To evaluate the double integral I = ∬D xy dA, where D is the triangular region with vertices (0,0),(1,0), (0,6), we need to set up the limits of integration for x and y.
Since D is a triangular region, we can integrate over the two sides that meet at the origin and then integrate over the third side. Let's integrate over the sides that form the right angle at (0,0).
For the side along the x-axis, y = 0 to y = 6x.
For the side along the y-axis, x = 0 to x = 1.
Thus, the double integral becomes:
I = ∫0^1 ∫0⁶x xy dy dx
Evaluating the inner integral with respect to y, we get:
I = ∫0^1 [x(y²/2)]0⁶x dx
Simplifying and evaluating the outer integral with respect to x, we get:
I = ∫0^1 18x⁴ dx
I = 18/5
Therefore, the value of the double integral I = ∬D xy dA over the triangular region with vertices (0,0),(1,0), (0,6) is 18/5.
To evaluate the double integral I = ∬_D xy dA for the triangular region D with vertices (0,0), (1,0), and (0,6), we first need to set up the limits of integration.
The base of the triangle lies on the x-axis, from x = 0 to x = 1. The height of the triangle lies on the y-axis, from y = 0 to the line y = 6(1-x), since the slope of the hypotenuse is -6 and passes through (1,0).
Now we can set up the integral:
I = ∬_D xy dA = ∫_(0 to 1) ∫_(0 to 6(1-x)) xy dy dx
Let's first integrate with respect to y:
∫_(0 to 6(1-x)) xy dy = [x(y²)/2]_(0 to 6(1-x)) = 18x(1-x)²
Next, integrate with respect to x:
I = ∫_(0 to 1) 18x(1-x)² dx
Using integration by substitution or expanding and integrating term by term, we get:
I = 2
So, the value of the double integral is 2.
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the variables x and y vary inversely. use the given values to write an equation relating i and y. then find y when i = i= 5, y = -4 an equation is y= when i = 3, y = 5
please help me!
When i (x) = 3, the value of y is approximately -6.67. The equation relating i (x) and y in this inverse variation is xy = -20.
The given information states that the variables x and y vary inversely. To write an equation relating i (assuming it's x) and y, we first need to understand the concept of inverse variation.
In inverse variation, the product of the two variables remains constant. Mathematically, it can be represented as xy = k, where k is the constant of variation. We are given the values i (x) = 5 and y = -4. Using these values, we can find the constant of variation, k:
5 * -4 = k
k = -20
Now that we have the constant of variation, we can write the equation relating i (x) and y as:
xy = -20
Next, we want to find the value of y when i (x) = 3. We can use the equation we just derived to find the value of y:
3 * y = -20
Now, we can solve for y:
y = -20 / 3
y ≈ -6.67
So, when i (x) = 3, the value of y is approximately -6.67. The equation relating i (x) and y in this inverse variation is xy = -20.
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3−(−1)+(−1)−33, minus, left parenthesis, minus, 1, right parenthesis, plus, left parenthesis, minus, 1, right parenthesis, minus, 3
The calculated value of the expression 3 - (-1) + (-1) - 3 using a calculator is 0
Finding the value of the expression 3 - (-1) + (-1) - 3From the question, we have the following parameters that can be used in our computation:
The expression 3 - (-1) + (-1) - 3
We can add the numbers using a calculator
So, we have the following representation
Value = 3 - (-1) + (-1) - 3
Using the above as a guide, we have the following:
Value = 0
This means that the value of the expression 3 - (-1) + (-1) - 3 using a calculator is 0
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Given that f(x) = (h(x))10 = h(-1) = 3 h'(-1) = 6 Calculate f'(-1).
The final value is f'(-1) = 16,777,2160.
We can use the chain rule and the power rule of differentiation to find f'(-1).
