Existence and uniqueness of solutions to systems of differential equations depend on the number of equations and the order of each equation. For example, a system of two first-order equations has a unique solution, while a system of three first-order equations may not have a unique solution.
Previously, we considered the scalar differential equation, which is a single differential equation involving only one independent variable. In this case, the solution to the equation is unique. This is because there is only one equation and only one variable, so the solution must be unique for a given set of initial conditions.
For a system of differential equations, the existence and uniqueness of solutions depend on the number of equations and the order of each equation.
Systems of differential equations with higher order equations or more equations than unknowns may not have a unique solution. On the other hand, if there are more unknowns than equations, then there is always a unique solution. To find the solution, the equations must be solved simultaneously.
In conclusion, A scalar differential equation always has a unique solution.
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In this warm-up activity, you will use your knowledge from the previous lesson on compound angle formulas to derive expressions for the double angle formulas.
Derive a general expression for sin(2θ) and cos(2θ). Hint: sin(2θ) = sin(θ + θ), and use the compound angle formula that was introduced in the previous lesson. Be sure to do this for both sin(2θ) and cos (2θ).
The general expressions for sin(2θ) and cos(2θ) are Sin(2θ) = 2sinθcosθ, and cos(2θ) = cos2θ − sin2θ.
The formula for deriving sin(2θ) and cos(2θ) is as follows:
To derive the sin(2θ) formula, use the following formula: sin (2θ) = 2sinθcosθ
And to derive the cos(2θ) formula, use the following formula: cos(2θ) = cos2θ − sin2θ
From the compound angle formulas, we know that:
Sin (α + β) = sinαcosβ + cosαsinβ, and Cos (α + β) = cosαcosβ − sinαsinβ
We may derive sin(2θ) from the above formulas by putting α = β = θ, which gives us:
Sin (2θ) = sinθcosθ + sinθcosθSin (2θ) = 2sinθcosθcos(2θ) can be derived from the above formula by following these steps:
Cos (2θ) = cosθcosθ − sinθsinθCos (2θ) = cos2θ − sin2θ
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Saving Sally makes $15 an hour, h, and spends $125 on her bills each week. Which expressions show how much Sally will have after 10 weeks? Select all that apply.
15(10h - 125)
150h - 1875
10(15h - 125)
150h - 1250
-1250 + 150h
10(15h + 125)
Answer:
10(15h-125)
Step-by-step explanation:
In this question, h is represented for hours. we want to see how much money sally will have after 10 weeks. We know each hour she makes $15 per h. This will look like 15h. We also know her bills cost $125 each week so we have to subtract 125 from 15h which the equation will now look like
15h-125.
Now we want to know how much she will have after 10 weeks so we will now the equation will look like 10(15h-125).
Answer: Pls I don't know this
Step-by-step explanation:
Pls I don't mean to hurt your feelings but my brother knows this and I have sent it to him
Would you help me with this question. I'm not sure what this answer is.
The area of the figure is 18 mm².
What is area?The area is the amοunt οf space within the perimeter οf a 2D shape. It is measured in square units, such as cm², m², etc.
Yοu can think οf area as the area inside a given shape οr space. It refers tο hοw much space is taken up. The larger the shape, the larger the area and perimeter οf the shape will be. Nοt tο be cοnfused with vοlume, area οnly refers tο space taken up by a flat οr 2D οbject.
We have given the figure, with all right angles,
Draw an imaginary rectangle of 5 × 6, that covers up the whole figure.
Now,
The area of full rectangle - area of small rectangle = area of the figure
⇒ (6 × 5) - (4 × 3)
⇒ 30 - 12
⇒ 18 mm²
Thus, the area of the figure is 18 mm².
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What percent of 32 is 48?
Answer: 150%
Step-by-step explanation:
Step-by-step explanation:
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Please help will give brainliest
The solution to the system of equations is (3, -7).
How to find the solution of equation of lines?To find the solution of these two equations, we need to find the values of x and y that satisfy both equations simultaneously.
