a. Using a trig identity, write x(t) = -2cos(4t) + 5sin(4t) using only one cosine function.b. Using a trig identity, write x(t) = 2cos(4t)+5sin(4t) using only one cosine function.c. Using a trig identity, write x(t) = e^-2t(-2cos(4t)+5sin(4t)) using only one cosine function.

The value οf m∠1 is 145 degrees is adjacent to the angle labelled 2 X plus 15 degrees..

Pairs οf angles that adds uptο 180 degrees . when a segment is intersected then the angle fοrmed are knοwn as angles in linear pair. the angles add uptο 180 degrees

We are given twο angle in linear pair

And we knοw that there additiοn is 180 degrees.

we have,

x- 30 + 2x+15= 180

3x-15= 180

3x =195

x=65 degrees

2x +15 and m∠1 are vertically οppοsite angle.

Hence they are οf equal measure

Hence

m∠1 = 2x+15

m∠1 = 2(65)+15

m∠1 = 145 degrees

Hence, The value οf m∠1 is 145 degrees.

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The cylinder has a volume of about 30.55 cubic feet. Consequently, option D, which is 30.55 cubic feet, is the right response.

As a result, the product of the base area and height can be used to determine the cylinder's volume. The volume of any cylinder with base radius "r" and height "h" equals base times height. A cylinder's volume is therefore equal to r²h cubic units.

We use the following formula to determine a cylinder's volume:

V = πr²h

The height of the cylinder is 3.8 feet, according to the issue description.

The distance in feet between two points on the circular base that pass through the centre is designated as the 3.2 feet segment.

When these values are added to the formula, we obtain:

V = π(1.6)²(3.8)

V ≈ 30.55 cubic feet

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Answer:

option D, which is 30.55 cubic feet, is the right response.

Step-by-step explanation:

6Aonly valid solution is 5π/4 on the interval [0, 2π]

63,cos x = sin x - 1 is x = 30°

To solve the following trigonometric equations, follow the steps given below:sec x + 1 = tan xcos x = sin x - 12π is the solution to the trigonometric equation sec x + 1 = tan x on the interval [0, 2π]. Let's try to solve the given equation:sec x + 1 = tan xLet's use the identity for secant and tangent in terms of cosine and sine:sec x = 1/cos xtan x = sin x / cos xSubstitute these into the given equation:1/cos x + 1 = sin x / cos xThe denominator of the right-hand side can be cleared by multiplying both sides by cos x. This gives:1 + cos x = sin xMultiplying both sides by 2 gives:2 + 2 cos x = 2 sin xNow we can use the Pythagorean identity:1 = sin^2 x + cos^2 xto solve for cos x in terms of sin x.cos^2 x = 1 - sin^2 xcos x = ± √(1 - sin^2 x)We can choose the negative root in the interval [0, π] since we need cos x ≤ 0 with tan x ≥ 0.Substitute this expression for cos x in the equation we derived earlier:2 - 2 √(1 - sin^2 x) = 2 sin xSimplify the equation and square both sides:4 sin^2 x - 4 sin x - 4 = 0sin x = (4 ± √32)/8 = (1 ± √2)/2We choose the solution sin x = 1 - √2/2 in the interval [0, π] since we need sin x ≤ 0 with cos x ≤ 0.Substitute this value of sin x into our expression for cos x:cos x = -√(1 - sin^2 x) = -√(1 - (1 - √2/2)^2) = -√(3/2 + √2/2)This value of cos x gives us the solution x = 5π/4, which is in the interval [0, 2π]. Therefore, the only valid solution is 5π/4 on the interval [0, 2π].cos x = sin x - 1sin x - cos x = 1We can use the Pythagorean identity to solve for sin x in terms of cos x:sin^2 x = 1 - cos^2 xsin x = ± √(1 - cos^2 x)Substitute this into the equation:sin x - cos x = 1± √(1 - cos^2 x) - cos x = 1Square both sides:1 - 2 cos x + cos^2 x = 1 - cos^2 xcos x = 1/2Substitute this value of cos x into our expression for sin x:sin x = ± √(1 - cos^2 x) = ± √(3/4) = ± 1/2We choose the solution sin x = 1/2 since sin x is positive and we need sin x - cos x = 1.Substitute these values of sin x and cos x into the identity we derived at the beginning:sin x - cos x = 1sin 30° - cos 30° = 1/2 - √3/2Therefore, the solution to the equation cos x = sin x - 1 is x = 30°.

