Finally, integrate with respect to u:
[4u](5 to 6) = 4(6) - 4(5) = 4
So, the double integral ∫∫R dydA is equal to 4.
To compute the double integral ∫∫R dydA, where D = Ф(R) and Ф(u, v) = (u, u + v), we first need to transform the integral using the given mapping.
The region R is defined as the set of all points (u, v) such that u ∈ [5, 6] and v ∈ [0, 4]. According to the transformation Ф, we have x = u and y = u + v.
Now we need to find the Jacobian determinant of the transformation:
J(Ф) = det([∂x/∂u, ∂x/∂v; ∂y/∂u, ∂y/∂v]) = det([1, 0; 1, 1]) = (1)(1) - (0)(1) = 1
Since the Jacobian determinant is nonzero, we can change the variables in the double integral using the transformation Ф:
∫∫R dydA = ∫∫D (1) dydx = ∫(5 to 6) ∫(u to u + 4) dydu
Now, compute the integral:
∫(5 to 6) ∫(u to u + 4) dydu = ∫(5 to 6) [y](u to u + 4) du
= ∫(5 to 6) [(u + 4) - u] du = ∫(5 to 6) 4 du
Finally, integrate with respect to u:
[4u](5 to 6) = 4(6) - 4(5) = 4
So, the double integral ∫∫R dydA is equal to 4.
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Willka can cover 13. 5 m² with 3 L of paint.
Complete the table using equivalent ratios.
Area covered (in)
Paint (L)
13. 5
3
1
10
Willka would need approximately 0.2222 L of paint to cover 1 m² and approximately 2.2222 L of paint to cover 10 m².
Willka can cover 13.5 m² with 3 liters of paint. To find equivalent ratios, we can determine how much paint is needed to cover 1 m² and then use that to find how much paint is required for other areas.
To find the amount of paint needed for 1 m², divide the area covered by the paint used:
1 m² = (13.5 m²)/(3 L) = 4.5 m²/L
Now, we can use this ratio to complete the table:
Area covered (m²) - Paint (L)
13.5 - 3
1 - (1/4.5) = 0.2222 L (approximately)
10 - (10/4.5) = 2.2222 L (approximately)
So, the completed table is:
Area covered (m²) - Paint (L)
13.5 - 3
1 - 0.2222
10 - 2.2222
Using equivalent ratios, Willka would need approximately 0.2222 L of paint to cover 1 m² and approximately 2.2222 L of paint to cover 10 m².
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please help i’ll give brainlist :)
1. A forest fire has been burning for several days. The burned area in acres, is given by
the equation y = (4. 800) 2 where d is the number of days since the area of the
fire was first measured
a. Complete the table
b. Look at the value of y = 4,800 2d when d = -1 what does it tell you about the area burned in the fire? what about when d = -3?
c. How much area had the fire burned a week before it measured 4,800 acres, explain your reasoning
The fire had burned half the area (2,400 acres) one week before it measured 4,800 acres.
How to evaluate Days (d) and Burned area (y), the area burned in the fire, and the area the fire has burned a week before it measured 4,800 acres?
a. To complete the table, we substitute the values of d and evaluate the expression for y.
Days (d) Burned area (y)
0 0
1 23,040
2 92,160
3 207,360
4 368,640
b. When d = -1, the value of y = (4,800)^(2*(-1)) = (4,800)^(-2) = 3.255e-8. This value is very small, indicating that the burned area was negligible or not measurable at this point in time.
When d = -3, the value of y = (4,800)^(2*(-3)) = (4,800)^(-6) = 1.130e-34. This value is extremely small, indicating that there was essentially no burned area at this point in time.
c. To determine the number of days before the fire measured 4,800 acres, we need to solve the equation y = (4,800)^2 for d.
4,800^2 = (4,800)^2d
1 = 2dlog(4,800)
d = log(4,800) / (2log(4,800)) = 0.5
Therefore, the fire had burned half the area (2,400 acres) one week before it measured 4,800 acres. This is because the area burned increases quadratically with time, so the area burned one week before is the square root of the area measured at the time.