We can set the two equations equal to each other to get:
-5x + 8 = x/3 - 8
Multiplying both sides by 3, we get:
-15x + 24 = x - 24
Simplifying, we get:
-16x = -48
Dividing both sides by -16, we get:
x = 3
Now that we know x = 3, we can substitute it into either of the original equations to find y. Let's use the equation y = -5x + 8:
y = -5(3) + 8 = -7
Therefore, the solution to the system of equations is (3, -7).
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Divide the polynomials using Long Division
The expression x^4 - 16 is divided by x^3 + 2x^2 + 4x + 8 is x - 2
How to divide the polynomialFrom the question, we have the following parameters that can be used in our computation:
x^4 - 16 is divided by x^3 + 2x^2 + 4x + 8
Using the long division method of quotient, we have
x^3 + 2x^2 + 4x + 8 | x^4 - 16
The division steps are as follows
x - 2
x^3 + 2x^2 + 4x + 8 | x^4 - 16
x^4 + 2x^3 + 4x^2 + 8x
--------------------------------------------------------------
-2x^3 - 4x^2 - 8x - 16
-2x^3 - 4x^2 - 8x - 16
--------------------------------------------------------------
0
Hence, the quotient is x - 2
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Area of quadrant of a circle with side 8cm and base 6cm
Answer:
A = (1/4)πr^2
where r is the radius of the circle.
If the side and base mentioned in the question refer to the radius of the circle, then the area of the quadrant can be calculated as follows:
Given, radius (r) = 8 cm
The area of the quadrant = (1/4)πr^2
= (1/4)π(8)^2
= 16π square cm
= 50.27 square cm
The diameter of a circle is 8 cm. Find its area to the nearest whole number.
Answer:
≈ 50 cm²
Step-by-step explanation:
Use the equation [tex]A= \frac{1}{4} \pi d^{2}[/tex] Where d is the diameter.
Answer:
The answer that you're looking for is approximately 50 (rounded), In terms of π, it is 16π.
Step-by-step explanation:
In order to find the area you need to use the formula: Area = [tex]\pi r^{2}[/tex].
Since The Diameter is double the amount of the radius you need to make sure to divide the diameter by 2 and replace "r" in the equation with the equation given.
8/2 gives you 4. Now you have the equation Area of Circle = [tex]\pi 4^{2}[/tex].
Following the rules of PEMDAS we do exponents since there is no parenthesis.
[tex]4^{2}[/tex] is the same as 16. In terms of pi, you just put pi next to your result giving 16π.
However, if you want to find out normally you can multiply with either 3.14 or π.
Both cases will give you different decimals, but when rounded to the nearest whole number they all give you 50.
Area of Circle = [tex]\pi r^{2}[/tex].
Area of Circle = [tex]\pi 4^{2}[/tex]
Area of Circle = 16π
Area of Circle = 50 (rounded).
I hope this was helpful!
Help me pls
The circumference of a circle is 171 cm.
What is exact the area of the circle?
Do not round. Include correct units.
Show all your work.
Answer:
72.25π cm²
Step-by-step explanation:
C = 2πr
17π cm = 2πr
Divide both sides by 2π.
8.5 cm = r
r = 8.5 cm
A = πr²
A = π × (8.5 cm)²
A = 72.25π cm²
Which table of values is defined by the function y=4-7x
The table of values for the function y=4-7x can be found by choosing any real value of x and we will get real value output from the function. So the table C is found to be correct for the function f(x)=4-7x.
For example, let's choose some values of x and calculate the corresponding values of y:
When x = 0, y = 4 - 7(0) = 4
When x = 1, y = 4 - 7(1) = -3
When x = 2, y = 4 - 7(2) = -10
When x = 3, y = 4 - 7(3) = -17
We can continue this process to find more values of y for different values of x.
The resulting table of values is:
x y
0 4
1 -3
2 -10
3 -17
So, this is the table of values defined by the function y=4-7x.
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The complete question is :
Which table of values is defined by the function y=4-7x
Table A Table B Table C Table D
x y x y x y x y
0 4 0 0 0 4 0 -4
1 3 1 1 1 -3 1 -3
2 10 2 2 2 -10 2 -10
3 17 3 3 3 -17 3 -17
The original price of a chair was $450.00. The tax on the chair was 5.5%. What is the exact price of the chair including tax?
The exact price of the chair including tax is $425.25. The solution has been obtained by using arithmetic operations.
