One type of firework splits 5 times after it is fired into the air. The number of new bursts after each split can be modeled by the function f (x ) = 2x where x is the number of times the firework splits. What is the domain of f ( x) in this context

You probably already have an idea of what a matrix is; it's a rectangular array of numbers. What they represent is a bit complicated to explain. There's a whole subject about it (see "linear algebra" for more info). Whatever they represent isn't important though, you don't need to know everything about matrices to compute their product (or whether it's even possible).

A quick definition: A matrix of dimension [tex]r[/tex]-by-[tex]c[/tex] is a matrix with [tex]r[/tex] rows and [tex]c[/tex] columns.

Matrix multiplication all comes down to an operation called the "dot product". It's defined by the sum of component-wise products of elements between two lists. What this means is, if [tex]x=\{1,2,0\}[/tex] and [tex]y=\{-1,0,3\}[/tex], then the dot product of [tex]x[/tex] and [tex]y[/tex] is

[tex]x\cdot y=(1)(-1)+(2)(0)+(0)(3)=-1+0+0=-1[/tex]

What we did was take the first elements of each list and multiplied them, and the same for the other two elements, then added them all together. Notice that the product can't be computed if [tex]x[/tex] and [tex]y[/tex] don't thave the same number of elements.

We write this product in matrix form as

[tex]\begin{bmatrix}1&2&0\end{bmatrix}\begin{bmatrix}-1\\0\\3\end{bmatrix}[/tex]

Notice the pattern here: on the left, a matrix with 1 row and 3 columns; on the right, a matrix with 3 rows and 1 column. The number of columns of the first matrix have to match the number of rows of the second.

The orientation makes a big difference. The product above returns a 1-by-1 matrix (or simply a scalar number):

[tex]\begin{bmatrix}1&2&0\end{bmatrix}\begin{bmatrix}-1\\0\\3\end{bmatrix}=\begin{bmatrix}(1)(-1)+(2)(0)+(0)(3)\end{bmatrix}=\begin{bmatrix}-1\end{bmatrix}=-1[/tex]

On the other hand, the alternate orientation would result in a 3-by-3 matrix.

[tex]\begin{bmatrix}1\\2\\0\end{bmatrix}\begin{bmatrix}-1&0&3\end{bmatrix}=\begin{bmatrix}(1)(-1)&(1)(0)&(1)(3)\\(2)(-1)&(2)(0)&(2)(3)\\(0)(-1)&(0)(0)&(0)(3)\end{bmatrix}=\begin{bmatrix}-1&0&3\\\-2&0&3\\0&0&0\end{bmatrix}[/tex]

So the number of rows of the first matrix and number of columns of the second matrix determine the number of rows and columns, respectively, of the matrix product.

I think we have enough information about matrix multiplication to answer this question. If [tex]A[/tex] has dimensions [tex]m\times n[/tex] and [tex]B[/tex] has dimensions [tex]n\times p[/tex], then the matrix product [tex]AB[/tex] exists ([tex]n[/tex] columns in [tex]A[/tex], [tex]n[/tex] rows in [tex]B[/tex]), but the matrix product [tex]BA[/tex] does not ([tex]p[/tex] columns in [tex]B[/tex], [tex]m[/tex] rows in [tex]A[/tex], but [tex]m\neq p[/tex]). So I is not true.

We know [tex]AB[/tex] exists, and with [tex]m[/tex] rows in [tex]A[/tex] and [tex] p[/tex] columns in [tex]B[/tex], we expect [tex]AB[/tex] to have [tex]m[/tex] rows and [tex]p[/tex] columns, so [tex]AB[/tex] has dimensions [tex]m\times p[/tex]. So II is true.

With dimensions [tex]r\times c[/tex], a matrix would contain [tex]rc[/tex] elements. [tex]m,n,p[/tex] are distinct, so [tex]mp\neq n^2[/tex]. So III is not true.

Answer:  The correct option is

(d) I and II only.

Step-by-step explanation:  Given that matrix A has dimensions m x n and matrix B has dimensions n x p where m, n, and p are distinct positive integers.

