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Bess [88]
3 years ago
6

Derek is 6.5 feet tall and casts a shadow that is 10 feet long. If a basketball hoop next to

Mathematics
1 answer:
Anna [14]3 years ago
8 0

Answer:

example lang po

Flagpole height is 50 ft.

Explanation:

This is a ratio problem

height of object

length of shadow

Let the height of the flagpole be

x

⇒

6

3

=

x

25

x

=

6

×

25

3

=

50

f

t

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How do the expressions 72 / 9 and -72/ (-9) compare when they are evaluated?
Artemon [7]

Answer:

D

Step-by-step explanation:

72/9=8

-72/-9=8 because negative/negative=positive.

7 0
3 years ago
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How do you know whether an equation is an identity?
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3 years ago
Pls help me I promise I’ll mark u brainleiest if I type in the fastest answer
Sever21 [200]

Answer:

1. $-50

2. $25

Step-by-step explanation:

1. If she had $150 in her bank account and bought a bike for $200, then that means she spent all of her money PLUS $50 extra then what she had.  That means $200-$150=$50.  Her $150 is spent and that $50 becomes negative because she paid $200 when she only had $150.

2.  If she deposits $75 in her account then it will be $75+(-50).  That translates to $75-$50 which is $25.  

+ and - = -

+ and + = +

- and - = +

4 0
3 years ago
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Let A be a given matrix below. First, find the eigenvalues and their corresponding eigenspaces for the following matrices. Then,
Rama09 [41]

It looks like given matrices are supposed to be

\begin{array}{ccccccc}\begin{bmatrix}3&2\\2&3\end{bmatrix} & & \begin{bmatrix}1&-1\\2&-1\end{bmatrix} & & \begin{bmatrix}1&2&3\\0&2&3\\0&0&3\end{bmatrix} & & \begin{bmatrix}3&1&1\\1&3&1\\1&1&3\end{bmatrix}\end{array}

You can find the eigenvalues of matrix A by solving for λ in the equation det(A - λI) = 0, where I is the identity matrix. We also have the following facts about eigenvalues:

• tr(A) = trace of A = sum of diagonal entries = sum of eigenvalues

• det(A) = determinant of A = product of eigenvalues

(a) The eigenvalues are λ₁ = 1 and λ₂ = 5, since

\mathrm{tr}\begin{bmatrix}3&2\\2&3\end{bmatrix} = 3 + 3 = 6

\det\begin{bmatrix}3&2\\2&3\end{bmatrix} = 3^2-2^2 = 5

and

λ₁ + λ₂ = 6   ⇒   λ₁ λ₂ = λ₁ (6 - λ₁) = 5

⇒   6 λ₁ - λ₁² = 5

⇒   λ₁² - 6 λ₁ + 5 = 0

⇒   (λ₁ - 5) (λ₁ - 1) = 0

⇒   λ₁ = 5 or λ₁ = 1

To find the corresponding eigenvectors, we solve for the vector v in Av = λv, or equivalently (A - λI) v = 0.

• For λ = 1, we have

\begin{bmatrix}3-1&2\\2&3-1\end{bmatrix}v = \begin{bmatrix}2&2\\2&2\end{bmatrix}v = 0

With v = (v₁, v₂)ᵀ, this equation tells us that

2 v₁ + 2 v₂ = 0

so that if we choose v₁ = -1, then v₂ = 1. So Av = v for the eigenvector v = (-1, 1)ᵀ.

• For λ = 5, we would end up with

\begin{bmatrix}-2&2\\2&-2\end{bmatrix}v = 0

and this tells us

-2 v₁ + 2 v₂ = 0

and it follows that v = (1, 1)ᵀ.

Then the decomposition of A into PDP⁻¹ is obtained with

P = \begin{bmatrix}-1 & 1 \\ 1 & 1\end{bmatrix}

D = \begin{bmatrix}1 & 0 \\ 0 & 5\end{bmatrix}

where the n-th column of P is the eigenvector associated with the eigenvalue in the n-th row/column of D.