Recall that the chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) h'(x). Applying this rule to f(x) = (h(x))^10, we get:
f'(x) = 10(h(x))^9 h'(x)
Now, we can substitute x = -1 into the above equation, since we are asked to find f'(-1). Thus, we have:
f'(-1) = 10(h(-1))^9 h'(-1)
We are given that h(-1) = 3 and h'(-1) = 6, so we can substitute these values to get:
f'(-1) = 10(3)^9 (6)
Simplifying, we get:
f'(-1) = 16,777,2160
Therefore, f'(-1) = 16,777,2160
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A cylinder has a volume of cubic centimeters and a height of 12 centimeters. What is the radius of the base of the cylinder, in centimeters?"
Answer:
Step-by-step explanation:
(PLEASE HELP + POINTS)
Select the correct graph.
Smith's Produce sells packages of pre-cut vegetables. The company has a tolerance level of less than or equal to y grams for a 250-gram
package. Which graph could be used to determine the variance levels that would result in a package of vegetables being rejected because of
its weight, X?
(Picture of graphs)
The answer of the given question based on the graph could be used to determine the variance levels that would result in a package of vegetables is histogram.
To determine the variance levels that would result in a package of vegetables being rejected because of its weight, X, consider the following:
1. The company has a tolerance level of less than or equal to y grams for a 250-gram package. This means that the graph must represent a relationship between the weight of the package (X) and the tolerance level (y).
2. Since the package is rejected if it weighs more than the allowed tolerance, the graph should show that as the weight (X) increases, the acceptance range decreases (y decreases).
3. The graph should ideally have a boundary line that represents the maximum tolerance level (y). Any points above this line would represent rejected packages.
Based on these criteria, you should select the graph that best represents this relationship between the weight of the package (X) and the tolerance level (y), where packages with a weight exceeding the tolerance level are rejected.
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During their team meeting, both managers shared their findings. Complete the statement describing their combined results.
Select the correct answer from each drop-down menu.
The initial number of video views was (more than, fewer than, the same as,)the initial number of site visits, and the number of video views grew by (a smaller factor, the same factor, a larger factor) the number of site visits.
The difference between the total number of site visits and the video views after 5 weeks is(15,625, 20,825, 36,450, 52,075)
The initial number of video views was (fewer than) the initial number of site visits, and the number of video views grew by (a larger factor) the number of site visits. The difference between the total number of site visits and the video views after 5 weeks is (20,825).
Video views refer to the number of times a video has been watched or viewed by viewers.
A site visit refers to a visit or session by a user to a website. It occurs when a user accesses and interacts with webpages or content on a particular website.
The initial number of video views was (fewer than) the initial number of site visits, and the number of video views grew by (a larger factor) the number of site visits. The difference between the total number of site visits and the video views after 5 weeks is (20,825).
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Let ⋆ be the binary operation on z (set of integers) defined by
a ⋆ b = 2ab + 5
show that ⋆ is commutative. hint: show that a ⋆ b = b ⋆ a
solution:
show that ⋆ is associative. hint: show that (a ⋆ b) ⋆ c = a ⋆ (b ⋆ c)
solution:
a. let ⋆ be the binary operation on z (set of integers) defined by
a ⋆ b = a + b + ab
1. show that ⋆ is commutative. hint: show that a ⋆ b = b ⋆ a
solution:
2.show that ⋆ is associative. hint: show that (a ⋆ b) ⋆ c = a ⋆ (b ⋆ c)
solution:
Since the expression is the same, we can conclude that the binary operation ⋆ is associative.
To show that the binary operation ⋆ is commutative, we need to demonstrate that a ⋆ b is equal to b ⋆ a for any integers a and b.
Let's start by evaluating a ⋆ b:
a ⋆ b = 2ab + 5.
Now let's evaluate b ⋆ a:
b ⋆ a = 2ba + 5.
By comparing the expressions for a ⋆ b and b ⋆ a, we can see that they are indeed equal:
2ab + 5 = 2ba + 5.
Since the expression is the same, we can conclude that the binary operation ⋆ is commutative.
