Let V be a finite-dimensional vector space and let S1 and S2 be two bases of V. To show that S1 and S2 have the same number of elements, we will assume, without loss of generality, that S1 has more elements than S2, i.e., n > m. We will then derive a contradiction.
Since S1 is a basis of V, every vector in V can be expressed as a linear combination of the vectors in S1. In particular, for each j = 1, 2, ..., m, we can express βj as a linear combination of the vectors in S1:
βj = c1,jα1 + c2,jα2 + ... + cn,jαn
where c1,j, c2,j, ..., cn,j are scalars. We can write this in matrix form as
| β1 | | c1,1 c1,2 ... c1,m | | α1 |
| β2 | | c2,1 c2,2 ... c2,m | | α2 |
| ... | = | ... ... ... ... | * | ... |
| βm | | cm,1 cm,2 ... cm,m | | αn |
where the matrix on the right is the matrix whose columns are the vectors in S1, and the matrix on the left is the matrix whose columns are the vectors in S2.
Since S2 is also a basis of V, the matrix on the left is invertible. Therefore, we can multiply both sides of the equation by the inverse of the matrix on the left, giving
| α1 | | b1,1 b1,2 ... b1,m | | β1 |
| α2 | | b2,1 b2,2 ... b2,m | | β2 |
| ... | = | ... ... ... ... | * | ... |
| αn | | bn,1 bn,2 ... bn,m | | βm |
where b1,j, b2,j, ..., bn,j are scalars.
Now consider the determinant of the matrix on the left-hand side of this equation. Since this matrix is obtained by multiplying the matrix whose columns are the vectors in S2 by the inverse of the matrix whose columns are the vectors in S1, its determinant is equal to the product of the determinants of these two matrices:
det([α1 α2 ... αn]) * det([β1 β2 ... βm]^-1) = det([α1 α2 ... αn] [β1 β2 ... βm]^-1)
The left-hand side is nonzero, since S1 and S2 are both bases of V and therefore their vectors are linearly independent, so the determinant of each matrix is nonzero. However, the right-hand side is zero, since the product of the two matrices on the right-hand side is the identity matrix, and the determinant of the identity matrix is 1.
This is a contradiction, so our assumption that S1 has more elements than S2 must be false. Therefore, S1 and S2 have the same number of elements, and the proof is complete.
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A pair of athletic shoes costs $85. If the inflation rate remains constant at 9%, write an algebraic rule to determine the cost, c(t), of the
shoes after t years.
c(t) =
(Simplify your answer. Use integers or decimals for any numbers in the expression.)
Answer:
c(t) = $85 × (1.09)^t
Step-by-step explanation:
Assuming that the inflation rate remains constant at 9% per year, the cost of the athletic shoes after t years can be determined by multiplying the original cost by the inflation factor for t years, which is given by (1 + 0.09)^t, or 1.09^t. Therefore, the algebraic rule to determine the cost, c(t), of the shoes after t years is:
c(t) = $85 × 1.09^t
Simplifying the expression, we get:
c(t) = $85 × (1.09)^t
where t is the number of years after the original purchase.
please help, i will give brainliest! please also explain how you got the answer!
Answer: 7.59
Step-by-step explanation: I added all of the miles he traveled up together. ( don't overthink questions like this)
green swam 2 laps every morning for 7 days.in addition to the laps he swam each morning, he swam 3 laps with his friends one Tuesday and Thursday.
The expression that shows the number of laps Glenn swam during the week is 2 * 7 + 3(2) = 20.
What distinguishes an equation from an expression?A mathematical expression is a grouping of numbers, variables, and operations without the equal sign. It may be condensed or given a single value. On the other hand, an equation is a declaration that employs the equal sign to demonstrate the equality of two expressions. To get the value of the variable that makes an equation true, equations can be solved.
Given that, 2 laps every morning for 7 days:
2 * 7
3 laps with his friends one Tuesday and Thursday.
