The graph of y=cos ⁡x is transformed to y=a cos⁡(x−c)+d by a vertical expansion by a factor of 3, then translated π/2 units left and 2 units up. The new equation is:

y = 3 cos (x + π/2) + 2

y = 1/3 cos (x - π/2) + 2

y = 3 cos (x - π/2) - 2

y = 3 cos (x - π/2) + 2

Answer:

The correct choice is B.

Step-by-step explanation:

The given function is

[tex]f(t)=-\frac{1}{3}\sin (4t-3\pi)[/tex]

The given function is of the form;

[tex]y=A\sin(Bt-C)[/tex]

where

[tex]|A|=|-\frac{1}{3}| =\frac{1}{3}[/tex] is the amplitude.

The period is calculated using the formula;

[tex]T=\frac{2\pi}{|B|}=\frac{2\pi}{|4|}=\frac{\pi}{2}[/tex]

The phase shift is given by;

[tex]\frac{C}{B}=\frac{-3\pi}{4}[/tex]

The correct choice is B

9)

Answer:

X=37/20

Explanation:

In a kite, one of the diagonals will bisect the other. So, 2(PT)=PR

By substitution, this becomes the equation

2(10x-1)=35

20x-2=35

20x=37

x=37/20

10)

Answer:

Angle TQR is 52 degrees

Explanation:

A kite only bisects one set of opposite angles, so angle TQR is congruent to angle TQP. To find angle TQP use this equation:

Angle TPQ+Angle TQP=90

This is possible because angle QTP is right(diagonals of kites are perpendicular) and because all triangles have interior angles that add up to 180 degrees. The remaining amount of degrees apart from the right angle should be 90 degrees(180-90=90).

Substitute:

38+ angle TQP= 90

Angle TQP= 52 degrees

Angle TQP is congruent to angle TQR, so

Angle TQR=52 degrees

Answer:

(2, 4)

Step-by-step explanation:

The center is the midpoint of the diameter.

(x, y) = ((-1 + 5)/2, (5 + 3)/2)

(x, y) = (2, 4)

ANSWER

The center is (2,4)

EXPLANATION

The given circle has a diameter with endpoints of (-1, 5) and (5, 3).

We use the midpoint formula to find the center of the circle

[tex]( \frac{x_1+x_2}{2} ,\frac{y_1+y_2}{2})[/tex]

We plug in the points to obtain;

[tex]( \frac{ - 1+5}{2} ,\frac{5+3}{2})[/tex]

This simplifies to ;

[tex]( \frac{ 4}{2} ,\frac{8}{2})[/tex]

[tex]( 2 ,4)[/tex]

Answer:

c. 1 and 3

Step-by-step explanation:

To quickly solve this problem, we can use a graphing tool or a calculator to plot each equation.

Please see the attached image below, to find more information about the graph s

The equations are:

1) y = sin (3x + π/6)

2) y = cos (3x - π/6)

3) y = cos (3x - π/3)

Looking at the graphs, we can see that the identical ones

are equations one and three

Correct option:

c. 1 and 3

The correct option is:

             option: c     c.   1 and 3

The first trignometric function is given by:

     [tex]y=\sin (3x+\dfrac{\pi}{6})[/tex]

and also we know that:

[tex]\sin \theta=\cos(\dfrac{\pi}{2}-\theta)[/tex]

This means that:

[tex]\sin (3x+\dfrac{\pi}{6})=\cos (\dfrac{\pi}{2}-(3x+\dfrac{\pi}{6})\\\\\\\sin (3x+\dfrac{\pi}{6})=\cos (\dfrac{\pi}{2}-3x-\dfrac{\pi}{6})\\\\\\\sin (3x+\dfrac{\pi}{6})=\cos (\dfrac{\pi}{2}-\dfrac{\pi}{6}-3x)\\\\\\\sin (3x+\dfrac{\pi}{6})=\cos (\dfrac{2\pi}{6}-3x)\\\\\\\sin (3x+\dfrac{\pi}{6})=\cos (\dfrac{\pi}{3}-3x)\\\\\\\sin (3x+\dfrac{\pi}{6})=cos (-(3x-\dfrac{\pi}{3}))[/tex]

As we know that:

[tex]\cos (-\theta)=cos(\theta)[/tex]

Hence, we have:

[tex]\sin (3x+\dfrac{\pi}{6})=\cos (3x-\dfrac{\pi}{3})[/tex]

Also,  by the graph we may see that the graph of 1 and 2 function do not match.