To show that the binary operation ⋆ is associative, we need to demonstrate that (a ⋆ b) ⋆ c is equal to a ⋆ (b ⋆ c) for any integers a, b, and c.
Let's evaluate (a ⋆ b) ⋆ c:
(a ⋆ b) ⋆ c = (2ab + 5) ⋆ c = 2(2ab + 5)c + 5 = 4abc + 10c + 5.
Now let's evaluate a ⋆ (b ⋆ c):
a ⋆ (b ⋆ c) = a ⋆ (2bc + 5) = 2a(2bc + 5) + 5 = 4abc + 10a + 5.
By comparing the expressions for (a ⋆ b) ⋆ c and a ⋆ (b ⋆ c), we can see that they are indeed equal:
4abc + 10c + 5 = 4abc + 10a + 5.
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Karissa wants to use the data to determine which brand is most absorbent. Based on the data collected at the beginning of the contest, enter the approximate number of paper towels needed to clean each square foot of the spill. Enter the number of Brand A paper towels needed per square foot in the first box. Enter the number of Brand B paper towels needed per square foot in the second box. Enter the number of Brand C paper towels needed per square foot in the third box
Answer:
Step-by-step explanation:
Jason wants to earn at least $250 each week working during the
summer.
Jason earns $6. 00 an hour babysitting.
Jason earns $7. 75 an hour working at a store.
He can work no more than 40 hours each week.
Let b equal hours of babysitting, and s equal hours working at the
store.
Which system of inequalities models the constraints?
O 6. 00b + 7. 755 2 250
b +5 s 40
O 6. 00b + 7. 75s 250
b +5 s 40
O 6. 000 + 7. 75s 40
b + s 250
O 6. 00b + 7. 75s 2 40
b + s 250
Answer:
The correct system of inequalities that models the constraints is:
6.00b + 7.75s ≥ 250
b + s ≤ 40
Explanation:
The first inequality represents the requirement that Jason needs to earn at least $250 each week. The earnings from babysitting are $6.00 per hour, and the earnings from working at the store are $7.75 per hour.
Therefore, the total earnings from both jobs can be represented by the expression 6.00b + 7.75s. This expression must be greater than or equal to $250, hence the first inequality.
The second inequality represents the constraint that Jason can work no more than 40 hours each week. The variables b and s represent the number of hours worked babysitting and working at the store, respectively.
The sum of these hours must be less than or equal to 40, hence the second inequality.
Therefore, the correct system of inequalities is:
6.00b + 7.75s ≥ 250
b + s ≤ 40
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A theater is selling tickets to a ''preview night'' of their new musical. The tickets cost $12 per adult and $7. 50 per child. Due to limit on seating, they can sell no more than 150 tickets. However, they would like to make at least $675 from ticket sales
A theater can make at least $675 from ticket sales by selling at least 100 adult tickets and up to 50 child tickets.
To solve this problem, we can use a system of equations. Let's define:
- x as the number of adult tickets sold
- y as the number of child tickets sold
We know that:
- x + y ≤ 150 (due to the limit on seating)
- 12x + 7.5y ≥ 675 (they want to make at least $675)
We can solve this system of equations using substitution or elimination. Let's use elimination:
- Multiply the second equation by 2 to get rid of the decimals: 24x + 15y ≥ 1350
- Multiply the first equation by 15: 15x + 15y ≤ 2250
- Subtract the second equation from the first: 9x ≥ 900
- Divide both sides by 9: x ≥ 100
So they need to sell at least 100 adult tickets to make at least $675. Let's see if that's possible:
- If they sell 100 adult tickets, that leaves 50 tickets for children
- 100 adult tickets * $12 = $1200
- 50 child tickets * $7.50 = $375
- Total ticket sales = $1575 (more than $675)
So they can make at least $675 from ticket sales by selling at least 100 adult tickets and up to 50 child tickets.
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“The mode of a data set is one of the values in the data set.” This statement is ____________.
