Eigenvalues and eigenvectors play a crucial role in the study of linear transformations and matrices. For a normal matrix A, it can be proven that eigenvectors corresponding to different eigenvalues are necessarily orthogonal.
To understand why eigenvectors corresponding to different eigenvalues are orthogonal for a normal matrix, we need to consider the properties of normal matrices. A matrix A is normal if it commutes with its conjugate transpose A* (i.e., A * A* = A* A).
Now, let's consider two eigenvectors v₁ and v₂ corresponding to different eigenvalues λ₁ and λ₂, respectively. We want to show that v₁ and v₂ are orthogonal, meaning their dot product is zero (v₁ · v₂ = 0).
Let's denote the conjugate transpose of A as A*, and the eigenvalues and eigenvectors as follows:
A * A = A * A* (1)
Multiplying both sides of equation (1) by v₂* (the conjugate transpose of v₂) from the left gives:
v₂* A * A = v₂* A * A* (2)
Since v₂ is an eigenvector of A, we can express it as:
A * v₂ = λ₂ v₂ (3)
Substituting equation (3) into equation (2) gives:
v₂* λ₂ A = v₂* A * A* (4)
Now, let's multiply equation (4) by v₁ from the right:
v₂* λ₂ A v₁ = v₂* A * A* v₁ (5)
Since v₁ is an eigenvector of A, we can express it as:
A * v₁ = λ₁ v₁ (6)
Substituting equation (6) into equation (5) gives:
v₂* λ₂ λ₁ v₁ = v₂* λ₁ A* v₁ (7)
Notice that λ₁ and λ₂ are scalars, so we can move them around. Taking the conjugate transpose of equation (7), we get:
(λ₂ λ₁) v₁* v₂ = (λ₁ v₁)* A v₂ (8)
Now, we have v₁* v₂ on the left-hand side and (λ₁ v₁)* A v₂ on the right-hand side. If v₁ and v₂ are not orthogonal (v₁ · v₂ ≠ 0), then v₁* v₂ ≠ 0. However, the right-hand side of equation (8) is proportional to (λ₁ v₁)* A v₂, which is proportional to A v₂. This implies that A v₂ is a scalar multiple of v₁, which contradicts the assumption that v₁ and v₂ correspond to different eigenvalues.
Therefore, we conclude that eigenvectors corresponding to different eigenvalues for a normal matrix are necessarily orthogonal.
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Select all the TRUE sentences!!!
Answer:
The last two.
Step-by-step explanation:
Answer:
A and C
Step-by-step explanation:
A circle has a radius of 13 cm what is the diameter of the circle? What is the circumference of the circle? What is the area of the circle? Step-by-step answer please
Answer:
Diameter: 26cm
Circumference: 26π / 81.68cm
Area: 169π / 531cm²
Step-by-step explanation:
Diameter is radius x 2, so 13 x 2 = 26
Circumference is diameter x π, so 26 x π = 26π / 81.68cm
Area is π x r², so π x 13² = 169π / 531cm²
Cody weighed 110 pounds when he started 6th grade but now weighs 150 pounds
as a 7th grader. What is the percent of increase in his weight?
answer: he increased 40 pound
Step-by-step explanation: if it gives negative option please select that
Evaluate the function at the given value.
G(x)= 10.2^x
what is g(-4)?
Answer:
5/8
Step-by-step explanation:
10 * 2^-4 = 10 * 1/16 = 10/16 = 5/8
Help with this ( math)
The equation r = a describes a right circular cylinder of radius a in the cylindrical (r, t, z)-coordinate system. Consider the points P : (r = a, t = 0, z = 0), Q: (r = a, t = tmax, z = h) on the cylinder, and let C be a curve on the cylinder that goes from P to Q. Suppose C is parametrized as a(t) = (a cost, a sin(t), p(t)), 0 ≤ t ≤ tmax, where p(0) = 0 and p(tmax) = h. • (4 pts) Express the length L(p) of C in terms of p. (Hint: You need to look up the formula for the length of a curve in cylindrical coordinates in your calculus textbook.) • (4 pts) Apply the Euler-Lagrange equation of the calculus of varia- tions to find a differential equation for the ☀ that minimizes L(p). • (4 pts) Solve that differential equation and conclude that the mini- mizing curve is a helix.