3(2)
Total laps = 2( 7) + 3(2) = 20
Hence, expression that shows the number of laps Glenn swam during the week is 2 * 7 + 3(2) = 20.
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I need help with that
Answer:
Step-by-step explanation:
Multiply each then divide by the first number you started off with.
quation of the circle centered at (-4,3) with radius 3. Fully simplify the equation.
The equation of the circle centered at (-4,3) with radius 3 is (x + 4)^2 + (y - 3)^2 = 9. Simplifying this equation means to simplify the terms in the equation by using the Pythagorean Theorem. The Pythagorean Theorem states that in a right triangle, the sum of the squares of the two sides is equal to the square of the hypotenuse.
In this equation, the hypotenuse is the radius, which is 3, and the sides are (x + 4) and (y - 3). Substituting the values into the Pythagorean Theorem, we get (x + 4)^2 + (y-3)^2 = 9, which is the equation of the circle centered at (-4,3) with radius 3. This equation can be simplified further by factoring the terms, which yields (x + 4) (x + 4) + (y - 3) (y - 3) = 9. This is the fully simplified equation of the circle.
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Demonstrate, using induction, that each of the equations
corresponding to the subsections are true for all n:
1) P(????) : 2 + 4 + 6 + ⋯+ 2????= ????(????+ 1), ∀ ????∈ℕ.
2) ∑????????= 1 �
This equation is true, so the equation is true for n = k+1. Therefore, by induction, the equation is true for all n. To demonstrate the equations using induction, we need to follow three steps: the base case, the induction hypothesis, and the induction step.
1) P(n) : 2 + 4 + 6 + ⋯+ 2n= n(n+ 1), ∀ n∈ℕ.
Base case: For n = 1, the equation becomes 2 = 1(1+1), which is true.
Induction hypothesis: Assume the equation is true for n = k, that is, 2 + 4 + 6 + ⋯+ 2k= k(k+ 1).
Induction step: We need to show that the equation is also true for n = k+1. That is, 2 + 4 + 6 + ⋯+ 2k + 2(k+1)= (k+1)(k+2).
Using the induction hypothesis, we can substitute k(k+1) for 2 + 4 + 6 + ⋯+ 2k:
k(k+1) + 2(k+1) = (k+1)(k+2)
Distributing the (k+1) on the right side of the equation gives us:
k(k+1) + 2(k+1) = k(k+1) + 2(k+1)
This equation is true, so the equation is true for n = k+1. Therefore, by induction, the equation is true for all n.
2) ∑i^2= n(n+1)(2n+1)/6, ∀ n∈ℕ.
Base case: For n = 1, the equation becomes 1^2 = 1(1+1)(2(1)+1)/6, which simplifies to 1 = 1, which is true.
Induction hypothesis: Assume the equation is true for n = k, that is, ∑i^2= k(k+1)(2k+1)/6.
Induction step: We need to show that the equation is also true for n = k+1. That is, ∑i^2 + (k+1)^2 = (k+1)(k+2)(2(k+1)+1)/6.
Using the induction hypothesis, we can substitute k(k+1)(2k+1)/6 for ∑i^2:
k(k+1)(2k+1)/6 + (k+1)^2 = (k+1)(k+2)(2(k+1)+1)/6
Multiplying both sides of the equation by 6 gives us:
k(k+1)(2k+1) + 6(k+1)^2 = (k+1)(k+2)(2(k+1)+1)
Distributing the (k+1) on both sides of the equation gives us:
k(k+1)(2k+1) + 6(k+1)^2 = k(k+1)(k+2)(2k+3)
This equation is true, so the equation is true for n = k+1. Therefore, by induction, the equation is true for all n.
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Which equation represents the formula for the general term, an, of an arithmetic sequence when a1=3 and a11=23
Answers:
an=3+2(n-1)
an=3+23(n-1)
an=23+10(n-1)
an=23+2(n-1)
Therefore , the solution of the given problem of arithmetic mean comes out to be option A an=3+2(n-1).