Hence, they are not equivalent.

Answer:

-6r+6

Step-by-step explanation:

8-(6r+2)

8-6r-2

-6r+6

Answer:

-6r +6.

Step-by-step explanation:

Given : 8 - (6r+2) .

To find : Which expression is equivalent .

Solution : We have given 8 - (6r+2) .

Remove the parenthesis

8 - 6r -2.

Combine like terms

-6r +8 -2.

-6r +6.

Therefore, -6r +6.

The last ones answer is six

Answer:

  62 newspapers

Step-by-step explanation:

Let s represent the number of Sunday papers sold. Then s/2 is the number of Friday papers sold. The total revenue is ...

  1.50s + 0.75(s/2) = 116.25

  1.875s = 116.25 . . . . simplify

  s = 116.25/1.875 = 62 . . . . . divide by the coefficient of s

62 Sunday papers were sold.

Answer:

Step-by-step explanation:

A

The vertex form: f(x) = a(x-h)^2 + k

f(x) = x^2 + 16x + 60 = (x^2 + 16x) + 60

We want to get a perfect square in the brackets, so we solve for our b^2 coefficient.

b^2 = (16/2)^2 = 64

f(x) = (x^2 + 16x + 64 - 64) + 60. Note we subtracted 60 right away to end up with an equivalent expression and not some other function.

f(x) = (x+8)^2 - 4, as you can see it matches the general vertex form.

The vertex form shows when the profit is minimal. The point (h, k) or f(h).

B. The x-intercepts or when the function is equal to 0, or the profit is 0 in the context of the problem.

f(x) = x^2 + 16x + 60 set = 0

x^2 + 16x + 60 = 0

[tex]x_{12} = \frac{-16 \pm \sqrt{256 - 4(1)(60)}}{2} = \frac{-16 \pm \sqrt{16}}{2} = \frac{-16 \pm 4}{2} = -8 \pm 2[/tex]

Answer:

the vertex is either the maximum or minimum value

since the leading coefinet is negative (the number in front of the x² term), the parabola opens down and is a maximum

so

A.

a hack version is to use the -b/(2a) form

if  you have f(x)=ax²+bx+c, then the x value of the vertex is -b/(2a)

so

given

f(x)=-1x²+16x-60

the x value of the vertex is -16/(2*-1)=-16/-2=8

the y value is f(8)=-1(8)²+16(8)-60=

-1(64)+128-60=

4

the vertex is (8,4)

so you selll 8 candels to make the max profit which is $4

B.

x intercepts are where the line crosses the x axis or where f(x)=0

solve

0=-x²+16x-60

0=-1(x²-16x+60)

factor

what 2 numbers multiply to get 60 and add to get -16

-6 and -10

0=-1(x-6)(x-10)

set each factor to 0

0=x-6

x=6

0=x-10

10=x

x intercepts are at x=6 and 10

that is where you make 0 profit

I hope u get what ur looking for and I wish u give me brainlist But I know u won't cuz everyone say that . But thank you so much if u put for me and It would be a very appreciated from you and again thank you so much

Thank you

sincerely caitlin

Answer:

B) 3/4

Step-by-step explanation:

3 /4

Set A interquartile range = 4

Difference between medians is 3.

Therefore,  3 /4

The difference between the median of set A and set B as a multiple of interquartile range of set A is 3/4.

Set A interquartile range is 4

Difference between their medians is 3

The interquartile range is a measure of the “middle fifty” in a data set in which a range is a measure of the beginning and end are in a set, an interquartile range is a measure of the bulk of the values lie.