Minimizing curve C is a helix, described by the equation :
C(t) = (a cos(t), a sin(t), C exp(-cot(t)) + h)
To express the length L(p) of curve C in terms of p, we can use the formula for the length of a curve in cylindrical coordinates. In cylindrical coordinates, the arc length element ds can be given by:
ds² = dr² + r² dt² + dz²
Since dr = 0 (as r = a is constant along the curve C), and dt = -a sin(t) dt (from the parametrization), we have:
ds² = a² sin²(t) dt² + dz²
Integrating ds over the curve C from t = 0 to t = tmax, we get:
L(p) = ∫[0,tmax] √(a² sin²(t) + p'(t)²) dt
where p'(t) denotes the derivative of p(t) with respect to t.
To find the differential equation for the function p(t) that minimizes L(p), we can apply the Euler-Lagrange equation of the calculus of variations. The Euler-Lagrange equation is given by:
d/dt (dL/dp') - dL/dp = 0
Differentiating L(p) with respect to p' and p, we have:
dL/dp' = 0 (since p does not appear explicitly in L(p))
dL/dp = d/dt (dL/dp') = d/dt (a² sin²(t) p'(t) / √(a² sin²(t) + p'(t)²))
Using the chain rule, we can simplify the expression:
dL/dp = (a² sin²(t) p''(t) - a² sin(t) cos(t) p'(t)²) / (a² sin²(t) + p'(t)²)^(3/2)
Setting the Euler-Lagrange equation equal to zero, we get:
(a² sin²(t) p''(t) - a² sin(t) cos(t) p'(t)²) / (a² sin²(t) + p'(t)²)^(3/2) = 0
Simplifying further, we have:
p''(t) - (sin(t) cos(t) / sin²(t)) p'(t)² = 0
This is the differential equation that the function p(t) must satisfy to minimize L(p).
To solve this differential equation, we can make the substitution u = p'(t). Then the equation becomes:
du/dt - (sin(t) cos(t) / sin²(t)) u² = 0
This is a separable first-order ordinary differential equation. By solving it, we can obtain the solution for u = p'(t). Integrating both sides and solving for p(t), we get:
p(t) = C exp(-cot(t)) + h
where C is a constant determined by the initial condition p(0) = 0, and h is the value of p at t = tmax.
Therefore, the minimizing curve C is a helix, described by the equation :
C(t) = (a cos(t), a sin(t), C exp(-cot(t)) + h)
where C is a constant determined by the initial condition, and h is the value of p at t = tmax.
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Let W = {(0, x, y, z): x - 6y + 9z = 0} be a subspace of R4 Then a basis for W is: a O None of the mentioned O {(0,-6,1,0), (0,9,0,1); O {(0,3,1,0), (0,-9,0,1)} O {(0,6,1,0), (0,-9,0,1)} Let w = {(:a+2c = 0 and b – d = 0} be a subspace of M2,2. 2 W d } Then dimension of W is equal to: 4. O 3 1 O 2 O None of the mentioned
The dimension of w is 1.
To find a basis for the subspace W = {(0, x, y, z) : x - 6y + 9z = 0} of R4, we can first find a set of vectors that span W, and then apply the Gram-Schmidt process to obtain an orthonormal basis.
Let's find a set of vectors that span W. Since the first component is always zero, we can ignore it and focus on the last three components. We need to find vectors (x, y, z) that satisfy the equation x - 6y + 9z = 0. One way to do this is to set y = s and z = t, and then solve for x in terms of s and t:
x = 6s - 9t
So any vector in W can be written as (6s - 9t, s, t, 0) = s(6,1,0,0) + t(-9,0,1,0). Therefore, {(0,6,1,0), (0,-9,0,1)} is a set of two vectors that span W.