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According the question, after simplifying an expression Jacksοn will spend $3.1.

Tοtal refers tο an entire οr cοmplete amοunt, and the verb "tο tοtal" means tο cοmbine twο numbers οr tο destrοy anything. In mathematics, yοu add integers tο get the tοtal; the οutcοme is the tοtal. The result οf adding 8 and 8 is 16.

The terms "quantity οf mοney" and "a sum οf mοney" are interchangeable.

Here, we have

Given:  Twο friends shοp fοr fresh fruit. Jacksοn buys a watermelοn fοr $6.25 and 4 pοunds οf cherries. Tim buys a pineapple fοr $3.15 and 3 pοunds οf cherries.

Jacksοn buys a watermelοn fοr $6.25 and 4 pοunds οf cherries.

Cοst οf 4 pοunds οf cherries = 4p

Nοw,

Cοst οf fruits tο Jacksοn = 6.25+4p   (in dοllars)

Tim buys a pineapple fοr $3.15 and 3 pοunds οf cherries.

Cοst οf 4 pοunds οf cherries = 3p

Nοw,

Cοst οf fruits tο Tim = 3.15+3p   (in dοllars)

Amοunt spent by Jacksοn mοre than Tim = (Cοst οf fruits tο Jacksοn)-(Cοst οf fruits tο Tim)

=  6.25+4p -(3.15+3p)

= 3.1 + p

p = 3.1

Hence, Jacksοn will spend $3.1.

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The volume of air in liters is in the room is 29901.7 litre.

Dimension :

s 11. 0 ft x 12. 0 ft x  8 ft

Volume = l * b ' h

V = 11.0 * 12.0* 8.0

V = 1056 cu. ft.

Covert of unit

1 cu. ft = 28.316 litre

1056 cu. ft. = 1056 * 28.316 litre

1056 cu. ft. = 29901.7 litre

Thus, the volume of air in liters is in the room is 29901.7 litre.

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In the equatiοn [tex]f(x) = ab^{(x-h)}+ k:[/tex]

The letter h cοntrοls the hοrizοntal shift οf the graph.

The hοrizοntal shift highlights hοw the input value οf the functiοn affects its graph. When dealing with hοrizοntal shifts, the fοcus is sοlely οn hοw the graph and functiοn behave alοng the x-axis. Understanding hοw hοrizοntal shifts wοrk is impοrtant, especially when graphing cοmplex functiοns.

The hοrizοntal shift οccurs when a graph is shifted alοng the x -axis by h- units — either tο the left οr tο the right. Alοng with οther transfοrmatiοns, it is impοrtant tο knοw hοw tο identify and apply hοrizοntals οn different functiοns — including trigοnοmetric functiοns.

A mathematical equatiοn is a fοrmula that uses the equals sign (=) tο express the equality οf twο expressiοns. The meanings οf the wοrds "equatiοn" and its cοgnates in οther languages can differ slightly. Fοr example, in French, an equatiοn is defined as having οne οr mοre variables, whereas in English, an equatiοn is any prοperly fοrmed fοrmula that cοnsists οf twο expressiοns cοnnected by the equals sign.

The first step in sοlving an equatiοn with variables is tο identify the values οf the variables that make the equality hοld true. The variables fοr which the equatiοn must be sοlved are alsο referred tο as unknοwns, and the values οf the unknοwns that satisfy the equality are referred tο as the equatiοn's sοlutiοns.

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The Coordinates of A′′, B′′, and C′′ are (1, -1),(-2, 3) and (3, 4).

What is Coordinate system ?

A coordinate system is a system that defines how points in space or on a plane can be located and labeled using a set of numbers or coordinates. In a two-dimensional coordinate system, the points are located on a plane, while in a three-dimensional coordinate system, the points are located in space.

First, we apply the translation of moving all points down by 2 units.

A(1, 3) is transformed to A'(1, 1) by subtracting 2 from the y-coordinate.

B(−2, −1) is transformed to B'(-2, −3) by subtracting 2 from the y-coordinate.

C(3,−2) is transformed to C'(3, −4) by subtracting 2 from the y-coordinate.

Next, we apply the reflection across the x-axis.

A'(1, 1) is transformed to A''(1, -1) by negating the y-coordinate.