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8. Brock placed a 20 foot ladder against the side of the house. The base of the ladder was 6 foot from the base of the house. How high does the ladder reach on the side of the house? Draw a picture and solve. (Round to tenth)
Step-by-step explanation:
1st i think u divide 20 and 6 and then round tht to the tenth place because we already know our answer is going to be a decimal bc 6 cant go into 20.
Figure A and B are similar. Figure A has a perimeter of 72 meters and one of the side lengths is 18 meters. Figure B has a perimeter of 120 meters Find The missing corresponding side length.
The missing corresponding side length in Figure B is 30 meters.
Perimeter is the total length of the boundary of a two-dimensional shape. It is found by adding up the lengths of all the sides of the shape.
How can we determine the missing corresponding side length ?Since Figure A and Figure B are similar, their corresponding side lengths are proportional.
Let's represent the missing side length in Figure B with x. Then, we can set up a proportion to solve for x:
18 / (72 - 3 × 18) = x / (120 - 3 × x)
Here, 72 - 3 × 18 represents the sum of the other three sides in Figure A, and 120 - 3 × x represents the sum of the other three sides in Figure B.
Simplifying the left-hand side, we get:
18 / (72 - 3 × 18) = 18 / 18 = 1
Substituting this into the proportion, we get:
1 = x / (120 - 3 × x)
Multiplying both sides by (120 - 3 × x), we get:
120 - 3 × x = x
Simplifying and solving for x, we get:
4x = 120
x = 30
Therefore, the missing corresponding side length in Figure B is 30 meters.
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PLS HELP! URGENT!!!! A circular flower bed is 20 m in diameter and has a circular sidewalk around it that is 3 m wide. Find the area of the sidewalk in square meters. Use 3. 14 for pi
The area of the sidewalk in square meters is 69π.
To find the area of the sidewalk, we need to subtract the area of the flower bed from the area of the outer circle formed by the sidewalk.
First, we need to find the area of the flower bed. We know that the diameter of the flower bed is 20 m, so the radius is half of that, which is 10 m. We can use the formula for the area of a circle, which is A = πr^2, where A is the area and r is the radius.
Area of flower bed = π(10m)^2 = 100π square meters
Next, we need to find the area of the outer circle formed by the sidewalk. Since the sidewalk is 3 m wide, the radius of the outer circle will be 10 + 3 = 13 m (10 m for the flower bed radius plus 3 m for the width of the sidewalk).
Area of outer circle = π(13m)^2 = 169π square meters
Finally, we can find the area of the sidewalk by subtracting the area of the flower bed from the area of the outer circle:
Area of sidewalk = Area of outer circle - Area of flower bed
Area of sidewalk = (169π) - (100π)
Area of sidewalk = 69π square meters
Therefore, the area of the sidewalk in square meters is 69π, or approximately 216.6 square meters (if we use 3.14 for π).
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Heights for 16-year-old boys are normally distributed with a mean of 68. 3 in. And a standard
deviation of 2. 9 in.
Find the z-score associated with the 96th percentile.
Find the height of a 16-year-old boy in the 96th percentile.
State your answer to the nearest inch
The height of a 16-year-old boy in the 96th percentile is approximately 73 inches. Rounded to the nearest inch, the answer is 73 inches.
Find out the height of a boy in the 96th percentile?To find the z-score associated with the 96th percentile, we need to find the z-score such that the area to the right of it under the standard normal distribution is 0.96. Using a standard normal distribution table or calculator, we find that the z-score is approximately 1.75.
Next, we can use the z-score formula to find the height of a 16-year-old boy in the 96th percentile:
z = (x - μ) / σ
where z is the z-score, x is the height we want to find, μ is the mean height, and σ is the standard deviation.
Plugging in the values we have:
1.75 = (x - 68.3) / 2.9
Multiplying both sides by 2.9, we get:
x - 68.3 = 5.075
Adding 68.3 to both sides, we get:
x = 73.375
So the height of a 16-year-old boy in the 96th percentile is approximately 73 inches. Rounded to the nearest inch, the answer is 73 inches.