What are arithmetic operations?
The four fundamental operations, also referred to as "arithmetic operations," are said to be able to explain all real numbers. The four mathematical operations following division, multiplication, addition, and subtraction are quotient, product, sum, and difference.
We are given that the original price of a chair was $450.00 and on it, there was tax of 5.5%.
So, using the arithmetic operations, we get
⇒Exact price = 450 - 5.5% of 450
⇒Exact price = 450 - 24.75
⇒Exact price = $425.25
Hence, the exact price of the chair including tax is $425.25.
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a man spent one-eighth of his spare change for a package of cigarettes, three times as much for a meal, and then had eighty cents left. how much money did he have at first?
The man had $1.60 at first.
What is an equation?
An equation is a statement that shows that two expressions are equal. It contains an equals sign "=" and may include variables, constants, coefficients, and mathematical operations such as addition, subtraction, multiplication, and division.
Let's assume the man had x amount of money at first.
Then he spent 1/8x on a package of cigarettes and 3 times as much, or 3/8x, on a meal.
So he spent a total of 1/8x + 3/8x = 1/2x of his money.
If he had 80 cents left, that means he spent x - 0.8.
So we can set up an equation:
1/2x = x - 0.8
Solving for x:
1/2x - x = -0.8
-1/2x = -0.8
x = (-0.8)/(-1/2)
x = 1.6
Therefore, the man had $1.60 at first.
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This is 1/6 problems finish them all each is 10 points 60 total.
The cosine of θ is the ratio of the length of the adjacent side to the length of the hypotenuse
What is the cosine of an angle?The cosine of an angle is a trigonometric function that relates the length of the adjacent side of a right triangle to the length of the hypotenuse. Specifically, it is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.
In mathematical terms, if we have a right triangle where one of the angles is labeled as theta (θ), then the cosine of theta is given by the formula:
cos(θ) = adjacent side / hypotenuse
1) Cos R =30/34 =15/17
Cos S = 16/34 = 8/17
2) Cos R = 24/26 = 12/13
Cos S = 10/26 = 5/13
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NO EXPLANATION JUST ANSWER!
Answer: 364 cm^3
Step-by-step explanation: Please mark brainliest and give thanks!
Answer:
[tex]364cm^3[/tex]
Hope this helps!
Brainliest and a like is much appreciated!
You want to buy a triangular lot measuring 1360 feet by 1850 feet by 2430 feet. The price of the land is $2200 per acre. How much does the land cost? (Hint: 1 acre =43,560 square feet. Round your answer to two decimal places.)
The land will cost approximately $52,800.00.
What is triangle ?
A triangle is a polygon with three sides and three angles. It is one of the simplest and most basic shapes in geometry. The three sides of a triangle can be of different lengths, and the three angles can also be of different sizes. The sum of the angles in a triangle is always 180 degrees.
To find the area of the triangular lot, we can use Heron's formula for the area of a triangle:
s = (1360 + 1850 + 2430)/2 = 2820
A = √[s(s-1360)(s-1850)(s-2430)] ≈ 1,046,482.74 square feet
To convert this to acres, we divide by 43,560:
A ≈ 24.00 acres
Finally, we can calculate the cost of the land by multiplying the area in acres by the price per acre:
cost = 24.00 acres × $2200/acre ≈ $52,800.00
Therefore, the land will cost approximately $52,800.00.
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15 The table shows values of s and t.
S
t
0.2
7.5
0.5
1.4
0.9
Is s inversely proportional to f? Explain why.
(2 marks)
Answer:
s is not inversely proportional to t
Step-by-step explanation:
This is an edited response. My first answer was incorrect.s is not inversely proportional to t. I had responded that they were, based on the fact that as s went up, t went down. But the question was not simply is there an inverse relationship, but are they inversely proportional.
The term proportional means that the relationship between s and t is a constant. That is:
t = s*(1/x)
Let's rewrite that to y*x = k and then check the numbers. See the attached spreadsheet. If the relationship were inversel proportioanl, thaen the product of t*s would be a contant for the series. The third set is different from the first two. The data has an is inverse relationship, but it is NOT proportional.