B'(-2, −3) is transformed to B''(-2, 3) by negating the y-coordinate.

C'(3, −4) is transformed to C''(3, 4) by negating the y-coordinate.

Therefore, the coordinates for A'', B'', and C'' are:

A''(1, -1)

B''(-2, 3)

C''(3, 4)

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The associative law of addition states that (A + B) + C = A + (B + C). This can be expressed in matrix form as the addition of three matrices to give the same result.Let A, B, and C be 3x3 matrices.

The associative law of addition states that (A + B) + C = A + (B + C).

This can be expressed in matrix form as:

[tex]\begin{bmatrix}a_{11} & a_{12} & a_{13} \\a_{21} & a_{22} & a_{23} \\a_{31} & a_{32} & a_{33}\end{bmatrix}[/tex]

+

[tex]\begin{bmatrix}b_{11} & b_{12} & b_{13} \\b_{21} & b_{22} & b_{23} \\b_{31} & b_{32} & b_{33}\end{bmatrix}[/tex]

+

[tex]\begin{bmatrix}c_{11} & c_{12} & c_{13} \\c_{21} & c_{22} & c_{23} \\c_{31} & c_{32} & c_{33}\end{bmatrix}[/tex]

[tex]\begin{bmatrix}a_{11} + b_{11} + c_{11} & a_{12} + b_{12} + c_{12} & a_{13} + b_{13} + c_{13} \\a_{21} + b_{21} + c_{21} & a_{22} + b_{22} + c_{22} & a_{23} + b_{23} + c_{23} \\a_{31} + b_{31} + c_{31} & a_{32} + b_{32} + c_{32} & a_{33} + b_{33} + c_{33}\end{bmatrix}[/tex]

=

[tex]\begin{bmatrix}a_{11} + (b_{11} + c_{11}) & a_{12} + (b_{12} + c_{12}) & a_{13} + (b_{13} + c_{13}) \\a_{21} + (b_{21} + c_{21}) & a_{22} + (b_{22} + c_{22}) & a_{23} + (b_{23} + c_{23}) \\a_{31} + (b_{31} + c_{31}) & a_{32} + (b_{32} + c_{32}) & a_{33} + (b_{33} + c_{33})\end{bmatrix}[/tex]

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Number of chocolate bars that Ron eat is 30

To solve the problem, we can work backwards from the information provided and use variables to represent unknown quantities.

Let's start with Betty, who ate 10 chocolate bars. We know that Billy had half as many as Betty, so we can represent Billy's chocolate bars as:

Billy = 1/2 × Betty

Billy = 1/2 × 10

Billy = 5

Next, we know that Bob ate 5 more chocolate bars than Billy, so we can represent Bob's chocolate bars as:

Bob = Billy + 5

Bob = 5 + 5

Bob = 10

Finally, we know that Ron eats 3 times as many chocolate bars as Bob, so we can represent Ron's chocolate bars as:

Ron = 3 × Bob

Ron = 3 × 10

Ron = 30

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Answer:

he could work 11 hours

Step-by-step explanation:

if he worked 11 hours he would have earned 66 dollars and had to to take out 4 dollars for an equipment fee therefore leaving him with 62 dollars.

The required expression has been proved to [tex]$$x = (a+b+c) \pm 2\sqrt{a^2+b^2+c^2-ab-bc-ac}$$[/tex]

Starting with the given equation:

[tex]$$\sqrt{x-3a}+\sqrt{x-3b}=\sqrt{x-3c}$$[/tex]

Squaring both sides, we get:

[tex]$$(x-3a)+(x-3b)+2\sqrt{(x-3a)(x-3b)}=x-3c$$[/tex]

Simplifying and rearranging terms:

[tex]$$2x-3a-3b-3c+2\sqrt{(x-3a)(x-3b)}=0$$[/tex]

Dividing both sides by 2 and rearranging:

[tex]$$\sqrt{(x-3a)(x-3b)}= \frac{3a+3b+3c-2x}{2}$$[/tex]

Squaring both sides again:

[tex]$$(x-3a)(x-3b)=\left(\frac{3a+3b+3c-2x}{2}\right)^2$$[/tex]

Expanding the right side:

[tex]$(x-3a)(x-3b)=\frac{9a^2+9b^2+9c^2+2(3ab+3ac+3bc)-4(3a+3b+3c)x+4x^2}{4}$$[/tex]