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Solich sandwich shop had the following long-term asset balances as of december 31, 2021: accumulated cost depreciation book value land $ 77,000 − $ 77,000 building 442,000 $ (83,980 ) 358,020 equipment 245,000 (46,400 ) 198,600 patent 160,000 (64,000 ) 96,000 solich purchased all the assets at the beginning of 2019 (3 years ago). the building is depreciated over a 20-year service life using the double-declining-balance method and estimating no residual value. the equipment is depreciated over a 10-year useful life using the straight-line method with an estimated residual value of $13,000. the patent is estimated to have a five-year service life with no residual value and is amortized using the straight-line method. depreciation and amortization have been recorded for 2019 and 2020. problem 7-7a part 1 required: 1. for the year ended december 31, 2021, record depreciation expense for buildings and equipment. land is not depreciated. (if no entry is required for a transaction/event, select "no journal entry required" in the first account field.)
No journal entry is required for the land since it is not depreciated.
To record depreciation expense for buildings and equipment for the year ended December 31, 2021, we need to calculate the depreciation amounts for each asset based on their respective methods.
For the building, we will use the double-declining-balance method. The annual depreciation expense is calculated as (2 / 20) x $442,000 = $44,200. Since depreciation has already been recorded for 2019 and 2020, the accumulated depreciation balance for the building as of December 31, 2020 is $83,980. Therefore, the 2021 depreciation expense for the building is $44,200 - $83,980 = $(-39,780). We record this as follows:
Building Depreciation Expense: $39,780
Accumulated Depreciation - Building: $39,780
For the equipment, we will use the straight-line method. The annual depreciation expense is calculated as ($245,000 - $13,000) / 10 = $23,200. Since depreciation has already been recorded for 2019 and 2020, the accumulated depreciation balance for the equipment as of December 31, 2020 is $46,400. Therefore, the 2021 depreciation expense for the equipment is $23,200, and we record it as follows:
Equipment Depreciation Expense: $23,200
Accumulated Depreciation - Equipment: $23,200
No journal entry is required for the land since it is not depreciated.
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If $5,500 is invested at 1.55% interest, find the value (in dollars) of the investment at the end of 6 years if the interest is
compounded as follows. Roung vour answers to the nearest cent.
A. Annualy
B. quarterly
C. Monthly
The value of the investment at the end of 6 years is $6,359.77 if the interest is compounded annually, $6,416.52 if it is compounded quarterly, and $6,437.70 if it is compounded monthly.
What formula is used to for compound interest?
[tex]A = P(1 + r/n)^{nt}[/tex]
where A is the final amount, P is the initial investment, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the number of years.
A. If the interest is compounded annually, we have:
A = [tex]5,500(1 + 0.0155/1)^{1*6}[/tex] = $6,359.77
B. If the interest is compounded quarterly, we have:
A = [tex]5,500(1 + 0.0155/4)^{4*6[/tex] = $6,416.52
C. If the interest is compounded monthly, we have:
A = [tex]5,500(1 + 0.0155/12)^{12*6[/tex] = $6,437.70
Therefore, the value of the investment at the end of 6 years is $6,359.77 if the interest is compounded annually, $6,416.52 if it is compounded quarterly, and $6,437.70 if it is compounded monthly.
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Sketch a profit function P(q) satisfying the following: the domain is 0 ?q? 20,000; marginal profit is negative for 10,000 < q < 15,000 and nowhere else; P has an inflection point at q = 13,000; the break-even point is q = 2000; and marginal profit is constant for q > 17,000.
P(q) has a concave down inflection at q=13,000, negative marginal profit between q=10,000 to 15,000, constant marginal profit after q=17,000, and breaks even at q=2000.
One possible profit function that satisfies the given conditions is:
P(q) = -0.002(q-13,000)^3 + 20q - 40,000
The domain is 0 ≤ q ≤ 20,000, and the marginal profit is negative for 10,000 < q < 15,000, meaning that increasing production within this range will result in decreasing marginal profit. The inflection point at q = 13,000 indicates a change in the concavity of the profit function.
The break-even point is q = 2000, which means that the profit function crosses the x-axis at this point.