To obtain an orthonormal basis, we can apply the Gram-Schmidt process. Let u1 = (0,6,1,0) and u2 = (0,-9,0,1). We can normalize u1 to obtain:
v1 = u1/||u1|| = (0,6,1,0)/[tex]\sqrt{37}[/tex]
Next, we can project u2 onto v1 and subtract the projection from u2 to obtain a vector orthogonal to v1:
proj_v1(u2) = (u2.v1/||v1||^2) v1 = (-6/[tex]\sqrt{37}[/tex]), -1/[tex]\sqrt{37}[/tex], 0, 0)
w2 = u2 - proj_v1(u2) = (0,-9,0,1) - (-6/[tex]\sqrt{37}[/tex]), -1/[tex]\sqrt{37}[/tex], 0, 0) = (6/[tex]\sqrt{37}[/tex]), -9/[tex]\sqrt{37}[/tex], 0, 1)
Finally, we can normalize w2 to obtain:
v2 = w2/||w2|| = (6/[tex]\sqrt{37}[/tex], -9/[tex]\sqrt{37}[/tex], 0, 1)/[tex]\sqrt{118}[/tex]
Therefore, a basis for W is {(0,6,1,0)/[tex]\sqrt{37}[/tex], (6/[tex]\sqrt{37}[/tex]), -9/[tex]\sqrt{37}[/tex], 0, 1)/[tex]\sqrt{118}[/tex]}.
For the subspace w = {(:a+2c = 0 and b – d = 0} of [tex]M_{2*2}[/tex], we can think of the matrices as column vectors in R4, and apply the same approach as before. Each matrix in w has the form:
| a b |
| c d |
We can write this as a column vector in R4 as (a, c, b, d). The condition a+2c = 0 and b-d = 0 can be written as the linear system:
| 1 0 2 0 | | a | | 0 |
| 0 0 0 1 | | c | = | 0 |
| 0 1 0 0 | | b | | 0 |
| 0 0 0 1 | | d | | 0 |
The augmented matrix of this system is:
| 1 0 2 0 0 |
| 0 1 0 0 0 |
| 0 0 0 1 0 |
The rank of this matrix is 3, which means the dimension of the solution space is 4 - 3 = 1. Therefore, the dimension of w is 1.
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Approximate the stationary matrix S for the transition matrix P by computing powers of the transition matrix P. P= [ 0.37 0.63] [ 0.19 0.81] S= (Type an integer or decimal for each matrix element Round to four decimal places as needed.)
To approximate the stationary matrix S for the transition matrix P, we need to compute powers of the transition matrix P until it reaches a stable matrix.
Starting with P = [0.37 0.63; 0.19 0.81], we can compute powers of P as follows:
P^2 = P * P
= [0.37 0.63; 0.19 0.81] * [0.37 0.63; 0.19 0.81]
= [0.2746 0.7254; 0.1538 0.8462]
P^3 = P * P^2
= [0.37 0.63; 0.19 0.81] * [0.2746 0.7254; 0.1538 0.8462]
= [0.2421 0.7579; 0.1873 0.8127]
P^4 = P * P^3
= [0.37 0.63; 0.19 0.81] * [0.2421 0.7579; 0.1873 0.8127]
= [0.2222 0.7778; 0.1941 0.8059]
Continuing this process, we find:
P^5 = [0.2149 0.7851; 0.1957 0.8043]
P^6 = [0.2124 0.7876; 0.1961 0.8039]
P^7 = [0.2117 0.7883; 0.1961 0.8039]
As we can see, the matrix P^7 is very close to the stationary matrix S. Therefore, we can approximate the stationary matrix S as:
S ≈ [0.2117 0.7883; 0.1961 0.8039]
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The expenses E and income I for making and selling T-shirts with a
school logo are given by the equations E = 535 +4.50n and I = 12n,
where n is the number of T-shirts.
Expenses: Slope=
Y-Intercept =
income: Slope=
Y-Intercept
Please help will mark brainliest!!
Find the surface area of each figure. Round to the nearest tenth if necessary.