Find the permineter of the square. leave answers in simplified radical form and label with correct units.
To find the perimeter of a square, you simply add up the lengths of all four sides. If we let "s" be the length of one side of the square, then the perimeter P can be found using the formula:
P = 4s
Since all four sides of a square are equal, we can simplify this expression to:
P = s + s + s + s = 4s
Therefore, the perimeter of the square is equal to 4 times the length of one side. If the length of one side is given in simplified radical form (such as √2 or √3), then the perimeter should also be expressed in simplified radical form.
For example, if the length of one side is 2√2 units, then the perimeter would be:
P = 4s = 4(2√2) = 8√2 units
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How do i solve for surface area on a rectangular prism
To solve for the surface area of a rectangular prism, you'll need to find the area of each of its six faces and add them together.
A rectangular prism has three pairs of faces: two each for length (L), width (W), and height (H).
First, find the area of the two faces with dimensions L x W. The area is calculated by multiplying length by width: A₁ = L * W. Since there are two such faces, the total area for these is 2A₁ = 2(L * W).
Next, find the area of the two faces with dimensions L x H. The area is calculated by multiplying length by height: A₂ = L * H. The total area for these faces is 2A₂ = 2(L * H).
Finally, find the area of the two faces with dimensions W x H. The area is calculated by multiplying width by height: A₃ = W * H. The total area for these faces is 2A₃ = 2(W * H).
To find the total surface area of the rectangular prism, add the areas of all six faces together: Surface Area = 2A₁ + 2A₂ + 2A₃ = 2(L * W) + 2(L * H) + 2(W * H).
So, the formula for the surface area of a rectangular prism is Surface Area = 2(L * W + L * H + W * H).
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Una presa se construye en un rio. El nivel del agua del estanque esta dado por n = 4,5t + 28, dónde t es el tiempo en años. Traza la gráfica y determina el nivel del agua que tenía la presa al ser construida. (ayuda por favor)
The initial water level is given as follows:
28 units.
How to define a linear function?The slope-intercept representation of a linear function is given by the equation shown as follows:
y = mx + b
The coefficients m and b have the meaning presented as follows:
m is the slope of the function, representing the increase/decrease in the output variable y when the input variable x is increased by one.b is the y-intercept of the function, representing the numeric value of the function when the input variable x has a value of 0. On a graph, it is the value of y when the graph of the function crosses or touches the y-axis.The function for this problem is defined as follows:
n = 4.5t + 28.
The intercept is of b = 28, representing the initial amount of water.
The graph is given by the image presented at the end of the answer.
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Triangle TUV, with vertices T(2,-8), U(9,-6), and V(6,-3), is drawn on the coordinate
grid below. what is the area. in square units, of triangle TUV
The area of the triangle TUV, with vertices T(2,-8), U(9,-6), and V(6,-3), is 13.58
How did we arrive at the above?First using distance calculator we derived the length of TV and the length of VU.
Since TV = Height; and
VU = Base
and the triangle is a right triangle,
Then, area is given by 1/2 base x Height
Length of TV usign distance calculator is 6.40312
Lenght of VU using distance calculator is 4.24264
So area = 1/2 * 6.40312 * 4.24264
Area = 13.58
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Gray made a $3,500 tax-deductible contribution to his individual retirement account (IRA). Assuming he is in a 28 percent tax bracket, how much will this contribution save him on his taxes?
Gray's $3,500 tax-deductible contribution to his IRA will save him $980 on his taxes.
When Gray contributes $3,500 to his individual retirement account (IRA), it is considered a tax-deductible contribution. This means that the amount contributed is deducted from his taxable income, reducing the amount of taxes he owes.