For q > 17,000, the marginal profit is constant, indicating that the profit function becomes linear beyond this point. This profit function satisfies all the given conditions and can be used to model the profit of a business or a product as a function of production quantity.
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Use the drawing tool(s) to form the correct answer on the provided graph.
Plot the axis of symmetry and the vertex for this function: h(x) = (x − 5)2 − 7
A graph of the axis of symmetry and the vertex for this function is shown below.
What is the graph of a quadratic function?In Mathematics and Geometry, the graph of a quadratic function would always form a parabolic curve because it is a u-shaped. Based on the graph of this quadratic function, we can logically deduce that the graph is an upward parabola because the coefficient of x² is positive and the value of "a" is greater than zero (0).
Since the leading coefficient (value of a) in the given quadratic function h(x) = (x - 5)² - 7 is positive 1, we can logically deduce that the parabola would open upward and the solution would be on the x-intercepts. Also, the value of the quadratic function f(x) would be minimum at -7.
In conclusion, the turning point and vertex is given by the ordered pair (5, -7).
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Wanda's front porch is 7 feet wide and 13 feet long. Wanda wants to stain the wood on the porch next weekend. The stain costs $2. 00 per square foot. How much will it cost to buy enough stain for the whole porch?
The amount it will cost to buy enough stain for the whole porch is $182.
To determine the cost of staining Wanda's front porch, we'll first need to calculate the area of the porch, which is a rectangle in shape. The area of a rectangle can be found using the formula: Area = length × width.
In this case, the length of Wanda's porch is 13 feet, and the width is 7 feet. So, we'll multiply these two measurements to find the area:
Area = 13 ft × 7 ft = 91 square feet
Now that we know the area, we can calculate the cost of the stain. The stain costs $2.00 per square foot, so we'll multiply the area of the porch (91 square feet) by the cost per square foot:
Cost = 91 sq ft × $2.00/sq ft = $182
Therefore, it will cost Wanda $182 to buy enough stain for the whole porch.
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If the volume of the cube is (4)(4)(4) =
64 cm3, what is the volume of the oblique
prism if it has been tilted at 60°?
The volume of the oblique prism is approximately 55.424 cm³.
The volume of the cube is given as 64 cm³, which means that each side of the cube has a length of 4 cm.
To find the volume of the oblique prism, we need to know the area of the base and the height. The base of the oblique prism is a parallelogram, and we can find its area using the formula:
area = base × height
where the base is the length of one side of the parallelogram, and the height is the perpendicular distance between the base and the opposite side.
Since the parallelogram is tilted at an angle of 60°, we need to find the perpendicular height by using trigonometry. The height of the parallelogram is given by:
height = (side length) × sin(60°)
height = 4 × sin(60°)
height = 4 × 0.866 = 3.464
Therefore, the area of the base is:
area = (side length) × height
area = 4 × 3.464 = 13.856 cm²
To find the volume of the oblique prism, we multiply the area of the base by the height of the prism. Since the height of the prism is also 4 cm (the same as the side length of the cube), we have:
volume = area of base × height
volume = 13.856 × 4 = 55.424 cm³
Therefore, the volume of the oblique prism is approximately 55.424 cm³.
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(65 points) ASAP!!! Warren is building shelves for his 3-D printed model collection. He has a piece of wood that is 4.5 feet long. After cutting five equal pieces of wood from it, he has 0.7 feet of wood left over.
Part A: Write an equation that could be used to determine the length of each of the five pieces of wood he cut. (1 point)
Part B: Explain how you know the equation from Part A is correct. (1 point)
Part C: Solve the equation from Part A. Show every step of your work. (2 points)
Answer:
Total wood = 4.5 Left over wood = 0.7 let 1 piece of wood = x
hence we have 5x+0.7=4.5
5x=4.5 - 0.7
5x = 3.8 x= 3.8/5 = 0.76 ft
Step-by-step explanation:
Answer:
A: 5x + 0.7 = 4.5
B: You know the equation is correct because it takes into account that Warren cut five pieces of wood that are equal in length from a 4.5 feet long piece of wood. The total length of the five pieces of wood should equal the length of the original piece of wood, minus the leftover wood he has. In other words, 5x represents the total length of the five pieces of wood cut, and 0.7 represents the amount of wood left over after the cutting.