We are to select the one that is true from the following :

I.   the product of BA doesn't exist

II.  the product of AB exists and has dimensions m x p

III. the product of AB exists and has dimensions n x n

We know that two matrices X and Y can be multiplied if the number of columns in X is equal to the number of rows in Y.

Also, if X has dimensions  a x b and Y has dimensions b x c, then the product XY is possible and it has dimensions a x c. Also, the product YX doesn't exist.

So, for the given matrices A and B, the following points are true :

(I) the product of BA doesn't exist.

(II) the product of AB exists and has dimensions m x p.

Thus, only I and II are TRUE.

Option (d) is CORRECT.

Answer:

[tex]|LR|=20[/tex]

[tex]|RO|=10[/tex]

Step-by-step explanation:

The centroid divides the median in the ratio [tex]2:1[/tex].

Thus [tex]|LR|:|RO|=2:1[/tex]

This implies that,

[tex]|LO|:|RO|=3:1[/tex]

We were given that,

[tex]|LO|=30[/tex]

We substitute this value to obtain,

[tex]30:|RO|=3:1[/tex]

or

[tex]\frac{30}{|RO|}=\frac{3}{1}[/tex]

We cross multiply to get,

[tex]3|RO|=30\times 1[/tex]

We divide through by 3, to get,

[tex]|RO|=10[/tex]

We can observe from the diagram that

[tex]|LR|+|RO|=|LO|[/tex]

[tex]|LR|+10=30[/tex]

[tex]|LR|=30-10[/tex]

[tex]|LR|=20[/tex]

Answer:

see explanation

Step-by-step explanation:

Under a clockwise rotation about the origin of 90°

a point (x , y) → (y , - x)

Answer:The answer is choice C. and the rotation is 180

Step-by-step explanation:i just got it right

Point-slope form: y-y1 = m(x-x1)

Standard form: ax + by = c

Slope-intercept form: y = mx+b

Start by finding the slope. We know it is negative since the line is decreasing. The slope is -4/3.

To create point-slope form, we need to get one point from the graph. Let's use (3,0).

[tex]y = -\frac{4}{3}(x-3)[/tex]

To create slope-intercept form, we need the slope and the y-intercept. The y-intercept is the point where our equation crosses the y-axis. For this equation, it is 4.

[tex]y = -\frac{4}{3}x + 4[/tex]

To get standard form, solve the equation in terms of C.

Point-slope form: y = -4/3(x-3)

Slope-intercept form: y = -4/3x + 4

Standard form: 4/3x + y = 4

Answer:

-2-p

Step-by-step explanation:

(p-2)x=4-p^2

(4-p^2)/(p-2)

-2-p

The solution of the expression for the given condition would be, - (p + 2).

Used the concept of trigonometry formula which stated that,

[tex](a - b)^2 = a^2 - 2ab + b^2[/tex]

Let us assume that,

When we multiply by x in (p - 2) the solution is (4 - p²).

So, it can be written in mathematical form as,

[tex]x \times (p - 2) = (4 - p^2)\\[/tex]

[tex]x \times (p - 2) = (2^2 - p^2)\\[/tex]

[tex]x \times (p - 2) = (2 - p) (2 + p)[/tex]

[tex]x \times (p - 2) = - (p - 2) (2 + p)[/tex]

Divide both sides by (p - 2),

[tex]x = - (p + 2)[/tex]

So, we get;

[tex](p - 2) \times - (p + 2) = (4 - p^2)[/tex]

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Answer:

A

Step-by-step explanation:

When solving for x as an exponent, we need to use logarithms in order to undo the operation and rearrange the terms. We use log rules to bring down the exponent and solve. Logarithms are the inverse operations to exponents and vice versa. We have one special kind of logarithm called the natural logarithm whose base is e. We write it as ln. Since our base is e here, we will use the natural logarithm to rearrange and isolate x.

[tex]e^{4x-1} =3[/tex]

We begin by applying the natural logarithm to each side.

[tex]ln(e^{4x-1}) =ln(3)[/tex]

Log rules allow use to rearrange the exponent as multiplication in front of the log.

[tex](4x-1)ln(e) =ln(3)[/tex]

ln e as an inverse simplifies to 1.

[tex](4x-1)(1)=ln(3)[/tex]

We now apply the inverse operations for subtraction and multiplication.