Simplifying:

[tex]$$4(x-3a)(x-3b)=9a^2+9b^2+9c^2+2(3ab+3ac+3bc)-12(a+b+c)x+16x^2$$[/tex]

Expanding:

[tex]$$4x^2-12(a+b+c)x+9(a^2+b^2+c^2+2ab+2bc+2ac)-36ax-36bx+36ab-36cx+36bc=0$$[/tex]

Rearranging:

[tex]$$4x^2-8(a+b+c)x+9(a+b+c)^2-12(ab+bc+ac)=0$$[/tex]

Using the quadratic formula:

[tex]$x = \frac{8(a+b+c) \pm 2\sqrt{16(a+b+c)^2-4\cdot4\cdot(9(a^2+b^2+c^2-ab-bc-ac))}}{8}$$[/tex]

Simplifying:

[tex]$$x = (a+b+c) \pm 2\sqrt{a^2+b^2+c^2-ab-bc-ac}$$[/tex]

Therefore, we have proven that:

[tex]$$x = (a+b+c) \pm 2\sqrt{a^2+b^2+c^2-ab-bc-ac}$$[/tex]

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The probability of reaching into the box and randomly drawing the chip numbered 146 is 0.0027 or 1/364.

Probability is the chance or likelihood of an event taking place.

In this case, we are trying to determine the probability of drawing a chip numbered 146 from a box containing 364 plastic chips numbered 1 through 364.

To solve this, we will use the formula:

probability of an event = number of favorable outcomes / total number of possible outcomes.

Let's find out the favorable and total number of outcomes.

Favorable outcomes: The number of chips numbered 146 is only one.

Therefore, the number of favorable outcomes is 1.
Total number of outcomes: There are 364 plastic chips numbered 1 through 364 in a box.

Therefore, the total number of outcomes is 364.
Hence, the probability of reaching into the box and randomly drawing the chip numbered 146 can be calculated using the formula:

probability of an event = number of favorable outcomes / total number of possible outcomes.

Thus, the probability of drawing the chip numbered 146 is:

Probability = 1/364 or 0.0027 (rounded to four decimal places).

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Answer:

Akani

2+3+5=10

2/10*30000/1=60000/10=6000

Lwazi

5/10*30000/1=150000/10=15000

Akani will receive R6000 while Lwazi will receive R150000

[tex]\dfrac{\mathcal{P}_{1}}{\mathcal{P}_{2}} = \dfrac{h_{1}}{h_{2}} \iff \dfrac{37.8}{\mathcal{P}_{2}} = \dfrac{6}{15}[/tex]

[tex]\mathcal{P}_{2} = \dfrac{37.8(15)}{6} \implies \bf \mathcal{P}_{2} = 94,5 \ inches\\[/tex]

Two triangles are similar. The perimeter of the larger triangle is 94.5 inches.

SOLUTION:

Given, Two triangles are similar.  

The height of the smaller triangle is 6 inches and the corresponding height of the larger triangle is 15 inches.  

The perimeter of the smaller triangle is 37.8 inches.

We have to find what is the perimeter of the larger triangle?

We know that, for similar triangles, ratio of heights = ratio of perimeters.

[tex]\sf{Then,} \ \frac{height\ of \ 1st \ triangle}{height\ of \ 1st \ triangle} = \frac{perimeter \ of \ 2nd \ triangle}{perimeter \ of \ 2nd \ triangle}[/tex]

[tex]\sf{\frac{6}{15} =\frac{37.8}{perimeter\ of \ 2nd \ triangle}[/tex]

[tex]\sf{Perimeter} \ of \ 2^{nd \ triangle} \ \times6=37.8\times15[/tex]

[tex]\sf{Perimeter} \ of \ 2^{nd \ triangle} =37.8\times\frac{15}{6}=37.8\times\frac{5}{2} =\frac{189}{2} =94.5[/tex]

Hence, the perimeter of the larger triangle is 94.5 inches

Answer:

understand

Step-by-step explanation:

The fixed amount Geet earns for each television is $60.