C: 0.76 feet
Step-by-step explanation:
Part A:
Let x be the length of each of the five pieces of wood cut from the 4.5 feet long piece of wood. The equation that could be used to determine the length of each piece is:
5x + 0.7 = 4.5
Part B:
You know the equation is correct because it takes into account that Warren cut five pieces of wood that are equal in length from a 4.5 feet long piece of wood. The total length of the five pieces of wood should equal the length of the original piece of wood, minus the leftover wood he has. In other words, 5x represents the total length of the five pieces of wood cut, and 0.7 represents the amount of wood left over after the cutting.
Part C:
5x + 0.7 = 4.5
5x = 4.5 - 0.7
5x = 3.8
x = 3.8/5
x = 0.76
Therefore, each of the five pieces of wood that Warren cut is 0.76 feet long.
There is a rope running from the top of a flagpole to a hook in the ground. The flagpole is 24 feet high, and the hook is 32 feet from its base. How long is the rope?
Answer:
Step-by-step explanation:
Write a division expression for each fraction 1/2
Answer: 1 ÷ 2
Step-by-step explanation: I think this is what you mean?
Gareth pays $60 for 9m of climbing rope. How much will Sophie pay for 15m at the same store?
Sophie will pay money equivalent to $100 for 15m at the same store.
What is Money?The term "money" in mathematics refers to a form of payment, such as bills, coins, and demand deposits, that is used to purchase goods and services. Money is used to pay for the worth or price of an item or service.
A country's monetary system is referred to as its currency.
In the case of Gareth,
Money paid for 9 m of climbing [tex]=\$60[/tex]
Money paid per m of climbing [tex]=\$60\div9[/tex]
Thus, money paid by Sophie for 15 m of climbing [tex]= 15 \times (60\div9)[/tex]
[tex]\boxed{\bold{= \$100}}[/tex]
Hence Sophie will pay money equivalent to $100 for 15m at the same store.
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Answer:
$100
Step-by-step explanation:
We know that for 9m of rope, Gareth had to pay $60.
The question is asking us to find out how much Sophie will pay for 15m of rope. To do this, we have to find out how much is paid per meter of rope.
[tex]60/9\\=6\frac{2}{3}[/tex]
For the sake of not using fractions, let's keep it as an improper fraction: 60/9
So, we can write an equation for the price of 15m of rope:
(60/9)·15
=100
So, Sophie will pay $100 for 15m of rope.
Hope this helps! :)
Laura is currently paying off her four-year car financing. when she purchased her car, it had a list price of $19,858. laura traded in her previous car, a good-condition 2000 honda insight, for 85% of the trade-in value listed below, financing the rest of the cost at 9.5% interest, compounded monthly. she also had to pay 9.27% sales tax, a $988 vehicle registration fee, and a $77 documentation fee. however, because laura wants to pay off her loan more quickly, she makes a total payment of $550 every month. how much extra is she paying monthly? round all dollar values to the nearest cent.
Laura is paying each month:
extra payment = $550 - monthly payment
To calculate how much extra Laura is paying each month, we first need to calculate the total cost of her car financing. Here are the steps:
Calculate the trade-in value of Laura's old car. We don't have the exact value, but we know that she received 85% of the trade-in value listed below, so we can set up an equation:
0.85 * trade-in value = value Laura received
Solving for the trade-in value, we get:
trade-in value = value Laura received / 0.85
Add the trade-in value to the list price of the new car to get the total cost before taxes and fees:
total cost before taxes and fees = $19,858 + trade-in value
Add the sales tax, registration fee, and documentation fee to get the total cost of the car financing:
total cost = (1 + 0.0927) * total cost before taxes and fees + $988 + $77
Calculate the monthly payment using the formula for a loan with monthly compounding:
monthly payment = (principal * monthly interest rate) / (1 - (1 + monthly interest rate)^(-number of months))
We know that Laura is financing the rest of the cost after her trade-in value, so:
principal = total cost - value Laura received
monthly interest rate = 0.095 / 12
number of months = 48 (since it's a four-year financing)
Substituting these values into the formula, we get:
monthly payment = ($19,858 + trade-in value - value Laura received) * 0.007916 / [tex](1 - (1 + 0.007916)^{(-48)})[/tex]
Now that we have the total monthly payment, we can calculate how much extra Laura is paying each month:
extra payment = $550 - monthly payment
Note that this assumes that Laura doesn't have any other fees or interest charges on her car financing, such as late payment fees or penalties for paying off the loan early. If there are any additional fees or charges, the calculation may be different.