Arithmetic mean : What is it?The usual values of a list are determined by adding up every single integer one of the items on the list, which are frequently referred to as the organization's objectives. The number of list items with the highest abundance is then used to adjust these average values. Mathematical growth trends are similar. The number 21 turns into seven by adding three to the true means of the digits 5, 7, and 9, which is 4.
Here,
The following is the formula for the general word "an" in an arithmetic sequence:
=> a = a1 + (n - 1)d
where the first term is denoted by a1, the term number is n, and the common difference is d.
A1 = 3 and A11 = 23 are provided. To determine the common difference, we can use the method for the eleventh term:
=> a11 = a1 + (11 - 1)d
=> 23 = 3 + 10d
=> d = 2
Inputting this number of d into the general term's formula yields the following results:
=> a = 3 + (n - 1)2
The right response is therefore: an=3+2(n-1).
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QUESTION 5 How many ways are there to construct a 3-digit code if numbers can be repeated?
There are 1,000 ways to construct a 3-digit code if numbers can be repeated. This is because there are 10 possible numbers (0-9) for each digit, and since numbers can be repeated, each digit has 10 options. So the total number of ways to construct a 3-digit code is 10 × 10 × 10 = 1,000.
Here is a step-by-step explanation:
1. Start with the first digit. There are 10 possible numbers (0-9) that can be used for this digit.
2. Move on to the second digit. Again, there are 10 possible numbers (0-9) that can be used for this digit, since numbers can be repeated.
3. Finally, move on to the third digit. There are 10 possible numbers (0-9) that can be used for this digit, since numbers can be repeated.
4. Multiply the number of options for each digit together to get the total number of ways to construct a 3-digit code: 10 × 10 × 10 = 1,000.
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How much water was in the cylinder before any marbles were dropped in
It is one of the most fundamental curvilinear geometric shapes, a cylinder has historically been thought of as a three-dimensional solid. It is considered a prism with a circle as its base in basic geometry.
What is Geometry?the area of mathematics that focuses on the characteristics and connections between points, lines, surfaces, solids, and their higher dimensional equivalents.
We can find the solution this way,
The volume of water in the Cylinder before marbles were dropped in =
The volume of water in the Cylinder After marbles were dropped in -
Volume of marbles
So, we can write it as,
Volume of water before = Volume of water after - Volume of Marbles
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A
man will go on a trip for 3 days, so he will take with him 3
shirts, if he has 7 shirts, how many combination of shirts can he
take, without repetition.
The man has 7 shirts in total, so he can take 3 of those shirts on his 3-day trip without repetition. This means that he can make 7 different combinations of shirts, since each shirt has only one choice. For example, he could take shirts A, B, and C; or he could take shirts D, E, and F; or he could take shirts G, A, and B. In total, he has 7 different combinations of shirts to choose from.
The number of combinations of shirts that the man can take without repetition can be found using the formula for combinations, which is:
C(n, r) = n! / (r! * (n-r)!)
In this case, n = 7 (the total number of shirts) and r = 3 (the number of shirts he will take with him).
Plugging in these values into the formula, we get:
C(7, 3) = 7! / (3! * (7-3)!)
C(7, 3) = 7! / (3! * 4!)
C(7, 3) = 5040 / (6 * 24)
C(7, 3) = 5040 / 144
C(7, 3) = 35
Therefore, the man can take 35 different combinations of shirts without repetition.
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Enter the value of n for the equation 35.3″ = 38.
The value of n for the given equation 3⁵. 3ⁿ = 3⁸ is n = 3.
What are laws of exponents?Several rules of exponents are presented according to the capacities they possess. The following rule governs multiplication: Add the exponents while maintaining the base's consistency.
When bases are raised by a power of two or more, multiply the exponents while maintaining the original base.
Division Rule: When dividing similar bases, take the exponent of the denominator and divide it by the exponent of the numerator, keeping the base constant.