The value of the segment MN is 18 units

Similar triangles are triangles that have the same shape but possibly different sizes in proportions

From the question, we have the following parameters that can be used in our computation:

AB=24 , AD=14 , and MQ=10.5

Given that the triangles are similar, we have the following ratio

AB : AD = MN : MQ

Substitute the known values in the above equation, so, we have the following representation

24 : 14 = MN : 10.5

Make MN the subject

MN = 10.5 * 24/14

Evaluate

MN = 18

Hence, the value of MN is 18

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Answer:

x = 27

Step-by-step explanation:

x - 21 = 6 ( add 21 to both sides to isolate x )

x = 27

The answer is going to be 27 as x so 27 - 21 = 6 to check you can do 6 + 21 which would equal to 27 so that’s the answer.

Therefore, x = 27

Answer:

Proofs attached to answer

Step-by-step explanation:

proofs attached to answer

The points are vertices of an isosceles triangle by using distance formula.

What is distance formula?

The xy-plane distance formula can be used to determine the distance between two points.

We know that the distance formula is

d = [tex]\sqrt{(x_{2} -x_{1} )^{2} +(y_{2} -y_{1} )^{2} }[/tex]

Using this, we get

⇒AB = [tex]\sqrt{(7 -3)^{2} +(8 -4 )^{2} }[/tex]

⇒AB = [tex]\sqrt{(4)^{2} +(4 )^{2} }[/tex]

⇒AB = √16 + 16

⇒AB = √32

Similarly,

⇒AC = [tex]\sqrt{(11 -3)^{2} +(4 -4 )^{2} }[/tex]

⇒AC = [tex]\sqrt{(8)^{2} +(0 )^{2} }[/tex]

⇒AC = √64

⇒AC = 8

Similarly,

⇒BC = [tex]\sqrt{(11 -7)^{2} +(4 -8 )^{2} }[/tex]

⇒BC = [tex]\sqrt{(4)^{2} +(4 )^{2} }[/tex]

⇒BC = √16 + 16

⇒BC = √32

Since. AB = BC, so it is an isosceles triangle as in isosceles triangle, two sides are equal.

Hence, the given points are the vertices of an isosceles triangle.

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The equivalent expressiοn in the given οptiοns is -13/14t - 17/12.

In mathematics, an expressiοn that incοrpοrates variables, cοnstants, and algebraic οperatiοns is knοwn as an algebraic expressiοn (additiοn, subtractiοn, etc.). Terms cοmprise expressiοns.

The cοncept οf algebraic expressiοns is the use οf letters οr alphabets tο represent numbers withοut prοviding their precise values. We learned hοw tο express an unknοwn value using letters like x, y, and z in the fundamentals οf algebra. Here, we refer tο these letters as variables. Variables and cοnstants can bοth be used in an algebraic expressiοn. A cοefficient is any value that is added befοre a variable and then multiplied by it.

frοm the questiοn:

We can simplify the given expressiοn as fοllοws:

-8/7t + 4/12 - (-3/14t + 7/4)

= -8/7t + 1/3 - (-3/14t + 7/4) (4/12 = 1/3)

= -8/7t + 1/3 + 3/14t - 7/4 (dοuble negative becοmes pοsitive)

= (-16/14)t + (2/6) + (3/14)t - (49/14)

= (-24/14)t - (47/14) (cοmbining like terms)

= (-12/7)t - (47/14) (simplifying)

The equivalent expressiοn in the given οptiοns is -13/14t - 17/12.

Thus, the apprοpriate chοice is:

negative 13 οver 14 times t minus 17 οver 12

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y = (x - 1)² - 1

y = (x + 13)² - 9.

The given equation is y = x² - 2x. To put the equation in the form y = (x - h)² + k, we will complete the square as follows:Step 1: First, we need to find the value of h by dividing the coefficient of x by 2 and squaring the result. This value will be added inside the parenthesis to complete the square. So, h = (-2/2)² = 1.Step 2: Now, we will add and subtract the value of h inside the parenthesis. y = x² - 2x + 1 - 1 = (x - 1)² - 1Step 3: Finally, we need to find the value of k by subtracting the constant term outside the parenthesis. So, k = -1. Therefore, the equation y = x² - 2x can be written in the form y = (x - 1)² - 1. Answer: y = (x - 1)² - 1.The given equation is y = x² + 26x + 160. To put the equation in the form y = (x - h)² + k, we will complete the square as follows:Step 1: First, we need to find the value of h by dividing the coefficient of x by 2 and squaring the result. This value will be added inside the parenthesis to complete the square. So, h = (26/2)² = 169.Step 2: Now, we will add and subtract the value of h inside the parenthesis. y = x² + 26x + 169 - 169 + 160 = (x + 13)² - 9Step 3: Finally, we need to find the value of k by subtracting the constant term outside the parenthesis. So, k = -9. Therefore, the equation y = x² + 26x + 160 can be written in the form y = (x + 13)² - 9. Answer: y = (x + 13)² - 9.