Find the
coordinates of the points on the graph of
ƒ(x) = ½ x³ − ¹⁄2x² − 8x + 7 where the gradient is 4.
Answer:
(-8/3, 19/27) and (3, -17/2).
Step-by-step explanation:
To find the coordinates of the points on the graph of ƒ(x) = ½x³ − ¹⁄₂x² − 8x + 7 where the gradient is 4, we need to find the points where the derivative of ƒ(x) is equal to 4.
First, we need to find the derivative of ƒ(x):
ƒ'(x) = 3/2x² - x - 8
Next, we need to set ƒ'(x) = 4 and solve for x:
3/2x² - x - 8 = 4
3/2x² - x - 12 = 0
Multiplying both sides by 2 to eliminate the fraction:
3x² - 2x - 24 = 0
Factoring the quadratic equation:
(3x + 8)(x - 3) = 0
So x = -8/3 or x = 3.
Now we can find the corresponding y-coordinates:
When x = -8/3:
ƒ(-8/3) = 1/2(-8/3)³ - 1/2(-8/3)² - 8(-8/3) + 7 = 19/27
So one point on the graph with gradient 4 is (-8/3, 19/27).
When x = 3:
ƒ(3) = 1/2(3)³ - 1/2(3)² - 8(3) + 7 = -17/2
So another point on the graph with gradient 4 is (3, -17/2).
Therefore, the coordinates of the points on the graph of ƒ(x) = ½x³ − ¹⁄₂x² − 8x + 7 where the gradient is 4 are (-8/3, 19/27) and (3, -17/2).
An element wIth mass 420 grams decays by 11.8% per minute. How much of the element is remainifg after 16 minutes, to the nearest 1oth of a gram?
Answer: To calculate the amount of the element remaining after 16 minutes, we can use the formula:
A = P * (1 - r)^t
where:
A = amount remaining after time t
P = initial amount
r = rate of decay per unit time (as a decimal)
t = time elapsed
In this case, we have:
P = 420 grams
r = 0.118 per minute
t = 16 minutes
Substituting these values into the formula, we get:
A = 420 * (1 - 0.118)^16
A ≈ 123.82 grams
Rounding this answer to the nearest tenth of a gram, we get:
A ≈ 123.8 grams
Therefore, approximately 123.8 grams of the element remain after 16 minutes.
For each of the following gas-phase reactions, indicate how the rate of disappearance of each reactant is related to the rate of appearance of each product.
Part A
H2O2(g)→H2(g)+O2(g)
Express your answer as an integer.
−Δ[H2O2]Δt =
Δ[H2]Δt
SubmitMy AnswersGive Up
Part B
Express your answer as an integer.
−Δ[H2O2]Δt = Δ[O2]Δt
SubmitMy AnswersGive Up
Part C
2N2O(g)→2N2(g)+O2(g)
Express your answer as an integer.
−Δ[N2O]Δt = Δ[N2]Δt
SubmitMy AnswersGive Up
Part D
Express your answer as an integer.
−Δ[N2O]Δt = Δ[O2]Δt
SubmitMy AnswersGive Up
Part E
N2(g)+3H2(g)→2NH3(g)
Express your answer using one decimal place.
−Δ[N2]Δt = Δ[NH3]Δt
SubmitMy AnswersGive Up
Part F
Express your answer using one decimal place.
−Δ[H2]Δt = Δ[NH3]Δt
SubmitMy AnswersGive Up
Part G
C2H5NH2(g)→C2H4(g)+NH3(g)
Express your answer as an integer.
−Δ[C2H5NH2]Δt = Δ[C2H4]Δt
SubmitMy AnswersGive Up
Part H
Express your answer as an integer.