Difference between the median = Median / Interquartile range of A

                                                       = 3/4

Thus, the difference between the median set A and set B as multiple of interquartile range is 3/4.

Learn more about the Interquartile range from:

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Answer:

10 kgs of 25% copper alloy and 40 kgs of 70% copper alloy

Step-by-step explanation:

Let a be kg of 25% copper alloy, and

b be kg of 70% copper alloy

We can write two equations:

1. a + b = 50

2. 0.25a+0.7b=0.61(50)

We can write #1 as b = 50 - a, and then plug it into #2. We have:

0.25a+0.7b=0.61(50)

0.25a+0.7(50 -  a) = 0.61(50)

0.25a + 35 - 0.7a = 30.5

-0.45a = 30.5 - 35

-0.45 a = -4.5

a = -4.5 / - 0.45

a = 10

Also, b = 50 - a, so b = 50 - 10 = 40

The metalworker should use 10 kgs of 25% copper alloy and 40 kgs of 70% copper alloy to make it.

Answer:

40

Step-by-step explanation:

Answer:

Domain = Range =  All real numbers

Step-by-step explanation:

To quickly solve this problem, we can use a graphing tool or a calculator to plot the equation.

Please see the attached image below, to find more information about the graph

The equation is:

f(x)=2x+cos x

From the plot, we can see the answer is

Option a.

Domain = Range = (-∞,∞)

Answer: a

Step-by-step explanation:

right on edg

Answer: A and D are both correct

Step-by-step explanation:

just took this test

In the straightedge and compass construction of the equilateral triangle above, the reasons can be used to prove that AC ≅ BC are:

A. AB and AC are radii of the same circle A, and AB and BC are radii of the same circle, so AB ≅ AC and AB ≅ BC, and AC ≅ BC

D. AB and AC are radii of the same circle and AB and BC are radii of the same circle, so AB ≅ AC and AB ≅ BC. AC and BC are both congruent to AB, so AC ≅ BC.

In Mathematics and Euclidean Geometry, an equilateral triangle can be defined as a special type of triangle that has equal side lengths and all of its three (3) interior angles are equal.

Since lines AB and AC are radii of the same circle and line AB and line BC are radii of the same circle, we can logically deduce that line AB would be congruent with line AC and line AB would be congruent with line BC.

This ultimately implies that line AC and line BC are both congruent to line AB, so based on the transitive property of equality, we have;

AC ≅ BC.

Answer:

The x intercepts are x=-4 and x=2

axis of symmetry x=-1

vertex (-1,-9)

D: {x: all real numbers}

R: {y: y≥ -9}

Step-by-step explanation:

f(x) = x^2+2x-8

y= x^2+2x-8

Factor the equation

What 2 numbers multiply to -8 and add to 2

4*-2 = -8

4+-2 =2

y = (x+4) (x-2)

The x intercepts are found when we set y =0

0=  (x+4) (x-2)

Using the zero product property

x+4=0   x-2 =0

x=-4 and x=2

The x intercepts are x=-4 and x=2

The axis of symmetry is halfway between the x intercepts.  It is symmetric must be in the middle of the x intercepts

1/2 (-4+2) = 1/2(-2) = -1

The axis of symmetry is at x=-1

To find the vertex, it is along the axis of symmetry.  Substitute x=-1 into the equation

y = (x+4) (x-2)

y = (-1+4) (-1-2)

  =3*-3

  =-9

The vertex is (-1,-9)

The domain is the values that x can take

x can be any number

D: {x: all real numbers}

The range is the values that y can take

since the parabola opens upward, the vertex is the minimum,Y must be greater than or equal to -9

R: {y: y≥ -9}

Answer:

983497

step-by-step explanation:

The sum formula of arithmetic sequence is given by:

[tex]S_n = \frac{n}{2}(2a_1 +(n - 1)d[/tex]

a_1 is the first term, n is the nth term and d is the common difference

From the given information

[tex]d = - 1 -( - 5) = - 1 + 5 = 4[/tex]

[tex]a_1 = - 5 \: and \: n = 703[/tex]