The given equation is:
3⁵. 3ⁿ = 3⁸
Using the product rule for exponents we have:
3⁽⁵ ⁺ ⁿ⁾ = 3⁸
The above expression can be written as:
5 + n = 8
n = 8 - 5
n = 3
Hence, the value of n for the given expression 3⁵. 3ⁿ = 3⁸ is n = 3.
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The complete question is:
2\34 The log cabin fire burns for the least amount of time It burns for 1/4 of a day. The average fire burns for times this length. What fraction of a day does the average fire burn?
The fraction of a day that the average fire burns is given as follows:
100%.
How to obtain the fraction?The fraction of a day that the average fire burns is obtained applying the proportions in the context of the problem.
For the log cabin, the fraction is given as follows:
1/4.
The average fire burns four times this length, hence the length is given as follows:
4 x 1/4 = 4/4 = 1 = 100% of a day.
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Help me pleaseeeeeeee
write the equation with both of them = 100
(3x + 50) + (2x + 15) = 100
combine like terms
5x + 65 = 100
subtract
5x = 35
x = 7
the question asks for <JTU = 2x + 15
plug in x = 7
x = 29
EDIT
Answer:
JTU = 29°
Step-by-step explanation:
I'm not sure if this is correct but I got 29°
(3x+50)° + (2x+15)° = 100°
x= 7°
(2x+15)°
((2x7)+15)° = 29°
please I need a solution
obtain the solution using the long division method
Three numbers form a gp. If the first and third numbers are 5 and 245 respectively, find two possible values for the middle number. Showing the workings
The two possible values for the middle numbers in the geometric progression are +35 and -35.
We know that the three numbers form a geometric progression. Let the middle number be x:
5, x, 245
Since these numbers form a geometric progression, we know that:
x^2 = 5 * 245
So we have:
x = ± √(5 * 245)
x = ± 35
Therefore, the two possible values for the middle number are +35 and -35. To find the two possible values for the middle number in the geometric progression, we used the fact that the product of the first and third terms of a geometric progression is equal to the square of the second term.
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The coach for the Lady Bugs basketball team kept track of the scores of their games. Lady Bugs Game Scores
57
50
57
53
53
62
57
What is the range of the scores of the games?
A. 62
B. 50
C. 57
D. 12
The range of the scores of the basketball games played by the Lady Bugs basketball team is equal to option D. 12.
Scores of the games played by Lady Bugs basketball team is equal to
57, 50, 57, 53, 53, 62, 57
Arrange the scores of the team in ascending order we get,
50, 53, 53, 57, 57, 57, 62
Highest score of the team = 62
Lowest score of the team = 50
Range = highest score - lowest score
Substitute the value in the formula we get,
⇒ Range = 62 - 50
⇒ Range = 12
Therefore, the range of the score of the game played by basketball team is equal to option D. 12.
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Will a line passes through (2,2)if it is intersects the axes (2,0)and (0,2)
A line intersecting at the axes (2,0)and (0,2) will not pass through (2,2).
Given, a line intersects at the axis (2,0)and (0,2)
let the line intercept be expressed as
[tex]ax+by=1[/tex] where a and b are the x & y intercept.
since the intercept points are the axis (2,0)and (0,2)
a=2 and b=2
[tex]2x+2y=1[/tex]
when the point (2,2) is considered and put in equation
2(2)+2(2)=4≠1
Therefore, point (2,2) doesn't satisfy the equation and line doesn't pass through (2,2).
From the graph also, we can say that the line passing through (2,0) and (0,2) intersecting the axes do not pass through the point (2,2).
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A parabola has x-intercepts -2 and -8, and has vertex (-5,-18). Determine the equation of this parabola in the form y=a(x-r)(x-s)
The equation of the parabola in the form y = a(x - r)(x - s) is y = -2(x + 2)(x + 8).
To determine the equation of the parabola in the form y = a(x - r)(x - s), we need to find the values of a, r, and s. We are given the x-intercepts and the vertex, so we can use this information to find these values.