−Δ[C2H5NH2]Δt = Δ[NH3]Δt
Part A: −Δ[[tex]H2O2[/tex]]/Δt = Δ[[tex]H2[/tex]]/Δt + Δ[[tex]O2[/tex]]/Δt
Part B: −Δ[[tex]H2O2[/tex]]/Δt = Δ[[tex]O2[/tex]]/Δt
Part C: −Δ[[tex]N2O[/tex]]/Δt = 1/2 Δ[[tex]N2[/tex]]/Δt + Δ[[tex]O2[/tex]]/Δt
Part D: −Δ[[tex]N2O[/tex]]/Δt = Δ[[tex]O2[/tex]]/Δt
Part E: −Δ[[tex]N2[/tex]]/Δt = 1/2 Δ[[tex]NH3[/tex]]/Δt
Part F: −Δ[[tex]H2[/tex]]/Δt = Δ[[tex]NH3[/tex]]/Δt
Part G: −Δ[[tex]C2H5NH2[/tex]]/Δt = Δ[[tex]C2H4[/tex]]/Δt + Δ[[tex]NH3[/tex]]/Δt
Part H: −Δ[tex][C2H5NH2][/tex]/Δt = Δ[tex][NH3][/tex]/Δt
The rate of disappearance of [tex]H2O2[/tex]is equal to the sum of the rates of appearance of H2 and O2.
Part B: −Δ[[tex]H2O2[/tex]]/Δt = Δ[[tex]O2[/tex]]/Δt
The rate of disappearance of [tex]H2O2[/tex] is equal to the rate of appearance of O2.
Part C: −Δ[[tex]N2O[/tex]]/Δt = 1/2 Δ[[tex]N2[/tex]]/Δt + Δ[[tex]O2[/tex]]/Δt
The rate of disappearance of N2O is equal to half the rate of appearance of N2 plus the rate of appearance of O2.
Part D: −Δ[[tex]N2O[/tex]]/Δt = Δ[[tex]O2[/tex]]/Δt
The rate of disappearance of [tex]N2O[/tex] is equal to the rate of appearance of O2.
Part E: −Δ[[tex]N2[/tex]]/Δt = 1/2 Δ[[tex]NH3[/tex]]/Δt
The rate of disappearance of N2 is equal to half the rate of appearance of[tex]NH3.[/tex]
Part F: −Δ[[tex]H2[/tex]]/Δt = Δ[[tex]NH3[/tex]]/Δt
The rate of disappearance of H2 is equal to the rate of appearance of NH3.
Part G: −Δ[[tex]C2H5NH2[/tex]]/Δt = Δ[[tex]C2H4[/tex]]/Δt + Δ[[tex]NH3[/tex]]/Δt
The rate of disappearance of[tex]C2H5NH2[/tex] is equal to the sum of the rates of appearance of[tex]C2H4[/tex] and [tex]NH3.[/tex]
Part H: −Δ[tex][C2H5NH2][/tex]/Δt = Δ[tex][NH3][/tex]/Δt
The rate of disappearance of [tex]C2H5NH2[/tex] is equal to the rate of appearance of [tex]NH3.[/tex]
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what is the probability of rolling a three
Answer:
1/6
Step-by-step explanation:
I am assuming you mean the probability of rolling a 3 on a die.
If you are rolling only 1 die, with the numbers 1 through 6, then the chance of landing on any number is 1/6.
For example, the chance of landing on a 2 or 5 are equal, both 1/6.
Therefore, the chance of landing on a die is 1/6.
Hope this helped!
6m^2-5my-y^2/12m+2y
Simplify the following rations expression and express in expanded form
The simplified expression, expressed in expanded form, is (3m - y)/(12m + 2y).
To simplify the expression (6m² - 5my - y²)/(12m + 2y), we can factor the numerator and denominator, if possible, and then simplify the expression by canceling out common factors.
The numerator can be factored as follows:
6m² - 5my - y² = (3m - y)(2m + y)
The denominator can also be factored by factoring out a common factor of 2:
12m + 2y = 2(6m + y)
Now we can substitute these factorizations back into the original expression:
(6m² - 5my - y²)/(12m + 2y) = [(3m - y)(2m + y)]/[2(6m + y)]
We can now cancel out the common factor of (2m + y) in the numerator and denominator:
[(3m - y)(2m + y)]/[2(6m + y)] = (3m - y)/(2(6m + y))
Expanding this expression, we get:
(3m - y)/(2(6m + y)) = (3m - y)/(12m + 2y)
Therefore, the simplified expression, in expanded form, can be written as (3m - y)/(12m + 2y).
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A jogger is running at 8m/s and their mass is 64kg. What is their momentum?
The jogger's momentum is 512 kg·m/s.