Answer:
9 - 166.42m²
10 - 308.8cm²
Step-by-step explanation:
The first figure shown is a triangular prism
We can find the surface area using this formula
[tex]SA=bh+L(S_1+S_2+h)[/tex]
where
B = base length
H = height
L = length
S1 = base length
S2 = slant height ( base's hypotenuse )
The triangular prism has the following dimensions
Base Length = 4m
Height = 5.7m
Length = 8.6m
S1 = 4m
S2 = 7m
Having found the needed dimensions we plug them into the formula
SA = ( 4 * 5.7 ) + 8.6 ( 4 + 7 + 5.7 )
4 * 5.7 = 22.8
4 + 7 + 5.7 = 16.7
8.6 * 16.7 = 143.62
22.8 + 143.62 = 166.42
Hence the surface area of the triangular prism is 166.42m²
The second figure shown is a pyramid
The surface area of a pyramid can be found using this formula
[tex]SA = A+\frac{1}{2} ps[/tex]
Where
A = Area of base
p = perimeter of base
s = slant height
The base of the pyramid is a square so we can easily find the area of the base by multiplying the base length by itself
So A = 8 * 8
8 * 8 = 64
So the area of the base (A) is equal to 64 cm^2
The perimeter of the base can easily be found by multiplying the base length by 4
So p = 4 * 8
4 * 8 = 32 so p = 32
The slant height is already given (15.3 cm)
Now that we have found everything needed we plug in the values into the formula
SA = 64 + 1/2 32 * 15.3
1/2 * 32 = 16
16 * 15.3 = 244.8
244.8 + 64 = 308.8
Hence the surface area of the pyramid is 308.8cm²
Reading Improvement Program To help students improve their reading, a school district decides to implement a reading program. It is to be administered to the bottom 14% of the students in the district, based on the scores on a reading achievement exam. If the average score for the students in the district is 124.5, find the cutoff score that will make a student eligible for the program. The standard deviation is 15. Assume the variable is normally distributed. Round 2-value calculations to 2 decimal places and the final answer to the nearest whole number.
Rounding the cutoff score to the nearest whole number, the cutoff score that will make a student eligible for the reading program is approximately 108.
To find the cutoff score that will make a student eligible for the reading program, we need to determine the score below which the bottom 14% of students fall.
Since the variable is normally distributed and we know the average score and standard deviation, we can use the Z-score formula to find the cutoff score.
The Z-score formula is:
[tex]\[Z = \frac{X - \mu}{\sigma}\][/tex]
Where:
Z is the Z-score,
X is the raw score,
[tex]\mu[/tex] (mu) is the mean, and
[tex]\sigma[/tex] (Sigma) is the standard deviation.
We want to find the Z-score that corresponds to the bottom 14% of students, which means the area to the left of the Z-score is 0.14.
Using a standard normal distribution table or calculator, we can find the Z-score that corresponds to an area of 0.14, which is approximately -1.08.
Now we can rearrange the Z-score formula to solve for X, the cutoff score:
[tex]\[X = Z \cdot \sigma + \mu\][/tex]
Substituting the values we have:
[tex]\[X = -1.08 \cdot 15 + 124.5\][/tex]
Calculating the expression:
[tex]\[X = -16.2 + 124.5\]\\X = 108.3[/tex]
Rounding the cutoff score to the nearest whole number, the cutoff score that will make a student eligible for the reading program is approximately 108.
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If I work for $7.25 an hour and work for 35 day how much money do I make
Answer:
1776.25
Step-by-step explanation:
35x7=245. 245x7.25=1776.25
Answer:
35 x 24 = 840, 840 hours
840 x 7.25 = $6090 for 35 days
Step-by-step explanation:
2. Write the equation of a circle that has a diameter of 12 units if its center is at (4,7).
O (x – 4)2 + (y – 7)2 = 144
O (x +4)2 + (y + 7)2 = 144
O (x – 4)2 + (y-7)2 = 36
O (x+4)2 + (y + 7)2 = 36
Answer:
The Answer is B
Step-by-step explanation:
The equation of a circle in standard form is
(x - h)² + (y - k)² = r²
where (h, k) are the coordinates of the centre and r is the radius
Here diameter = 12, thus radius = 12 ÷ 2 = 6 and (h, k) = (2.5, - 3.5), thus
(x - 2.5)² + (y - (- 3.5))² = 6², that is
(x - 2.5)² + (y + 3.5)² = 36 ← equation of circle
Let A and B be two events in a specific sample space. Suppose P(A) = 0,4; P(B) = x and P(A or B) = 0,7 For which values of x are A and B mutually exclusive? For which values of x are A and B independent?
For A and B to be mutually exclusive, the value of x must be 0. For A and B to be independent, the value of x can be any value between 0 and 0.3, inclusive.
Two events A and B are said to be mutually exclusive if they cannot occur at the same time, meaning that the intersection of A and B is an empty set. In probability terms, if A and B are mutually exclusive, then P(A and B) = 0.
Given that P(A) = 0.4 and P(A or B) = 0.7, we can use the formula for the probability of the union of two events: P(A or B) = P(A) + P(B) - P(A and B). Since we want to find the values of x for which A and B are mutually exclusive, we set P(A and B) = 0:
0.7 = 0.4 + x - 0
0.7 = 0.4 + x
x = 0.3
Therefore, for A and B to be mutually exclusive, the value of x must be 0. For any other value of x, A and B will have a non-empty intersection and therefore will not be mutually exclusive.