The x-intercepts are -2 and -8, so we know that r=-2 and s=-8. The vertex is (-5,-18), so we can plug these values into the equation and solve for a:
y = a(x - r)(x - s)
-18 = a(-5 + 2)(-5 + 8)
-18 = a(-3)(3)
-18 = 9a
a = -2
So the equation of the parabola is y = -2(x + 2)(x + 8).
In standard form, this equation is y = -2x² - 20x - 32.
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Quinn has 21 coins in his pocket, all of which are dimes and quarters. If the total value of his change is 435 cents, how many dimes and how many quarters does he have?
Quinn has ___ dimes and ___ quarters in his pocket.
Quinn has 6 dimes and 15 quarters in his pocket.
Let's use a system of two equations to represent the given information:
d + q = 21 (1) // The number of dimes and quarters adds up to 21
10d + 25q = 435 (2) // The total value of the coins in cents is 435
where d represents the number of dimes and q represents the number of quarters.
We can use equation (1) to solve for d in terms of q:
d = 21 - q
Substituting this expression for d into equation (2), we get:
10(21 - q) + 25q = 435
210 - 10q + 25q = 435
15q = 225
q = 15
So, Quinn has 15 quarters. Substituting this value into equation (1), we get:
d + 15 = 21
d = 6
So, Quinn has 6 dimes.
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Someone please answer my question
Answer:
Point D is the solution to the system of equations.
Step-by-step explanation:
When asked for a solution involving 2 equations, the goal is to find a point (x,y) that would be a solution to both equations. Any point on a line that is defined by an equation is a solution to that equation. For an equation of y = 2x + 2, possible solutions are (1,4), (2,6), (5,12) etc. These points all lie on the line formed by that equation. There are an infinite number of possible solutions. If a second eqaution is added, there is now a constraint on the possible answers. The goal is to find a point that satisfies both equations.
If a seond equation of y = 1x + 3 were matced with y=2x+2, both are straight lines, but with different slopes. So they will intersect at some point. One may either solve mathematically using substitution, or by graphing, as was done here.
Matematically:
y = 2x + 2
y = 1x + 3
Rearrange either equation to isolate a variable, x or y. These are already isolated (since I made them up) so go to the next step of substituting one expression of y into the other:
y = 1x + 3
2x + 2 = 1x + 3
x = 1
Now use this value of x to find y:
y = 2x + 2
y = 2*(1) + 2
y = 4
The point these two lines intersect is (1,4) and is the "solution" to this series of equations.
See the attached graph.
The Lions won 35 out of 40 games this season.
1) What fraction of games played did the Lions win? Write your answer in simplest form.
2) Write a decimal and a percent equivalent to the fraction of games the Lions won. Show your work.
Answer:
35/40 = (7*5)/(8*5) = 7/8
So the Lions won 7/8 of the games they played.
2) To write a decimal equivalent to 7/8, we can divide 7 by 8 using long division:
0.875
___________
8 | 7.000
- 6.4
_______
60
-56
____
40
-40
____
0
So 7/8 is equivalent to the decimal 0.875.
To write a percent equivalent to 7/8, we can multiply the decimal equivalent by 100:
0.875 * 100 = 87.5%
So the Lions won 87.5% of the games they played.
1. 1 8 Use the formula SA = 2πrh + 2πr2 to find is the surface area of the cylindrical food storage container. Use 3. 14 for π. Round your answer to the nearest hundredth of a square inch
The Surface Area of Container is 954. 56 inch².
What is Surface Area?The area is the area occupied by a two-dimensional flat surface. It has a square unit of measurement. The surface area of a three-dimensional object is the space taken up by its outer surface. Square units are used to measure it as well.
For each three-dimensional geometrical shape, surface area and volume are determined. The area or region that an object's surface occupies is known as its surface area.
As, we Know the Surface Area of Cylinder
= 2πr² + 2πrh
Radius of the base = 8 inches
Height of the cylinder = 11 inches.