The momentum of an object is defined as the product of its mass and velocity.
Mathematically, momentum (p) can be expressed as:
p = m x v
where m is the mass of the object and v is its velocity.
Momentum is an important concept in physics because it describes the quantity of motion an object possesses. The momentum of an object can be changed by applying a force to it for a certain period of time. This change in momentum is called impulse and is equal to the force multiplied by the time it acts on the object.
In this case, the jogger's mass is given as 64 kg and their velocity is given as 8 m/s. To find their momentum, we can simply multiply these values:
p = m x v
p = 64 kg x 8 m/s
p = 512 kg*m/s
Therefore, the momentum of the jogger is 512 kg*m/s.
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Emery bought 3 cans of beans that had a total weight of 2. 4 pounds. If each can of beans weighed the same amount, which
model correctly illustrates the relationship? Check all that apply
Answer:
You dis not add the option to check all that apply
A sandbox is shaped like a regular hexagon. The side lengths are 3 ft and the apothem 33√ ft.
What is the area of the hexagonal sandbox?
Enter your answer as a decimal to the nearest hundredth
The area of the regular hexagon shaped sandbox is found to be about 28.38 ft².
The sandbox is of the shape of regular hexagon, the area of the hexagon is given by the formula, 3√3a²/2, where, a is the side of the hexagon, the side of the hexagon is give to be 3 ft and the apothem is 3√3 ft.
Now, putting the value in the formula,
Area = 3√3(3)²/2
Area = 23.38 ft²
So, the area of the sandbox is found to be 23.38 ft².
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Find the distance between points A (2, 4) and B (-4, 0).
distance A B =
Answer:
7.21 units-------------------------
Use the distance formula to find the distance between two points.
[tex]d=\sqrt{(y_2-y_1)^2+(x_2-x_1)^2}[/tex]Substitute coordinates and find the length of AB:
[tex]AB=\sqrt{(0-4)^2+(-4-2)^2}=\sqrt{16+36}=\sqrt{52} =7.21[/tex]Question :-
Find the distance between points A(2, 4) and B(-4, 0).Answer :-
The distance between the two points is 7.21 units.[tex] \rule{200pt}{3pt}[/tex]
Solution :-
As per the provided information in the given question, we have been given that :-
[tex](x_1, y_1) = (2, 4)[/tex][tex](x_2, y_2) = (-4, 0)[/tex]To calculate the distance between the two points, we will apply the formula below :-
[tex] \bigstar \: \: \: \boxed{ \sf{ \: \: AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \: \: }}[/tex]
Substitute the given values into the above formula and solve for AB :-
[tex]\sf:\implies{ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}}[/tex]
[tex]\sf:\implies{AB = \sqrt{( - 4 - 2)^2 + (0 - 4)^2}}[/tex]
[tex]\sf:\implies{AB = \sqrt{(-6)^2 + (-4)^2}}[/tex]
[tex]\sf:\implies{AB = \sqrt{36 + 16}}[/tex]
[tex]\sf:\implies\bold{AB = \sqrt{52} \approx 7.21 \: units}[/tex]
Therefore :-
The distance between the two points is 7.21 units.[tex]\\[/tex]
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Turner needs to buy a bathroom mirror that is 4 feet wide and 5 feet long. If the mirror sells for $0.49 per square foot, what will the total cost of the mirror be?
The total cost of the mirror based on its area is obtained as $9.80.
What is area?
An object's area is how much space it takes up in two dimensions. It is the measurement of the quantity of unit squares that completely cover the surface of a closed figure.
The area of the bathroom mirror can be found by multiplying its length and width -
Area = Length × Width
Area = 5 feet × 4 feet
Area = 20 square feet
Since the mirror sells for $0.49 per square foot, we can find the total cost of the mirror by multiplying the area of the mirror by the cost per square foot -
Total Cost = Area × Cost per square foot
Total Cost = 20 square feet × $0.49 per square foot
Total Cost = $9.80
Therefore, the total cost of the bathroom mirror will be $9.80.