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Can anyone help with this question on this picture
The distance it would take to travel across the river on the bridge than to take the ferry is 4√6 units.
How to determine the distance between the coordinates for each points?In Mathematics and Geometry, the distance between two (2) end points that are on a coordinate plane can be calculated by using the following mathematical equation:
Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]
Where:
x and y represent the data points (coordinates) on a cartesian coordinate.
By substituting the given end points into the distance formula, we have the following;
Distance AC = √[(-4 - 0)² + (2 - 2)²]
Distance AC = √[(-4)² + (0)²]
Distance AC = √[16 + 0]
Distance AC = √16
Distance AC = 4 units.
Distance AB = √[(2 - 0)² + (0 + 2)²]
Distance AB = √[(2)² + (2)²]
Distance AB = √[4 + 4]
Distance AB = √8
Distance AC = 2√2 units.
From Pythagorean Theorem, the length of BC is given by;
BC² = (2√2)² + 4²
c² = 8 + 16
c = √24
c = 4√6 units.
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11 term of 7 -28 112
The 11th term of this geometric sequence 7 -28, 112, .... include the following: 7,340,032.
How to calculate the nth term of a geometric sequence?In Mathematics, the nth term of a geometric sequence can be calculated by using this mathematical equation (formula):
aₙ = a₁rⁿ⁻¹
Where:
aₙ represents the nth term of a geometric sequence.r represents the common ratio.a₁ represents the first term of a geometric sequence.Next, we would determine the common ratio as follows;
Common ratio, r = a₂/a₁
Common ratio, r = -28/7
Common ratio, r = -4
For the 11th term, we have:
a₁₁ = 7(-4)¹¹⁻¹
a₁₁ = 7,340,032.
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Suppose a square had an area measured in square inches, with a numerical value that is 12 more than that of its perimeter, measured in inches.
Write an equation that can be used to find the dimensions of the square
The dimensions of the square are 6 inches by 6 inches.
Let's assume that the side of the square is equal to x inches. Then, the area of the square can be expressed as x^2 square inches. The perimeter of the square is equal to 4 times the length of the side, or 4x inches.
According to the problem, the area of the square is 12 more than its perimeter, so we can write:
x^2 = 4x + 12
This is the equation we can use to find the dimensions of the square. To solve for x, we can rearrange the equation:
x^2 - 4x - 12 = 0
Then, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
where a = 1, b = -4, and c = -12.
Plugging in these values, we get:
x = (-(-4) ± √((-4)^2 - 4(1)(-12))) / 2(1)
x = (4 ± √64) / 2
x = (4 ± 8) / 2
Therefore, the two possible solutions are:
x = 6 or x = -2
Since the side of a square cannot be negative, the only valid solution is x = 6.
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A feeder at the zoo in the shape of a cone has a radius of 3 inches. It holds about 113. 04 cubic inches of food. Approximate it's height to the nearest inch. Use 3. 14 to approximate the value of pi
12
18
36
The height of the feeder is approximately 4 inches.
We can use the formula for the volume of a cone, which is V = (1/3)π[tex]r^{2}[/tex]h, where V is the volume, r is the radius, and h is the height. We know that V = 113.04 and r = 3, so we can solve for h:
113.04 = (1/3)*π*([tex]3^{2}[/tex])h
113.04 = 9πh
h = 113.04 / (9π)
h ≈ 4 inches
The y- intercept of a direct equation is the point where the line crosses the y- axis. It's the value of y when x is equal to 0. To graph a direct equation, you can compass the y- intercept on the y- axis, and also use the pitch to find other points on the line.