Now, putting the values we get
= 2πr² + 2πrh
= 2 x 3.14 x 8 x 8 + 2 x 3.14 x 8 x 11
= 401.92 + 552.64
= 954. 56 inch²
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Write an explanation on why you woukd lerfer to use Radians or
Degrees. List your pros and cons to each ones.
write an explanation on why uou would perfer to use Radians or
Degrees ? list pros and c
Radians and degrees are two units of measurement for angles. Each one has its own pros and cons, and the choice between the two largely depends on the situation and personal preference.
Radians:
Pros:
- Radians are the natural unit of measurement for angles in mathematics and physics. They are closely related to the concept of a circle, and many formulas and equations involving angles are simpler when using radians.
- Radians are unitless, which can make calculations easier and more intuitive.
Cons:
- Radians are not as familiar to most people as degrees, and can be harder to visualize and understand.
- Radians can be more difficult to work with when measuring small angles, since they use fractions or decimals instead of whole numbers.
Degrees:
Pros:
- Degrees are the most commonly used unit of measurement for angles, and are more familiar to most people.
- Degrees are easier to work with when measuring small angles, since they use whole numbers instead of fractions or decimals.
Cons:
- Degrees are not the natural unit of measurement for angles, and many formulas and equations involving angles are more complicated when using degrees.
- Degrees require the use of a special symbol (°) and are not unitless, which can make calculations more difficult and less intuitive.
Overall, the choice between radians and degrees deends on the situation and personal preference. Radians are generally more natural and simpler to work with in mathematics and physics, but degrees are more familiar and easier to visualize.
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Complete the statement.
y=cos −1
x means that x= for 0≤y≤π
y=cos −1
x means that x=, for 0≤y≤π
This statement means that the inverse cosine function, cos −1x, gives the angle y whose cosine is x, within the range of 0≤y≤π.
Complete the statement. y=cos −1x means that x=cos y, for 0≤y≤π
This statement means that the inverse cosine function, cos −1x, gives the angle y whose cosine is x, within the range of 0≤y≤π. In other words, if we know the value of x, we can use the inverse cosine function to find the angle y that has a cosine of x. This is useful for solving equations involving cosine, such as finding the angles of a triangle given the lengths of its sides.
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A triangle can be formed into a parallelogram as shown in the diagram below. Which equation can be used to find the area of the triangle in the diagram?
F.A = 4⋅6
G.A = 6÷2
H.A = 12
(2⋅6
)
J.A = 12
(4⋅6
)
Answer: J. A = 12 (4⋅6).
Step-by-step explanation:
Answer: J. A = 12 (4⋅6).
This equation can be used to find the area of the triangle in the diagram because it uses the formula for the area of a triangle, which is A = 1/2 * b * h, where b is the base and h is the height. Since the triangle in the diagram has a base of 4 and a height of 6, the equation A = 12 (4⋅6) can be used to find the area.
Answer:
The diagram is not provided, so it's difficult to determine the exact dimensions of the triangle and parallelogram. However, we can make some general observations to determine which equation can be used to find the area of the triangle.
First, we know that the area of a triangle is given by the formula:
A = 1/2 * base * height
We also know that the area of a parallelogram is given by the formula:
A = base * height
In the diagram, the triangle can be formed into a parallelogram by taking one of its sides and using it as the base of the parallelogram. The height of the parallelogram is the same as the height of the triangle.
Based on these observations, we can conclude that the equation that can be used to find the area of the triangle is:
A = 1/2 * base * height
where the base is one of the sides of the triangle, and the height is the height of the parallelogram (which is the same as the height of the triangle).
None of the answer choices provided match this equation, so the correct answer is not given.
Rotate the given figures around the origins as indicated above the graph.
A graph of the resulting image after the given triangle is rotated 90° counterclockwise about the origin is shown in the image below.
What is a rotation?In Geometry, the rotation of a point 90° about the center (origin) in a counterclockwise (anticlockwise) direction would produce a point that has these coordinates (-y, x).