[tex]4x-1+1=1+ln(3)\\4x=1+ln(3)\\\frac{4x}{4} =\frac{1+ln3}{4} \\x =\frac{1+ln3}{4}[/tex]

Option A is correct.

Answer: 5

Step-by-step explanation:

The length of the midsegment is one-half the length of the base.  So,

Base = 2(midsegment)

7x + 1 = 2(4x - 2)

7x + 1 = 8x - 4

       1 =  x - 4

       5 = x

The slope-intercept form:

[tex]y=mx+b[/tex]

We have:

[tex]3x+7y=14[/tex]              subtract 3x from both sides

[tex]7y=-3x+14[/tex]             divide both sides by 7

[tex]\boxed{y=-\dfrac{3}{7}x+2\to\boxed{C.}}[/tex]

Answer:

Our answer is [tex]x=-2+\sqrt{2}[/tex].

Step-by-step explanation:

To solve for x, we need to isolate the variable. We can do so by subtracting 2 to both sides.

[tex]x+2=\sqrt{2} \\x+2-2=-2+\sqrt{2} \\\\x=-2+\sqrt{2} \\[/tex]

Our answer is [tex]x=-2+\sqrt{2}[/tex].

To find the decimal, calculate with a calculator.

Answer:

A. 4 groups of 6 students in each group.

B. The smaller group will have 5 students.        

Step-by-step explanation:  

We have been given that Ms.Rochelle wants to put her 29 students into groups of 6.  

A. To find the number of groups of 6 that can be made from 29 students, we will find the greatest multiple of 6 from 29.    

Multiples of 6 are: 6, 12, 18, 24, 30, 36,...  

We can see that the greatest multiples of 6 in 29 in 24.  

[tex]\text{Number of groups that can be made with 29 students}=\frac{24}{6}[/tex]          

[tex]\text{Number of groups that can be made with 29 students}=4[/tex]

Therefore, Ms. Rochelle can form 4 groups from 29 students with 6 students in each group.            

B. To find the number of students in small group, we will subtract 24 from 29.  

[tex]\text{Number of students in the small group}=29-24[/tex]

[tex]\text{Number of students in the small group}=5[/tex]  

Therefore, If Ms. Rochelle puts any remaining students in a smaller group, the number of students in smaller group will be 5.  

Answer:

S22 for the arithmetic sequence is:

First option: 15.4

Step-by-step explanation:

a12=2.4

d=3.4

S22=?

Sn=(a1+an)n/2

n=22

S22=(a1+a22)22/2

S22=(a1+a22)11

ak=aj+(k-j)d

a12=a1+(12-1)d

2.4=a1+11(3.4)

2.4=a1+37.4

Solving for a1: Subtracting 37.4 both sides of the equation:

2.4-37.4=a1+37.4-37.4

Subtracting:

-35=a1

a1=-35

a22=a12+(22-12)d

a22=2.4+10(3.4)

a22=2.4+34

a22=36.4

S22=(a1+a22)11

S22=(-35+36.4)11

S22=(1.4)11

S22=15.4

Answer:

EF = 5 units

Step-by-step explanation:

If ΔABC and ΔDFE are congruent, then

AB = DF

BC = FE

CA = ED

and

angles A and D are congruent

angles B and F are congruent

angles C and E are congruent

Answer:

One brownie costs $1.25

Step-by-step explanation:

We need to solve this with simultaneous equations. First, write the equations out in two lines:

c = cookies and b = brownies.

10c + 3b = 11.25

6c + 9b = 15.75

We need to make either the two c values the same or the two b values the same. Lets change the two b values to the same by multiplying the first equation by three:

10c + 3b = 11.25

10c * 3 = 30c

3b * 3 = 9b

11.25 * 3 = 33.75

30c + 9b = 33.75

Now we can subtract the second equation from this:

30c - 6c = 24c

9b - 9b = 0

33.75 - 15.75 = 18

24c = 18

Now divide both sides by c to find 1c:

c = 0.75

Now put this back into the first equation:

10(0.75) + 3b = 11.25

7.50 + 3b = 11.25

Move the +7.50 over to the other side making it a negative:

3b = 11.25 - 7.50

3b = 3.75

Divide both sides by 3:

b = 1.25

Answer: I TINK 77 rice balls

Step-by-step explanation:

Answer:77

Step-by-step explanation:

Question 4:

  Statement                                           Reason

1. [tex]\overline{AC}[/tex] and [tex]\overline{BD}[/tex] bisect at O                    1. Given

2. O is the midpoint of [tex]\overline{AC}[/tex] and [tex]\overline{BD}[/tex]   2. Definition of bisector

3. [tex]\overline{AO}[/tex] ≅ [tex]\overline{CO}[/tex] and [tex]\overline{BO}[/tex]  ≅ [tex]\overline{DO}[/tex]                3. Definition of midpoint

4. ∠AOB ≅ ∠COD                                4. Vertical Angle theorem

5. ΔAOB ≅ ΔCOD                                5. SAS postulate

6. ∠A ≅ ∠C                                           6. CPCTC

*****************************************************************************

Question 5:

  Statement                                    Reason

1.  [tex]\overline{BD}[/tex] ≅ [tex]\overline{BC}[/tex] and [tex]\overline{AD}[/tex] ≅ [tex]\overline{CD}[/tex]         1. Given

2. [tex]\overline{BD}[/tex] ≅ [tex]\overline{BD}[/tex]                                 2. Reflexive Property

3. ΔABD ≅ ΔCBD                         3. SSS Postulate

4. ∠1 ≅ ∠2 and ∠3 ≅ ∠4               4. CPCTC

5. [tex]\overline{BD}[/tex] bisects ∠ABC and ∠ADC  5. Definition of angle bisector

The measure of 2 yards in centimeters is 182.88 cm

Using the conversions given:

1 yard = 3 × 12 inches

1 yard = 36 inches

1 yard = 36 × 2.54 cm

1 yard = 91.44 cm

Hence,

For 2 yards : (91.44 × 2) = 182.88 cm

The measure of 2 yards in centimeters is 182.88 cm

Answer:

1) f(x) = -1/2x+1

I'm sorry I don't know the 2nd one...

Step-by-step explanation:

The y-intercept is always plotted first. from there, you go down one, and to the right too. and so one, and in reverse, up one, and left two.

Answer:1. f(x) = -1/2x+1

Answer:

2x + 3

Step-by-step explanation:

product means to multiply, hence

the product of 2 and a number x = 2 × x = 2x

3 more than means add on 3, thus

expression is 2x + 3

Answer:

While you do not have the coordinates included, you can use the point-slope form of the equation and input the slope. This would be:

y - y1 = -1/7(x - x1)

In this equation, simply put the ordered pair in at x1 and y1. This will give you the full point-slope equation.

This is from the answer key to this question;

N= {4, 5, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19.}

Final answer:

To solve the division problem 14.2 divided by 0.5, you can use the long division method. The quotient is 28.4.

Explanation:

To solve the division problem 14.2 divided by 0.5, we can perform the long division method. Here's the step-by-step process:

Since there is no remainder anymore, the division is complete. The final quotient is 28.4. Therefore, 14.2 divided by 0.5 is equal to 28.4.

Answer:

Cost of 72 ounces of grapes is $13.5

Step-by-step explanation:

In order to solve his problem, we first need to get all the values in the same units.

  1 pound = 16 ounces

∴ 6 pounds= 16×6 = 96 ounces

Now by unitary method, we solve the question as follows:

price of 96 ounces of grapes = $ 18

price of 1 ounce of grapes = $ (18÷96)

∴ price of 72 ounces of grapes = $ [(18÷96)×72]

= (0.1875×72) =  $ 13.5

Jenny's profit per bracelet is about $1.22. If she sells all 45 bracelets, her total profit is approximately $54.77.

The subject of this question is profit, which in this case will occur when Jenny is able to sell her bracelets for more than she paid for the materials to make them. She pays $12.75 for the materials to make 45 bracelets, meaning each bracelet costs her $12.75 / 45 = $0.283 to make. If she sells each bracelet for $1.50, that means she earns $1.50 - $0.283 = $1.217 in profit for each bracelet. If she sells all 45 bracelets, her total profit would therefore be $1.217 * 45 = $54.765. However, since we typically round money to the nearest cent, her total profit will be approximately $54.77.

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