On the other hand, two events A and B are considered independent if the occurrence of one event does not affect the probability of the other event. In probability terms, if A and B are independent, then P(A and B) = P(A) * P(B).
Since P(A) = 0.4 and P(B) = x, we can set up the equation:
P(A) * P(B) = 0.4 * x
For A and B to be independent, this equation must hold for any value of x. Therefore, A and B are independent for any value of x between 0 and 0.3, inclusive.
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Which graphs have a line of symmetry? Check all of the boxes that apply.
Answer:
The last one is symmetrical.
The first one and the last one are correct.
Matt bought 7 shirts for a total of $38. Tee shirts cost $5 and long sleeve shirts cost $6. How many of each type of shirt did he buy? HELP PLZ 100 POINTS
Answer:
tee shirt:4
sleeve shirt:3
Step-by-step explanation:
we are given two conditions
Matt bought 7 shirts for a total of $38Tee shirts cost 5 dollars and long sleeve shirts cost 6 dollarswe want to figure out how many each type of shirt he bought
let tee and sleeve shirts be t and s respectively
according to the first condition
[tex] \displaystyle t + s = 7[/tex]
according to the second condition
[tex] \displaystyle5t + 6s = 38[/tex]
therefore
our system of linear equation is
[tex] \displaystyle\begin{cases}t + s = 7 \\ 5t + 6s = 38 \end{cases}[/tex]
so
now we need our algebra skills to figure out t and s
to do so we can use substitution method
cancel s from both sides of the first equation:
[tex] \displaystyle t = 7 - s \: \cdots \: i[/tex]
now substitute the value of i equation to the second equation:
[tex] \displaystyle 5(7 - s) + 6s = 38[/tex]
distribute:
[tex] \displaystyle 35 - 5s+ 6s = 38[/tex]
collect like terms:
[tex] \displaystyle s + 35 = 38[/tex]
cancel 35 to both sides:
[tex] \displaystyle \therefore s = 3[/tex]
now substitute the value of s to the i equation:
[tex] \displaystyle t = 7 - 3 \\ \therefore \: t = 4[/tex]
hence,
he bought tee shirt 4 and sleeve shirt 3
When rolling a die why is the probability of rolling a 2 or 3
Answer:
1/3
Step-by-step explanation:
If you're talking about a 6-sided die, then there are 6 sides. Rolling a 2 or a 3 would be a 2/6 chance. To simplify if from there, you can also say that there is a 1/3 chance.
The diameter of a circle is 8 inches. What is the area?
d=8 in
Give the exact answer in simplest form.
Step-by-step explanation:
Radius = Half of Diameter
r = 8/2
= 4 in.
Area of Circle =
[tex]\pi {r}^{2} \\ = \pi( {4}^{2} ) \\ = 16\pi \: {in}^{2} [/tex]
what is the area of a 3.3in circle
Answer:
3.3/2 = 1.65² x 3.14 = 8.54865
Please help me, GodBless.
Answer:
-6
Step-by-step explanation:
To find the slope, you do y₂ - y₁ / x₂ - x₁
y₂ - y₁ / x₂ - x₁
= -35 - 11 / 5 - 1
= -24 / 4
= -6
The slope is -6
Answer:
-6
Step-by-step explanation:
Hi,
To find the slope when given a table, just pick two points, subtract the y values, and then divide them by the x values after you subtract them as well. Here's what I mean...
Let's use 1, -11 and 5, -35
So...
-35 - (-11)
This is the change in y. -35 - (-11) is the same thing as -35 + 11 (subtracting negative switches to adding it)
You get -24
Now, the change in x.
5 - 1 = 4
So, -24/4 and you get the slope of : -6
I hope this helps :)
what is the mistake below? solve the system equations by substitution: {4x − y = 20−2x − 2y = 10
The solution to the system of equations by substitution is x = 3 and y = -8.