By applying a rotation of 90° counterclockwise to the vertices of triangle ABC, the coordinates of the vertices of the image are as follows:
(x, y) → (-y, x)
Ordered pair A = (9, 3) → Ordered pair A' = (-(3), 9) = (-3, 9).
Ordered pair B = (3, 0) → Ordered pair B' = (-(0), 3) = (0, 3).
Ordered pair C = (8, 0) → Ordered pair C' = (-(0), 8) = (0, 8).
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You buy a new smartphone for $700 and sell it 2 years later for $185. Assume that the resale value of the smartphone decays exponentially with time. Write an equation that represents the resale value V (in dollars) of the smartphone as a function of the time t (in years) since it was purchased.
Step-by-step explanation:
The equation that represents the resale value V (in dollars) of the smartphone as a function of the time t (in years) since it was purchased is:
V = 700e^(-0.2t)
The equation of the plane passing through the origin, containing the vectors \( \left[\begin{array}{c}1 \\ -1 \\ 1\end{array}\right] \) and \( \left[\begin{array}{l}1 \\ 0 \\ 3\end{array}\right] \) ha
(-3x + 2y + z = 0 )
The equation of the plane passing through the origin and containing the vectors \( \left[\begin{array}{c}1 \\ -1 \\ 1\end{array}\right] \) and \( \left[\begin{array}{l}1 \\ 0 \\ 3\end{array}\right] \) can be found by taking the cross product of the two vectors and using the resulting vector as the normal vector of the plane. The cross product of the two vectors is given by:
\( \left[\begin{array}{c}1 \\ -1 \\ 1\end{array}\right] \times \left[\begin{array}{l}1 \\ 0 \\ 3\end{array}\right] = \left[\begin{array}{c}(-1)(3) - (1)(0) \\ (1)(3) - (1)(1) \\ (1)(0) - (1)(-1)\end{array}\right] = \left[\begin{array}{c}-3 \\ 2 \\ 1\end{array}\right] \)
Therefore, the equation of the plane is given by:
\( -3x + 2y + z = 0 \)
This is the equation of the plane passing through the origin and containing the two given vectors.
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Solve the following trigonometric equation on the interval
[0,2π).Express your answers in exact form if possible. Otherwise,
round to two decimal places.2 cos2θ+ 5 cosθ+ 2 = 0
The solutions to the given trigonometric equation on the interval [0,2π) are θ = 2π/3 and θ = 4π/3. These values can also be expressed in decimal form as θ = 2.09 and θ = 4.19, rounded to two decimal places
To solve the given trigonometric equation on the interval [0,2π), we need to use the quadratic formula.
First, let us rewrite the equation in terms of x:
2x^2 + 5x + 2 = 0
Next, we can apply the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)
Plugging in the values from the equation:
x = (-(5) ± √((5)^2 - 4(2)(2)))/(2(2))
Simplifying:
x = (-5 ± √(25 - 16))/4
x = (-5 ± √9)/4
x = (-5 ± 3)/4
This gives us two possible values for x:
x = (-5 + 3)/4 = -2/4 = -0.5
x = (-5 - 3)/4 = -8/4 = -2
Now we need to convert these values back to θ by using the inverse cosine function:
θ = cos^-1(-0.5)
θ = cos^-1(-2)
The first value, θ = cos^-1(-0.5), gives us two solutions on the interval [0,2π):
θ = 2π/3 and θ = 4π/3
The second value, θ = cos^-1(-2), does not give us any solutions on the interval [0,2π) because the cosine function only takes on values between -1 and 1.
Therefore, the solutions to the given trigonometric equation on the interval [0,2π) are θ = 2π/3 and θ = 4π/3. These values can also be expressed in decimal form as θ = 2.09 and θ = 4.19, rounded to two decimal places.
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Marcus is painting all the walls in his apartment which she estimated to be a total of 1450 ft.² he computered that he has finished with 60% of the painting approximately how many square feet of of wall space or me remain to be painting
Answer:
580
Step-by-step explanation:
He painted 60/100 x 1450 = 870
Unpainted = 1450-870
Remaining = 580ft²