Since Gray is in the 28 percent tax bracket, this means that for every dollar of taxable income, he pays 28 cents in taxes. To calculate the tax savings from his $3,500 IRA contribution, we need to multiply the contribution amount by his tax rate:
$3,500 (contribution) x 0.28 (tax rate) = $980 (tax savings)
In this case, Gray's $3,500 tax-deductible contribution to his IRA will save him $980 on his taxes. By contributing to his IRA, Gray not only invests in his future retirement but also takes advantage of the tax benefits associated with these accounts. In the end, he reduces his taxable income and, consequently, the amount of taxes he needs to pay.
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According the April 12, 2017 Pew Research survey, 58% of Americans approve of U. S. Missile strikes in
Syria in response to reports of the use of chemical weapons by Bashar al-Assad's government (the
Syrian government). A sample of 50 Americans are surveyed. Let o be the sample proportion of
Americans who approve the U. S. Missile strikes.
1. What is the population proportion?
(decimal form)
2. What is the sample size?
3. Can the normal approximation be used with this distribution?
4. What is the mean of the sampling proportion?
Answer:
The population proportion is given as 58% or 0.58 in decimal form.
The sample size is given as 50 Americans.
Yes, the normal approximation can be used with this distribution because the sample size is sufficiently large (n=50) and the underlying population is assumed to be large enough to satisfy the independence requirement.
The mean of the sampling proportion (o) can be calculated using the formula:
mean = population proportion = 0.58
Therefore, the mean of the sampling proportion is 0.58 or 58%.
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Tell which measure of central tendency best describes the data.
Weights of books (oz):
12 10 9 15 16 10
Mean
Median
Mode
Last week Deion ran a total of 32 miles. This week, he increased his running distance by 6. 4 miles. By what percentage did he increase the distance he ran? Please help waaaaaa
Deion increased the distance he ran by 20%.
To discover the percentage increase within the distance Deion ran, we need to first calculate the amount of increase.
The increase in distance that Deion ran this week compared to final week is:
6.4 miles
To find the proportion increase, we need to divide the increase by means of the original value (the distance he ran last week),
Then multiply by using a hundred to express the result as a percent.
The original price (last week's distance) is:
32 miles
Therefore, the percentage increase within the distance he ran is:
(6.4 miles / 32 miles) x 100% = 20%
So, Deion increased the distance he ran by 20%.
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A standard number cube is rolled to play a certain board game. What is the Sample Space? Use proper notation as necessary and no extra spaces.
Answer: 1.3
Step-by-step explanation:
Solve the initial value problem. Dy/dx = 4x^-3/4, y(1) = 3 a. y = 16x^1/4 - 13 b. y = 16x1/4 + 48 c. y = -3/4^x7/4-13/4 d. y= 4x^1/4 - 1
The solution to the given initial value problem is (d) y = 4x^(1/4) - 1.
Given the initial value problem,
dy/dx = 4x^(-3/4), y(1) = 3
Integrating both sides with respect to x, we get
∫dy = ∫4x^(-3/4)dx
y = -8x^(-1/4) + C
where C is the constant of integration.
To find the value of C, we use the initial condition y(1) = 3
3 = -8(1)^(-1/4) + C
C = 3 + 8 = 11
Therefore, the solution to the initial value problem is
y = -8x^(-1/4) + 11
Simplifying further,
y = 11 - 8/x^(1/4)
Hence, the correct option is d) y = 4x^(1/4) - 1 is not the solution to the given initial value problem.
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Admission Charge for Movies The average admission charge for a movie is $5. 81. If the distribution of movie admission charges is approximately normal with a standard deviation of $0. 81, what is the probability that a randomly selected admission charge is less than $3. 50
The probability that a randomly selected admission charge is less than $3. 50 is 0.23% or 0.0023.
To find the probability that a randomly selected admission charge is less than $3.50, we will use the z-score formula and a standard normal table. The z-score formula is:
Z = (X - μ) / σ
Where X is the value we are interested in ($3.50), μ is the average admission charge ($5.81), and σ is the standard deviation ($0.81).
Z = (3.50 - 5.81) / 0.81 ≈ -2.84
Now, look up the z-score (-2.84) in a standard normal table, which gives us the probability of 0.0023. Therefore, the probability that a randomly selected admission charge is less than $3.50 is approximately 0.23% or 0.0023.
