subtraction, multiplication, and division. You may compare expressions to sentences in your mind. An expression, also known as a formula, is a limited collection of signs that are appropriately constructed in accordance with context-dependent requirements.
[tex](f+g)(x) = f(x) + g(x)[/tex]
[tex]= (x^3 - 1/2 x^2 + x) + (x^2 - 1/4 x - 5)[/tex]
[tex]= x^3 - 1/2 x^2 + x + x^2 - 1/4 x - 5[/tex]
[tex]= x^3 + 1/2 x^2 + 3/4 x - 5[/tex]
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=) Jordan surveys students to find out if they prefer blue or green as their school color. The two-way
table shows the results.
Blue
Green
Total
Grade 6 Grade 7 Grade 8
17
27
41
39
27
14
56
54
55
There ?
Is there an association between grade level and preference for the school color?
an association. As grade level increases, there appears
Total
85
80
165
to be a greater preference for blue
Based on the two-way table, it appears that there is an association between grade level and preference for school color.
As the grade level increases, there is a trend towards a greater preference for blue. Among sixth graders, there were 17 students who preferred blue and 39 who preferred green. Among seventh graders, 27 preferred blue and 27 preferred green. Finally, among eighth graders, 41 preferred blue and 14 preferred green. It is important to note, however, that this trend may not necessarily imply causation. There could be other factors at play that are contributing to this association, such as the age of the students or the characteristics of the school environment. Additionally, the sample size in each grade level is relatively small, which could lead to some variability in the results.
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Question 2 of 10
What is the standard form of the line that passes through the points (-4, 6) and (0, -7)?
This is the slope-intercept form of the equation of the line. To convert it to the standard form, we need to rearrange it so that the x and y terms are on the same side of the equation:= (13/4)x + y = -7
How to find standard form?
To find the standard form of the line that passes through the given points (-4, 6) and (0, -7), we need to first find the slope of the line using the slope formula:
slope = (y₂ - y₁) / (x₂- x₁)
where (x₁, y₁) and (x₂, y₂) are the coordinates of the two points.
Using the given points, we get:
slope = (-7 - 6) / (0 - (-4))
slope = -13/4
Now that we have the slope of the line, we can use the point-slope form of the equation of a line:
y - y₁= m(x - x₁)
where m is the slope and (x1, y1) is one of the points on the line. Using the point (-4, 6), we get:
y - 6 = (-13/4)(x - (-4))
y - 6 = (-13/4)x - 13
y = (-13/4)x - 7
This is the slope-intercept form of the equation of the line. To convert it to the standard form, we need to rearrange it so that the x and y terms are on the same side of the equation:
(13/4)x + y = -7
This is the standard form of the equation of the line that passes through the given points.
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What is the weight (in grams) of a liquid that exactly fills a 182.8 milliliter container if the density of the liquid is 0.135grams over milliliter? Round to the nearest hundredth when necessary, and only enter numerical values, which can include a decimal point.
The weight of the liquid can be calculated as follows:
Weight = Volume x Density
The volume is given as 182.8 milliliters.
The density is given as 0.135 grams over milliliter.
Weight = 182.8 x 0.135 = 24.67 grams
Therefore, the weight of the liquid that exactly fills a 182.8 milliliter container if the density of the liquid is 0.135 grams over milliliter is 24.67 grams.
Probably this
an elementary school teacher measures the heights of the students in her classroom. the shortest student was 51 inches tall and the tallest student was 63 inches tall. what would happen to the range if there were a new student added to the class who was 58 inches tall? question 13 options:
The range of the class between shortest student and longest student is 12 inches.
Range is the difference between highest and lowest possible values.
R= (higest value)-(lowest value)
Tallest student= 63 inches
Shortest student= 51 inches
So the range before additon of new student is the difference between the tallest and shortest student.
R= 63(tallest student) - 51(shortest student) = 12inches
Now the newly added student has a height of 58 inches which serves no purpose in range as it does not lies as the extreme-most value.
So the range will still be 63 inches (the height of the tallest student) minus 51 inches (the height of the shortest student), which is 12 inches.
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Show that the following transformation is not linear: T: R ^ 3 longrightarrow R^ 2 define by T(x_{1}, x_{2}, x_{3}) = (|x_{1}|, 0)
The transformation T: R^3 → R^2 defined by T(x1, x2, x3) = (|x1|, 0) is not linear transformation as it violates the additivity and homogeneity properties of linearity.
To show that the transformation T: R^3 → R^2 defined by T(x1, x2, x3) = (|x1|, 0) is not linear, we need to demonstrate that it violates at least one of the two properties of linearity, namely, additivity and homogeneity.
Additivity: T(u + v) = T(u) + T(v) for all vectors u and v in R^3.
Let u = (1, 2, 3) and v = (4, 5, 6) be two vectors in R^3. Then,
T(u) = T(1, 2, 3) = (|1|, 0) = (1, 0)
T(v) = T(4, 5, 6) = (|4|, 0) = (4, 0)
T(u + v) = T(1+4, 2+5, 3+6) = T(5, 7, 9) = (|5|, 0) = (5, 0)
However,
T(u) + T(v) = (1, 0) + (4, 0) = (5, 0)
Since T(u + v) ≠ T(u) + T(v), the transformation T is not additive and hence not linear transformation.
Homogeneity: T(cu) = cT(u) for all vectors u in R^3 and all scalars c.
Let u = (1, 2, 3) be a vector in R^3, and let c = 2 be a scalar. Then,
T(u) = T(1, 2, 3) = (|1|, 0) = (1, 0)
T(cu) = T(2, 4, 6) = (|2|, 0) = (2, 0)
However,
cT(u) = 2(1, 0) = (2, 0)
Since T(cu) ≠ cT(u), the transformation T is not homogeneous and hence not linear.
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