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At a real estate agency, an agent sold a house for $315000. the commission rate is %6.5 for the real estate agency and the commission rate for the agent is %20 of the amount the real estate agency gets. how much did the agency make on the house? how much did the agent earn in commission?
part 2
the agent earned $ in commission.
If the commission rate is %6.5 for the real estate agency, the real estate agency made $20,475 on the house. The agent earned $4,095 in commission.
First, let's determine the commission earned by the real estate agency. To do this, we'll multiply the house price ($315,000) by the commission rate (6.5%).
$315,000 * 6.5% = $20,475
The real estate agency made $20,475 on the house.
Next, let's determine the commission earned by the agent. We'll multiply the commission earned by the real estate agency ($20,475) by the agent's commission rate (20%).
$20,475 * 20% = $4,095
The agent earned $4,095 in commission.
So, the real estate agency made $20,475 on the house, and the agent earned $4,095 in commission.
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Consider the series Σ(1) с (where c is a constant). For which values of c will the series converge, and for which it diverge? Justify your answer, and show all your work. (Hint: Use the root test)
To determine whether the series Σ(1) с converges or diverges, we can use the root test. The root test states that if the limit of the absolute value of the nth root of the terms of the series approaches a value less than 1, then the series converges. If the limit approaches a value greater than 1, the series diverges. If the limit equals 1, the test is inconclusive and another test should be used.
Using the root test, we have:
lim┬(n→∞)〖|1^(1/n) c| = lim┬(n→∞)|c| = |c|〗
If |c| < 1, then the limit approaches a value less than 1 and the series converges. If |c| > 1, then the limit approaches a value greater than 1 and the series diverges. If |c| = 1, then the test is inconclusive.
Therefore, the series Σ(1) с converges if |c| < 1, and diverges if |c| > 1. If |c| = 1, then another test should be used to determine convergence or divergence.
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Given the differential equation dy/dx = x-3/y, find the particular solution, y = f(x) with initial condition f(6) = -2
The particular solution of the given differential equation with the initial condition f(6) = -2.
To find the particular solution of the given differential equation dy/dx = (x-3)/y with the initial condition f(6) = -2, we first need to solve the differential equation. This is a first-order separable equation, so we can rewrite it as:
y dy = (x - 3) dx
Now, integrate both sides:
∫y dy = ∫(x - 3) dx
(1/2)y^2 = (1/2)x^2 - 3x + C
Now, apply the initial condition f(6) = -2:
(1/2)(-2)^2 = (1/2)(6)^2 - 3(6) + C
(1/2)(4) = (1/2)(36) - 18 + C
2 = 18 - 18 + C
C = 2
So the particular solution is:
(1/2)y^2 = (1/2)x^2 - 3x + 2
This is the particular solution of the given differential equation with the initial condition f(6) = -2.
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5. Betty will spend $375. 00 on a new lawnmower. She will use her credit card to withdraw $400
cash to pay for the lawnmower. The credit card company charges a $6. 00 cash-withdrawal
fee and 3% interest on the borrowed amount, but not including the cash-withdrawal fee. How
much will Betty owe after one month?
The amount Betty will owe after one month is $418,
Betty will owe more than $400 after one month because of the cash-withdrawal fee and interest charges. The cash-withdrawal fee is a one-time charge of $6.00, which means Betty's total borrowed amount is $406.00.
The interest on this amount at a rate of 3% for one month is calculated by multiplying the borrowed amount by the interest rate and time, giving $12.18.
Therefore, Betty will owe $418.18 after one month, which is the borrowed amount plus the cash-withdrawal fee and interest charges.
It's important to be aware of the additional fees and charges associated with borrowing on a credit card, as they can significantly increase the amount owed and lead to financial difficulties if not managed properly.
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Trudy takes out an easy access loan for $500. It cost her $10 for every $100 and a one-time fee of
$150. How much did it cost Trudy to get the loan for $500?