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Use the following bond listing for Pacific Bell to answer the following: A 5-column table with 1 row. Column 1 is labeled Bonds with entry PacBell 6 and StartFraction 5 Over 8 EndFraction 34. Column 2 is labeled current yield with entry 6. 55. Column 3 is labeled Volume with entry 5. Column 4 is labeled Close with entry 99 and one-fourth. Column 5 is labeled net change with entry + StartFraction 1 Over 8 EndFraction. How many bonds were traded during this session?
5 bonds were traded during this session.
Based on the provided bond listing for Pacific Bell, the number of bonds traded during this session is 5. Here's the breakdown of the information in the 5-column table:
- Column 1 (Bonds): PacBell 6 5/8 34
- Column 2 (Current Yield): 6.55
- Column 3 (Volume): 5
- Column 4 (Close): 99 1/4
- Column 5 (Net Change): +1/8
The "Volume" column indicates the number of bonds traded during the session. In this case, the volume entry is 5. Therefore, 5 bonds were traded during this session.
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Where should one start to learn maths if they're really bad at it
Consistent practice is key to improving your math skills over time.
How to learn math if you feel like you're really bad at it?If you feel like you're really bad at math, it's important to start with the basics. This might mean reviewing concepts like arithmetic, fractions, decimals, and percentages. You can find resources online or in books that can help you with this. Once you have a solid foundation, try to identify your strengths and weaknesses so you can focus your efforts on the areas where you need the most improvement. Find a learning style that works best for you, whether it's working independently, with a tutor, or in a study group. Finally, remember that consistent practice is key to improving your math skills over time. Don't give up, and don't be afraid to ask for help when you need it.
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Jacinta compares the volume of two boxes. Both boxes have a width of 2. 5 inches, and a height of 10 inches. The larger box has a length of 8 inches. The smaller box has a length that is 75 % of the length of the larger box.
Volume of large box =
Volume of small box =
What is the difference in the volumes of the two boxes?
Which units should be used for each of these answers?
The volume of the larger box is 200 cubic inches, and the volume of the smaller box is 150 cubic inches.
To find the volume of each box, we use the formula for the volume of a rectangular prism, which is V = lwh, where l is the length, w is the width, and h is the height.
For the larger box, we have l = 8 inches, w = 2.5 inches, and h = 10 inches, so
Volume of large box = = 8 x 2.5 x 10 = 200 cubic inches.
For the smaller box, we have l = 0.75 x 8 = 6 inches, w = 2.5 inches, and h = 10 inches, so
Volume of small box= 6 x 2.5 x 10 = 150 cubic inches.
The difference in the volumes of the two boxes is
Volume of large box - Volume of small box = 200 - 150 = 50 cubic inches.
The units for the volumes are cubic inches, since we are dealing with three-dimensional space.
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i need to know the full quadratic equation answer
Required Answer :
x(2x + 4)= x²+ 4 (9 - 3x)
2x² + 4x = x² + 36 - 12x
2x² - x² + 4x + 12x - 36=0
x² + 16x - 36=0
Quadratic equation in Standard form,
a = 1, b = 16c = -36Using Quadratic formula,
[tex] \implies \sf x = \dfrac{ - b \pm \sqrt{ {b}^{2} - 4ac} }{2a} \\ \\ \implies \sf x = \dfrac{ - 16 \pm \sqrt{ {16}^{2} - 4(1)( - 36)} }{2 \times 1} \\ \\ \implies \sf x = \dfrac{ - 16\pm \sqrt{256 - ( - 144)} }{2} \\ \\ \implies \sf x = \frac{ - 16 \pm \sqrt{256 + 144} }{2} \\ \\ \implies \sf x = \frac{ - 16 \pm \sqrt{400} }{2} \\ \\ \implies \sf x = \dfrac{ - 16 \pm20}{2} \\ \\ \implies \sf x = \dfrac{ - 16 \pm 20}{2} \\ \\ \implies \sf x = \dfrac{4}{2} \: or \: \dfrac{ - 36}{2} \\ \\ \implies \sf x = 2 \: or \: - 18[/tex]
个
Work out the volume of this prism.