(b) Consult part (a) for specific details. You would find that the eigenvalues are i and -i, as in i = √(-1). The corresponding eigenvectors are (1 + i, 2)ᵀ and (1 - i, 2)ᵀ, so that A = PDP⁻¹ if

P = \begin{bmatrix}1+i & 1-i\\2&2\end{bmatrix}

D = \begin{bmatrix}i&0\\0&i\end{bmatrix}

(c) For a 3×3 matrix, I'm not aware of any shortcuts like above, so we proceed as usual:

\det(A-\lambda I) = \det\begin{bmatrix}1-\lambda & 2 & 3 \\ 0 & 2-\lambda & 3 \\ 0 & 0 & 3-\lambda\end{bmatrix} = 0

Since A - λI is upper-triangular, the determinant is exactly the product the entries on the diagonal:

det(A - λI) = (1 - λ) (2 - λ) (3 - λ) = 0

and it follows that the eigenvalues are λ₁ = 1, λ₂ = 2, and λ₃ = 3. Now solve for v = (v₁, v₂, v₃)ᵀ such that (A - λI) v = 0.

• For λ = 1,

\begin{bmatrix}0&2&3\\0&1&3\\0&0&2\end{bmatrix}v = 0

tells us we can freely choose v₁ = 1, while the other components must be v₂ = v₃ = 0. Then v = (1, 0, 0)ᵀ.

• For λ = 2,

\begin{bmatrix}-1&2&3\\0&0&3\\0&0&1\end{bmatrix}v = 0

tells us we need to fix v₃ = 0. Then -v₁ + 2 v₂ = 0, so we can choose, say, v₂ = 1 and v₁ = 2. Then v = (2, 1, 0)ᵀ.

• For λ = 3,

\begin{bmatrix}-2&2&3\\0&-1&3\\0&0&0\end{bmatrix}v = 0

tells us if we choose v₃ = 1, then it follows that v₂ = 3 and v₁ = 9/2. To make things neater, let's scale these components by a factor of 2, so that v = (9, 6, 2)ᵀ.

Then we have A = PDP⁻¹ for

P = \begin{bmatrix}1&2&9\\0&1&6\\0&0&2\end{bmatrix}

D = \begin{bmatrix}1&0&0\\0&2&0\\0&0&3\end{bmatrix}

(d) Consult part (c) for all the details. Or, we can observe that λ₁ = 2 is an eigenvalue, since subtracting 2I from A gives a matrix of only 1s and det(A - 2I) = 0. Then using the eigen-facts,

• tr(A) = 3 + 3 + 3 = 9 = 2 + λ₂ + λ₃   ⇒   λ₂ + λ₃ = 7

• det(A) = 20 = 2 λ₂ λ₃   ⇒   λ₂ λ₃ = 10

and we find λ₂ = 2 and λ₃ = 5.

I'll omit the details for finding the eigenvector associated with λ = 5; I ended up with v = (1, 1, 1)ᵀ.

• For λ = 2,

\begin{bmatrix}1&1&1\\1&1&1\\1&1&1\end{bmatrix}v = 0

tells us that if we fix v₃ = 0, then v₁ + v₂ = 0, so that we can pick v₁ = 1 and v₂ = -1. So v = (1, -1, 0)ᵀ.

• For the repeated eigenvalue λ = 2, we find the generalized eigenvector such that (A - 2I)² v = 0.

\begin{bmatrix}1&1&1\\1&1&1\\1&1&1\end{bmatrix}^2 v = \begin{bmatrix}3&3&3\\3&3&3\\3&3&3\end{bmatrix}v = 0

This time we fix v₂ = 0, so that 3 v₁ + 3 v₃ = 0, and we can pick v₁ = 1 and v₃ = -1. So v = (1, 0, -1)ᵀ.

Then A = PDP⁻¹ if

P = \begin{bmatrix}1 & 1 & 1 \\ 1 & -1 & 0 \\ 1 & 0 & -1\end{bmatrix}

D = \begin{bmatrix}5&0&0\\0&2&0\\0&2&2\end{bmatrix}

3 0
2 years ago
A cable company charges $35 for basic subscription and $4 per premium channel. Write a function rule that represents the money c
vampirchik [111]

Answer:

c = 4x + 35

Step-by-step explanation:

Imagine c = cost

$35 is base price so you add that to whatever else you spend.

Let x = number of channels you buy. ($4/CHANNEL)

so cost is equal to $4*channels plus 35

Therefore

c = 4x+35

5 0
2 years ago
Read 2 more answers
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