By substitution we obtain:

[tex]S_{703}= \frac{703}{2}(2( - 5) +(703- 1)4)[/tex]

[tex]S_{703}= \frac{703}{2}( - 10 + 2808)[/tex]

[tex]S_{703}= \frac{703}{2}(2798)[/tex]

[tex]S_{703}=98397[/tex]

Answer:

S = 983,497

Step-by-step explanation:

We are given the following sequence and we are to find the sum of the first 703 terms of this sequence:

[tex]-5, -1, 3, 7, ...[/tex]

Finding the common difference [tex]d[/tex] = [tex]-1-(-5)[/tex] = [tex]4[/tex]

[tex]a_1=-5[/tex]

[tex]a_n=?[/tex]

[tex]a_n=a_1+(n-1)d[/tex]

[tex] a_n = - 5 + ( 7 0 3 - 1 ) 4 [/tex]

[tex] a _ n = 2803 [/tex]

Finding the sum using the formula [tex]S_n = \frac{n}{2}(a_1+a_n)[/tex].

[tex]S_n = \frac{703}{2}(-5+2803)[/tex]

S = 983,497

Answer:

Sanjay watched television last weekend for 180 minutes.

Step-by-step explanation:

Sanjay watched 3 television shows and one movie on television last week.

Time duration of 1 television shows = 30 mins

Time duration of 3 television shows = 3* 30 = 90 mins

Time duration of movie = 90 mins

Total number of minutes Sanjay watched television last weekend = Time duration of 3 television shows + Time duration of movie

= 90 + 90

= 180 min

So, Sanjay watched television last weekend for 180 minutes.

Answer:

D

Step-by-step explanation:

Substitute x = g(x) into f(x)

f(2x + 3) = (2x + 3)² ← expand factors

f(g(x)) = (2x)² + 6x + 6x + 3² = 4x² + 12x + 9 → D

Answer:

The measure of angle BCD is [tex]68.5\°[/tex]

Step-by-step explanation:

step 1

Find the value of x

In this problem

[tex]m<ABD+m<BCD+m<EDF=180\°[/tex] -----> is a straight line

substitute the values

[tex](3x+4)+(6x-8)+(7x-20)=180\°[/tex]

[tex](16x-24)=180\°[/tex]

[tex]16x=180\°+24\°[/tex]

[tex]x=12.75\°[/tex]

step 2

Find the measure of angle BCD

[tex]m<BCD+=(6x-8)\°[/tex]

substitute the value of x

[tex]m<BCD+=(6(12.75)-8)=68.5\°[/tex]

Answer:

1440

Step-by-step explanation:

The answer is: The third option, the functions cos(θ) and sin(θ) will have negative values for 180°<θ<270°.

To answer the question we must remember where the trigonometric functions have positive and negative values. We can remember it by considerating where the coordinates of any point are positive or negative along the coordinate plane (x and y).

The primary trigonometric functions are:

[tex]sin(\alpha)\\cos(\alpha)[/tex]

Where,

[tex]Tan(\alpha)=\frac{sin(\alpha)}{cos(\alpha)}[/tex]

Also, we need to remember the quadrants of the coordinate plane.

First quadrant: I, 0°<θ<90°

We can find the first quadrant between 0° and 90° , taking the values from 0 to the positive numbers for the x-axis and the y-axis, the points located on this quadrant, will always have positive coordinates, meaning that the functions sine, cosine and tangent will always have positive values.

Second quadrant: II, 90°<θ<180°

We can find the second quadrant between 90° and 180°, taking the values from 0 to the negative numbers for the a-axis, and from 0 to the positive numbers, the points located on this quadrant, will have negative coordinates along the x-axis and positive coordinates along the y-axis, meaning that the function cosine and tangent will always have negative values, while the sine function will always have positive values.

Third quadrant: III, 180°<θ<270°

We can find the third quadrant between 180° and 270°, taking values from 0 to the negative numbers for both x-axis and y-axis, where the points located on this quadrant, will always have negative coordinates along the x-axis and the y-axis, meaning that both functions sine and cosine will always have negative values, while the tangent function will have positive values.