Area =
20 cm²
9cm
The diagram is not drawn to scale.
cm³
The volume of the prism is 180 cm³.
How to work out the volume of a prism?A prism is a 3D (three-dimensional) solid which has faces that are identical at both ends. The other faces are flats. A prism is named after its base.
The volume of any prism can be calculated using the formula:
V = A[tex]_{B}[/tex] * h
where A[tex]_{B}[/tex] is area of base and h is height of prism
In this case, we have the following information about the prism:
A[tex]_{B}[/tex] = 20 cm²
h = 9cm
V = 20 * 9
V = 180 cm³
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Complete Question
Check attached image
Mike receives a bonus every year. His bonus is calculated as 3 percent of his company's total profits. If he estimates his company's total profits to be between $500,000 and $650,000, which inequality best represents Mike's bonus, B, for the year?
Mike's bonus for the year is between $15,000 and $19,500.
The inequality that best represents Mike's bonus, B, for the year is:
$15,000 [tex]\leq B \leq[/tex] 19,500$
to see why, we are able to use the given data that Mike's bonus is calculated as 3 percent of his corporation's overall profits.
If we let P be the organization's general income, then Mike's bonus B can be expressed as:
$B = 0.03P$
We recognise that the organization's total profits are between $500,000 and $650,000, so we will write:
$500,000 [tex]\leq P \leq[/tex] 650,000$
Substituting this inequality into the equation for Mike's bonus, we get:
$15,000 [tex]\leq B \leq[/tex] 19,500$
Therefore, Mike's bonus for the year is between $15,000 and $19,500.
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why is 101 not in the sequence of 3n-2
101 is not in the sequence of 3n-2 because it cannot be obtained by multiplying a positive integer n by 3 and subtracting 2 from the product.
The sequence 3n-2 is a set of numbers obtained by taking a positive integer n, multiplying it by 3 and then subtracting 2 from the product. For example, if n = 1, then 3n-2 = 1. If n = 2, then 3n-2 = 4. If n = 3, then 3n-2 = 7, and so on.
Now, you may wonder why the number 101 is not in the sequence of 3n-2. To understand this, we need to determine whether there exists a positive integer n such that 3n-2 is equal to 101.
Let's start by assuming that such an n exists. Then we can write:
3n-2 = 101
Adding 2 to both sides, we get:
3n = 103
Dividing both sides by 3, we get:
n = 103/3
This means that n is not a whole number, which contradicts our assumption that n is a positive integer. Therefore, there cannot exist any positive integer n such that 3n-2 equals 101.
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Read and imagine what is happening in this problem. Hannah mixed 6. 83 lb of pretzels with 3. 57 lb of popcorn. After filling up 6 bags that were the same size with the mixture, she had 0. 35 lb left.
Hannah mixed 6.83 lb of pretzels with 3.57 lb of popcorn to make 10.4 lb of mixture. She then filled up 6 bags with an average of 1.68 lb of mixture per bag, leaving her with 0.35 lb of mixture left over.
In this problem, Hannah mixed 6.83 lb of pretzels with 3.57 lb of popcorn. This means that she had a total of 10.4 lb of mixture. She then filled up 6 bags that were the same size with the mixture, which means that each bag had approximately 1.73 lb of mixture (10.4 lb / 6 bags).
After filling up all 6 bags, Hannah had 0.35 lb of the mixture left over. This means that she used a total of 10.05 lb of mixture for the bags (10.4 lb - 0.35 lb).
To find out how much mixture was used per bag, we can divide the total amount of mixture used (10.05 lb) by the number of bags (6). This gives us an average of approximately 1.68 lb per bag.
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