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Using the biased wheel, "Tisch 1", it was determined that the probability for one of the numbers was about 0.03776, which is higher than normal. Suppose you bet on this number for 36 rounds. Use this probability to fill in the blanks in the biased wheel column. (Round your answers to four significant figures.) X, the Number of Winning Rounds Net Profit from X Wins Probability of X Wins with Biased Wheel 0 −$36 1 $0 2 $36 3 $72 ... ... ... 36 $1,260
Probability of X Wins with Biased Wheel: 0.0378
X, the Number of Winning Rounds: 36
Net Profit from X Wins: $1,260
Probability of X Wins with Biased Wheel: 0.0378 (rounded to 4 significant figures)
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A pharmaceutical company has developed a new drug to reduce cholesterol. A regulatory agency will recommend the new drug for use if there is convincing evidence that the mean reduction in cholesterol level after one month of use is more than 20 milligrams/deciliter (mg/dl), because a mean reduction of this magnitude would be greater than the mean reduction for the current most widely used drug.
The pharmaceutical company collected data by giving the new drug to a random sample of 50 volenteers having high cholestrol. The reduction in cholestrol level after one month was recorded for each individual, resulting in a sample mean reduction of 24 mg/dl and a standard deviation of 15 mg/dl.
(a) The regulatory agency decided o use a confidence interval estimate for the population mean reduction in cholestrol level for the new drug. Provide a 95% confidence inerval for the mean reduction in cholestrol level
A 95% confidence interval for the population mean reduction in cholesterol level is (19.78, 28.22) mg/dl, based on a sample mean of 24 mg/dl and a standard deviation of 15 mg/dl.
Using the information, we can compute a confidence interval for the population mean decrease in cholesterol level as follows:
1. Calculate the standard error of the mean:
standard error = standard deviation/sqrt(sample size)
= 15/sqrt(50)
= 2.12
2. Calculate the margin of error using a t-distribution with (n-1) degrees of freedom at 95% confidence level:
margin of error = t_(n-1, 0.025) * standard error
= t_(49, 0.025) * 2.12 (using a t-table)
= 2.009 * 2.12
= 4.26
3. Calculate the confidence interval by subtracting and adding the margin of error to the sample mean:
CI = sample mean +/ - margin of error
= 24 +/ - 4.26
= (19.74, 28.26)
In this manner, we can say with 95% certainty that the true mean reduction in cholesterol level following one month of drug of the new medication is between 19.74 mg/dl and 28.26 mg/dl. Since the lower limit reaches the confidence interval (19.74 mg/dl) is greater than 20 mg/dl, we can reason that there is persuading proof that the new medication is powerful in lessening cholesterol level following one month of purpose.
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What is the equation of the line that passes through the point (-4, 2) and has a
slope of -1?
Answer:
Step-by-step explanation:
The equation of a line in slope-intercept form is y = mx + b, where m is the slope of the line and b is the y-intercept.
We are given that the line passes through the point (-4, 2) and has a slope of -1. This means that we can substitute the values of the point and slope into the equation and solve for b.
y = mx + b
2 = (-1)(-4) + b
2 = 4 + b
b = -2
Now we know the slope and y-intercept of the line, so we can write the equation in slope-intercept form:
y = -x - 2
Therefore, the equation of the line that passes through the point (-4, 2) and has a slope of -1 is y = -x - 2.
Find the volume of the sphere.
Either enter an exact answer in terms of π or use 3.14 for π and round your
final answer to the nearest hundredth.
4
units3
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The required volume of the given sphere is 904.32 cm³.
What is a sphere?A sphere is a geometrical object that resembles a two-dimensional circle in three dimensions.
In three-dimensional space, a sphere is a collection of points that are all located at the same distance from a single point.
The radius of the sphere is denoted by the letter r, and the specified point represents its center.
All of the points on a circle are equally spaced apart from the center along a plane, but all of the points on a sphere are equally spaced apart from the center along any of the axes.
So, we must ascertain the sphere's volume. With a radius of 6 cm, we have:
V = 4/3 π r^3
V = 4/3 x 3.14 x 6^3
V = 4/3 x 3.14 x 216
V = 904.32
Therefore, the required volume of the given sphere is 904.32 cm³.
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Complete question:
Find the volume of the sphere.
Either enter an exact answer in terms of \piπpi or use 3.143.143, point, 14 for \piπpi and round your final answer to the nearest hundredth.