To solve the system of equations by substitution:
Solve one equation for one variable in terms of the other variable. Let's solve the first equation for x:
4x - y = 20
4x = y + 20
x = (y + 20)/4
Substitute this expression for x into the second equation:
-2x - 2y = 10
-2((y + 20)/4) - 2y = 10
(y + 20)/2 - 2y = 10
(y + 20) - 4y = 20
-y - 20 - 4y = 20
-5y = 40
y = -8
Substitute the value of y back into the first equation to find x:
4x - (-8) = 20
4x + 8 = 20
4x = 20 - 8
4x = 12
x = 12/4
x = 3
Therefore, the solution to the system of equations is x = 3 and y = -8.
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Sydney is building a doghouse. The house will be 3 and one-half feet tall. How
many inches tall will the house be?
Answer:
42 inches
Step-by-step explanation:
1 FT=12 inches
One half FT=6 inches
12×3=36
36+6=42
Answer:
42 inches
Step-by-step explanation:
The question is basically asking you to convert 3 1/2 ft into inches. Since there are 12 inches in a feet you have to multiply 3 1/2 by 12. It easier to do it when you turn 3 1/2 into a decimal, which would be 3.5. So then, you multiply 3.5 by 12, to get 42 inches.
12 inches = 1 feet
3 1/2 = 3.5
3.5 x 12 = 42
The house will be 42 inches tall.
How did Clay and Webster fight against Jackson’s dislike for the bank?
Answer:
Jackson's opponents planned to use the bank to defeat him in the 1832 presidential campaign. Senators Henry Clay and Daniel Webster were friends of Biddle. ... They thought that if Jackson tried to veto, or reject, the renewal of the charter, he would lose support.
Step-by-step explanation:
Tyler and his friends wanted to watch a movie on opening night. They bought tickets online for $9 each. They paid an additional $5 handling fee for the order. The costs was more than $150. How many tickets could they have purchased?
Answer:
caca
Step-by-step explanation:
Help pleaseeeeeeeeeee
Answer:
d
Step-by-step explanation:
A square dance floor has a perimeter of 120 yards.
What is the length of a diagonal of the dance floor?
O 38.3 yd
O 30 yd
O 60 yd
O 42.4 yd
OMG HELP PLS IM PANICKING OMG OMG I GOT A F IN MATH AND I ONLY HAVE 1 DAY TO CHANGE MY GRADE BECAUSE TOMORROW IS THE FINAL REPORT CARD RESULTS AND I DONT WANNA FAIL PLS HELP-
Answer:
it would be $1800
Step-by-step explanation:
if each hoop costs $600, and they buy 3, 600 x 3 = 1800
Answer:
1800
Step-by-step explanation:
600 * 3 = 1800
The table shows the transactions for one week for Tasha and Jamal Who was the greater account balance at the end of the week How Much greater?
______ has the greater balance at the end of the week. This balance is $_______ greater than the other balance
Answer:
Tasha has the greater balance at the end of the week.This balance is $381
The account balance is the remaining amount after the deduction of withdrawals from the deposits.
Tasha has the greater balance at the end of the week. This balance is $1 greater than the other balance.
What is account balance?It is the remaining amount after the deduction of withdrawals from the deposits.
We have,
Tasha:
Initial deposits = $250
Withdrawals = $20
Deposits = $65
Debit card purchases = $46
Balance = $250 - $20 + $65 - $46 = $249
Jamal:
Initial deposits = $200
Withdrawals = $60
Deposits = $135
Debit card purchases = $27
Balance = $200 - $60 + $135 - $27 = $248
We see that Tasha has a greater balance than Jamal and Tasha's balance is greater than Jamal's by $1.
Thus,
Tasha has the greater balance at the end of the week. This balance is $1 greater than the other balance.
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Can someone help me you’re correct me it only let me type one number plz
Answer:
9
Step-by-step explanation:
Hello There!
The circumference is equal to the diameter times π
The diameter given is 9m
NOTE: it says "leave answer in terms of π" meaning that we do not apply π to the diameter
So the answer would be just 9π
Answer:
9
Step by step:
C= 2 x pi x r
c = 2 x pi x 4.5
c= 9 x pi
Kathy bought a total of 60 cupcakes. Supreme cupcakes cost $12 each and regular cupcakes cost $3 each. She spent a total of $90. Which system of equations will determine the number of supreme (s) and regular (r) cupcakes that Kathy purchased?
Answer:
s + r = 60 and 12s + 3r = 90
Step-by-step explanation:
I just took the quiz and it said this is correct