A $250
B $300
C$200
D Not Here
It cost Trudy $200 to get the loan for $500. The correct answer is C) $200.
Trudy has taken a loan of $500, and the cost of the loan is $10 for every $100 borrowed. Therefore, the cost of borrowing $500 will be:
Cost of borrowing $500 = ($10/$100) * $500 = $50
In addition to the above cost, there is a one-time fee of $150 to be paid. So, the total cost of the loan will be:
Total cost of the loan = Cost of borrowing + one-time fee
= $50 + $150
= $200
Hence, it cost Trudy $200 to get the loan for $500.
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Which set of ordered pairs represents a proportional relationship? A. {(4, 1), (0, 0), (6, 2), (8, 4)} B. {(2, 1), (4, 3), (8, 9), (16, 27)} C. {(2, 3), (6, 9), (10, 15), (22, 33)} D. {(4, 9), (7, 12), (10, 15), (18, 23)}
Answer:
C.
Step-by-step explanation:
3 = 1.5 * 2
9 = 1.5 * 6
15 = 1.5 * 10
33 = 1.5 * 22
5)
The population of Sanibel, Florida can
be modeled by P = 6191 · 1. 05t,
where t is the number of years since
2016. What was the population in
2016? What percent did the
population increase by each year?
The population increased by a percentage of 5%.
The population of Sanibel, Florida in 2016 can be determined using the given population model P = 6191 * 1.05^t, where t represents the number of years since 2016. To find the population in 2016, we set t to 0 since there are 0 years since 2016.
Step 1: Set t to 0 in the equation:
P = 6191 * 1.05^0
Step 2: Calculate the population P:
P = 6191 * 1
P = 6191
So, the population in Sanibel, Florida in 2016 was 6,191.
Regarding the percent population increase each year, the given model uses an exponential growth formula with a constant factor of 1.05. The factor (1.05) represents a 5% increase in the population each year.
In summary, the population in Sanibel, Florida in 2016 was 6,191, and the population increases by 5% each year. This exponential growth model demonstrates how the population continues to grow at a steady rate, contributing to the overall population increase in the area.
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2. Iwo functions t and g are definea on the set R of real numbers by f: x x² - 2x - 4. g: x → X - 1 Find the value fx for which f(x) = (x) = m - 4.
Answer:
We are given that:
- f(x) = x² - 2x - 4
- g(x) = x - 1We want to find fx for which f(x) = g(x) - 4, or in other words:
- f(x) = x - 1 - 4
- f(x) = x - 5
We can solve forTo find the value of x for which f(x) = g(x) - 4 (which is what I assume you meant by "f(x) = (x) = m - 4"), we can set up the following equation:
f(x) = g(x) - 4
Substituting the given expressions for f(x) and g(x), we get:
x² - 2x - 4 = x - 1 - 4
Simplifying, we have:
x² - 3x - 3 = 0
We can solve for x using the quadratic formula:
x = (-(-3) ± sqrt((-3)² - 4(1)(-3))) / (2(1))
x = (3 ± sqrt(21)) / 2
Therefore, the two values of x for which f(x) = g(x) - 4 are:
- x = (3 + sqrt(21)) / 2
- x = (3 - sqrt(21)) / 2
The function f(x) = 454 + 0.9x gives the cost, in dollars, of manufacturing x vinyl records. select all the true statements. the initial cost for manufacturing records depends on the quantity ordered. the initial cost for the manufacturing is $454, regardless of the number of records ordered. in addition to the initial cost, each record costs $0.90 to manufacture. in addition to the initial cost, each record costs $454 to manufacture. each record costs a total of $454.90 to manufacture.
The true statements are:
The initial cost for manufacturing records is $454, regardless of the number of records ordered.
In addition to the initial cost, each record costs $0.90 to manufacture.
Each record costs a total of $454.90 to manufacture.
As according to the question the function f(x)= 454+0.9x gives the cost, in dollars, of manufacturing x vinyl records.
Hence the incorrect statements are:
"The initial cost for manufacturing records depend on the quantity ordered".
"in addition to the initial cost, each record costs $454 to manufacture".
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