Fourth quadrant: IV, 270°<θ<360°

We can find the fourth quadrant between 270° and 360°, taking values from 0 to the positive numbers for the x-axis, and from 0 to the negative numbers for the y-axis, the points located at this quadrant will always have positive coordinates along the x-axis and negative coordinates along the y-axis, meaning that the sine and tangent function will always have negative values, while the cosine function will always have positive values.

Hence, the answer to the question is the third option, the functions cos(θ) and sin(θ) will have negative values for 180°<θ<270°.

Have a nice day!

Final answer:

For angles between 180° and 270°, the trigonometric functions cosθ and sinθ have negative values while tanθ is positive.

Explanation:

The primary trigonometric functions under consideration are sin, cos, and tan. The question pertains to angles that fall in the third quadrant, specifically for 180° < θ < 270°.

According to the unit circle and trigonometric properties, cosθ and sinθ have negative values in this range.

This is because in the third quadrant, the x-coordinates (cosine values) and the y-coordinates (sine values) are both negative, while the division of two negative values (sinθ/cosθ for tanθ) gives a positive result for tanθ.

Hence, the correct answer is that cosθ and sinθ may have negative values for the specified range of θ.

4x➕6(503-x)=2296

Distribute the 6 into the parentheses

Let x represent the students

503-x will represent the non-students

4x➕3018➖6x=2296

Then combine like terms

-2x➕3018=2296

Now since you move your constant your sign has to change as well,

-2x=2296➖3018

Then subtract

-2x=-722

Then divide both sides by -2

You are left with x=361

This represents the students

503➖361= 142 which equal the non-students

Therefore your answer is:

361=students

142=non-students

Hope this helps! :3

Answer:

361 students and 503-361=142 non students

Step-by-step explanation:

Let x = number of students. And 503-x = the number of non students. Therefore, 4x+6(503-x)=2296

Answer:

The answer is -8.

Step-by-step explanation:

You cannot divide any number by zero.

When you try to divide something by zero the answer becomes undefined.

Answer:

See below.

Step-by-step explanation:

Dividing  a number  n where n is not zero gives an Undefined result. No matter how many zeros you add together the result is zero - it can never be equal to n.

The expression 0/0 is  referred to as Indeterminate.

Answer:

Step-by-step explanation:

1. cos(π) = -1

2. sin(π) = 0

_____

It is useful to memorize the table below.

Answer:

  y = e^x -4

Step-by-step explanation:

The function has increasing slope so is exponential, not logarithmic. That eliminates the first two choices. The horizontal asymptote is -4, so the function is shifted down 4 units (not 3). The appropriate choice is ...

  y = e^x -4

The mathematical expectation for the number of spaces moved in one turn is:

           A.    3 spaces forward.

Th result or the sample space on spinning a spinner twice is:

           (1,1)    (1,4)    (1,6)    (1,8)

           (4,1)   (4,4)    (4,6)  (4,8)

           (6,1)   (6,4)    (6,6)  (6,8)

           (8,1)   (8,4)    (8,6)   (8,8)

Total number of outcomes= 16

The number of outcomes whose sum is even= 10

( Since the outcomes are: {(1,1) , (4,4) , (4,6) , (4,8) , (6,4) , (6,6) , (6,8) , (8,4) , (8,6) , (8,8)}  )

The number of outcomes whose sum is odd= 6

( Since, the outcomes are: { (1,4) , (1,6) , (1,8) , (4,1) , (6,1) , (8,1) }

Probability(sum even)=10/16

Probability(sum odd)=6/16

Hence, the expectation is:

 [tex]E(X)=\dfrac{10}{16}\times (+6)+\dfrac{6}{16}\times (-2)\\\\\\E(X)=\dfrac{60-12}{16}\\\\\\E(X)=\dfrac{48}{16}\\\\\\E(X)=+3[/tex]

                 Hence, the answer is:

              A.  3 spaces forward.

ANSWER

D. 13

EXPLANATION

The given function is:

[tex]f(x) = \frac{2x + 1}{x -4} [/tex]

Let

[tex]y = \frac{2x + 1}{x -4} [/tex]

Interchange x and y.