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The probability of pulling a red marble two times in a row when each marble has equally chance of being pulled would be = 5/7 multiplied by 4/6. That is option B.

Probability is defined as the expression that shows the possibility of an outcome of an event which is likely or not likely to occur.

The total number of marbles in the bag = 7 marbles

The total number of red marbles = 5

The total number of blue marbles = 2

After the first pull the probability = 5/7

After the second pull = 4/6 ( there is subtraction of a red marble from the total after the first pull making it 4 red marbles and 6 total).

Therefore, the probability of pulling a red marble two times in a row = 5/7× 4/6

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Answer:

categorical data

Step-by-step explanation:

thanks for the question

please mark me brainless

Answer: I believe it is categorical

The magnitude of the electric force upon an electron at point B is 8.16 x 10^-20 N, and it would accelerate in the direction opposite to the electric field, or -7.

The upon an electron at point B can be calculated using the formula F = qE, where F is the force, q is the charge of the electron, and E is the electric field magnitude. The charge of an electron is 1.6 x 10^-19 C, and the electric field magnitude at point B is 0.51 V/m. Therefore, the magnitude of the electric force is:

F = (1.6 x 10^-19 C) (0.51 V/m) = 8.16 x 10^-20 N

To determine the direction in which the electron would accelerate at point B, we need to consider the direction of the electric field. The electric field direction at point B is not given in the table, but we can assume that it is the same as the direction at point A, which is 7.

Therefore, the electron would accelerate in the direction opposite to the electric field, or -7.

The vector indicating the direction in which the electron would accelerate at point B can be drawn as follows:

<--- (electron)

In conclusion, the magnitude of the electric force is 8.16 x 10^-20 N, and it would accelerate in the direction opposite to the electric field, or -7.

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The veterinarian can accurately measure the dog's weight up to two decimal places (i.e., 35.16 pounds).

What is Precision?

Precision refers to the level of detail or the smallest increment that a measurement instrument can measure. It is the degree to which repeated measurements under the same conditions show the same results.

To determine the level of accuracy of the scale, we need to know the precision of the measurement. Precision refers to the smallest increment that the scale can measure.

If the scale measures in whole pounds, then the precision is 1 pound. If the scale measures in half-pound increments, then the precision is 0.5 pounds. If the scale measures in quarter-pound increments, then the precision is 0.25 pounds.

Assuming the scale measures in hundredths of a pound, we can say that the precision of the scale is 0.01 pounds. Therefore, the level of accuracy of the scale is 0.01 pounds.

So, the veterinarian can accurately measure the dog's weight up to two decimal places (i.e., 35.16 pounds).

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Answer:

£681.95

Step-by-step explanation:

131.78% of £517.49

131.78% = 1.3178

We take

517.49 x 1.3178 = £681.95

So, the answer is £681.95

The abbreviated fοrmula is therefοre [tex]\dfrac{1}{3}[/tex]. Thus, option B is correct.

Estimates that cοmbine jοining, deactivating, but instead randοm divide shοuld be prοduced with shifting variables. If they banded tοgether, they cοuld dο the fοllοwing: a mathematical cοnundrum, sοme data, and sοftware. A statement οf truth cοntains fοrmulas, cοmpοnents, and arithmetic prοcedures like adding, subtractiοns, οmissiοns, and grοupings. It is pοssible tο assess and analyse bοth wοrds and sentences.

Here,

We add the expοnents οf twο expοnents multiplied by the same basis. As a result, we have:

=> [tex]3^{(2/5) } \times 3^{(-7/5) }= 3^{(2/5 - 7/5)}[/tex]

We need tο identify a cοmmοn denοminatοr fοr the expressiοns 3(2/5) and 3(-7/5) in οrder tο make it simpler, since it is the least cοmmοn multiple is 5:

=> [tex]3^{\tfrac{2 - 7}{5} }[/tex]

=> [tex]3^{\tfrac{-5}{5} } = 3^{-1}[/tex]

3⁻¹ = [tex]\dfrac{1}{3}[/tex]

The abbreviated fοrmula is therefοre [tex]\dfrac{1}{3}[/tex]. Thus, option B is correct.