[tex]x= \frac{2y + 1}{y-4} [/tex]

Solve for y.

[tex]x(y - 4)=2y + 1[/tex]

[tex]xy - 4x=2y + 1[/tex]

[tex]xy - 2y = 1 + 4x[/tex]

[tex](x - 2)y = 1 + 4x[/tex]

[tex]y = \frac{1 + 4x}{x - 2} [/tex]

[tex] {f}^{ - 1}(x) = \frac{1 + 4x}{x - 2} [/tex]

We put x=3,

[tex]{f}^{ - 1}(3) = \frac{1 + 4 \times 3}{3- 2}[/tex]

[tex]{f}^{ - 1}(3) = \frac{13}{1} = 13[/tex]

Answer:

a) Each corral should be 33⅓ ft long and 25 ft wide

b) The total enclosed area is 1666⅔ ft²

Step-by-step explanation:

I assume that the corrals have identical dimensions and are to be fenced as in the diagram below

Let x = one dimension of a corral

and y = the other dimension

(a) Dimensions to maximize the area

The total length of fencing used is:

4x + 3y = 200

4x = 200 – 3y

x = 50 - ¾y

The area of one corral is A = xy, so the area of the two corrals is

A = 2xy

Substitute the value of x

A = 2(50 - ¾y)y

A = 100 y – (³/₂)y²

This is the equation for a downward-pointing parabola:

A = (-³/₂)y² + 100y

a = -³/₂; b = 100; c = 0

The vertex (maximum) occurs at  

y = -b/(2a)  = 100 ÷ (2׳/₂) = 100 ÷ 3 = 33⅓ ft  

4x + 3y = 100

Substitute the value of y

4x + 3(33⅓) = 200

4x + 100 = 200

4x = 100  

x = 25 ft

Each corral should measure 33⅓ ft long and 25 ft wide.

Step 2. Calculate the total enclosed area

The enclosed area is 50 ft long and 33⅓ ft wide.

A = lw = 50 × 100/3 = 5000/3 = 1666⅔ ft²

The maximum area is achieved when the shared fence is 50 feet and the other two sides are 75 feet each, yielding a maximum area of 3750 square feet.

This problem can be solved by the principles of calculus. Assuming that the two corrals share a common side, we can say the total length of fencing is divided into two lengths (x and y). The optimization problem can be formed as follows:

Since the total length available is 200 feet, 2y + x = 200. The area A = xy. Substitute y=(200-x)/2 into the area formula to get a quadratic A = x(200-x)/2. This graph opens downwards, meaning the vertex is the maximum point. The x-coordinate of the vertex of a quadratic given in standard form like Ax^2 + Bx + C is -B/2A. Therefore, x = -B/2A = 200/(2*2) = 50. Substitute x back into y = (200-2x)/2 to get y = 75. So, the maximum area is achieved with a common side of 50 feet and the other sides being 75 feet each.

The maximum area A can be found by substituying these values back into the area formula: A = 75*50 = 3750 square feet.

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Answer:

The length of QS and TV is 8 units

Step-by-step explanation:

we know that

If triangle QRS is congruent with triangle TUV

then

QS=TV

QR=TU

RS=UV

In this problem

we have

QS=3v+2

TV=7v-6

so

QS=TV

3v+2=7v-6

Solve for v

7v-3v=2+6

4v=8

v=2

Find the length of QS

substitute the value of v

QS=3(2)+2=8 units

so

TS=8 units

Final answer:

Triangle QRS is congruent to triangle TUV, so side QS is equal to side TV. By setting the expressions for QS and TV equal and solving for 'v', we find that both lengths are 8 units.