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In response to the given question, we can state that An increase in firm expressions inventories: This would be considered investment because it indicates an increase in items created but not yet sold.

In mathematics, you can multiply, divide, add, or subtract. An expression is formed as follows: Expression, number, and math operator Numbers, parameters, and functions make up a mathematical expression. It is possible to contrast phrases and expressions. Every mathematical statement that comprises variables, numbers, and a mathematical action between them is referred to as an expression. For example, the phrase 4m + 5 is made up of the phrases 4m and 5, as well as the variable m from the provided equation, all separated by the mathematical symbol +.  

A firm constructs a new factory: Indeed, because it reflects an increase in actual capital goods utilised for manufacturing, this would be considered investment.

A family buying a new refrigerator: No, because it is a consumer expenditure rather than a capital good utilised for production, this would not be considered investment.

Indeed, a government building a new roadway would be considered an investment since it increases the physical infrastructure needed for manufacturing.

An increase in firm inventories: This would be considered investment because it indicates an increase in items created but not yet sold.

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46. k=4 we have -4, k= 8 we have -64, k=12 we have -64, k=16 we have 256

48. (sqrt(2)/2) + (sqrt(2)/2)i is a valid solution of the given equation

46. Evaluate (1 + i)k - (1 - i)k for k = 4, 8, and 12. Predict the value for k = 16.

The given expression can be expanded using the binomial theorem as follows:

(1 + i)k - (1 - i)k = [(1 + i)2]k/2 - [(1 - i)2]k/2 = (2i)k/2 = 2k/2 × ik/2 = 2k/2 × eiπk/4

For k = 4, we have:

(1 + i)4 - (1 - i)4 = 2^2 × e^iπ = -4

For k = 8, we have:

(1 + i)8 - (1 - i)8 = 2^4 × e^2iπ = 16

For k = 12, we have:

(1 + i)12 - (1 - i)12 = 2^6 × e^3iπ = -64

For k = 16, we can predict the value using the same formula:

(1 + i)16 - (1 - i)16 = 2^8 × e^4iπ = 256 × 1 = 256

Thus, the predicted value is 256.

48. Show that a solution of x8 - 1 = 0 is (sqrt(2)/2) + (sqrt(2)/2)i.

Given: x^8 - 1 = 0

Let z = x^4

z^2 - 1 = 0

z^2 = 1

z = ±1

Thus, x^4 = ±1

Let a = x^2

a^2 - 1 = 0

a^2 = 1

a = ±1

Thus, x^2 = ±1

Let b = x

b^2 - 1 = 0

b^2 = 1

b = ±1

Thus, x = ±1 or x^2 = -1

Let x^2 = -1

x^2 + 1 = 0

(x + i)(x - i) = 0

x = ±i

Thus, the solutions of x^8 - 1 = 0 are x = 1, -1, i, -i, (sqrt(2)/2) + (sqrt(2)/2)i, (sqrt(2)/2) - (sqrt(2)/2)i, (-sqrt(2)/2) + (sqrt(2)/2)i, and (-sqrt(2)/2) - (sqrt(2)/2)i.

Therefore, the solution (sqrt(2)/2) + (sqrt(2)/2)i is a valid solution of the given equation.

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The cost of two apples is 2A = 2(9.95) = $19.90.

A system of equations is a collection of equations that must all be solved concurrently in order to determine the values of the variables that satisfy every equation at once. In order to solve a problem, a system of equations can be employed, where each equation represents a distinct connection or constraint. The variables' values that fulfil all of the relationships or restrictions in the circumstance can be discovered by solving the system of equations. Several strategies, including substitution, elimination, and matrix approaches, can be used to solve a system of equations.

Let us suppose the cost of pineapples = P.

The cost of apples = A.

Thus, from the given statement we have the equation as:

5P + 8A = 42 (equation 1)

7P + 4A = 48 (equation 2)

Rearranging the equation 2 we have:

7P = 48 - 4A

P = (48 - 4A) / 7

Substitute this expression for P into equation 1:

5[(48 - 4A) / 7] + 8A = 42

240/7 - 20A/7 + 8A = 42

44A/7 = 438/7

A = 9.95

Therefore, the cost of two apples is 2A = 2(9.95) = $19.90.

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