Explanation:

To solve the problem of finding the length of QS and TV given that triangle QRS is congruent to triangle TUV, and being given the expressions QS = 3v + 2 and TV = 7v - 6, we must recognize that congruent triangles have corresponding sides of equal length. Thus, we can set the expressions for QS and TV equal to each other:

3v + 2 = 7v - 6

We then solve for 'v' by subtracting 3v from both sides:

2 = 4v - 6

Next, we add 6 to both sides:

8 = 4v

Divide both sides by 4 to find 'v':

v = 2

Now that we have the value of 'v', we can substitute it back into the expressions for QS and TV to find their lengths:

QS = 3(2) + 2 = 8

TV = 7(2) - 6 = 8

Therefore, the length of QS and TV is 8 units each.

Answer:

  $18,925.99

Step-by-step explanation:

The sum of n=16 terms of the geometric series with first term a1=800 and common ratio r=1.05 will be ...

  Sn = a1·(r^n -1)/(r -1)

  S16 = $800·(1.05^16 -1)/(1.05 -1) ≈ $18,925.99

Answer: they have saved $18925.99 for his 16th birthday.

Step-by-step explanation:

We know that they save $800 per year, and in each year after the first, they add a 5% extra (0.05 in decimal form).

then, the first year the amount is $800.

the second year, they add $800 + 0.05*$800 = $800*1.05

the third year, they add: $800*1.05 + 0.05*$800*1.05 = $800*(1.05)^2

Now is easy to see that the relation is:

C(n)= $800*(1.05)^(n)

where n goes from 0 to 15, and represents the 16 years in which the parents are saving money.

now, we know that for a geometric series we have:

∑a*r^n = a*( 1 + r^N)/(1 + r)

where the sumation goes from 0 to N -1.

in our case, N - 1 = 15, so N = 16. a = $800 and r = 1.05

then the total of money is;

T = $800*(1 - 1.05^16)/( 1 - 1.05) = $18925.99

Answer:

Step-by-step explanation:

First you assume some complex number of the form [tex]a + bi[\tex] is the square root of [tex]3 - 4i[\tex].

Then, by the definition, that number squared is 3 - 4i.

And you end up with the following equation:

[tex](a+bi)^2 = 3 - 4i\\a^2 + 2abi - b^2 = 3 - 4i\\(a^2 - b^2) + (2ab)i = 3 - 4i[/tex]

Then you assume the real part of the left is equal to 3 and the complex part [tex]2abi[\tex] is equal to [tex]-4i[\tex].

You end up with a system of equations:

[tex]a^2 - b^2 = 3\\2ab = -4[/tex]

Then you simplify the 2nd equation to [tex]ab = -2[\tex], then you rewrite b in terms of a [tex]b = \frac{-2}{a}[\tex].

You plug your new definition into the first equation and you end up with:

[tex]a^2 - (\frac{-2}{a})^2 = 3\\a^2 - \frac{4}{a^2} = 3[/tex]

You multiply the whole equation by [tex]a^2[\tex] as it is not equal to 0.

[tex]a^4 - 4 = 3a^2\\a^4 - 3a^2 -4 = 0[/tex]

We let [tex]t = a^2[\tex] and we end up with:

[tex]t^2 -3t - 4 = 0\\t_{12} = \frac{3 \pm \sqrt{9 - 4(1)(-4)} }{2} = \frac{3 \pm \sqrt{25}}{2} = \frac{3 \pm 5}{2}\\t_1 = 4\\t_2 = -1[/tex]

We then go back to the definition of [tex]t[\tex]:

[tex]t = a^2\\a^2 = 4 \mid a^2 = -1[/tex]

But since a is a real number we only use the first result:

[tex]a^2 = 4\\a_{12} = \pm 2[/tex]

We then solve for [tex]b[\tex]:

[tex]ab = -2\\b_1 = \frac{-2}{a_1}\\b_2 = \frac{-2}{a_2}\\b_{12} = \pm 1[/tex]

We then write the newly achieved complex number:

[tex]a_1 + b_1i = \sqrt{3-4i} \mid a_2 +b_2i = \sqrt{3-4i} \\2-i = \sqrt{3-4i} \mid -2 + i = \sqrt{3-4i}[/tex].

Use which equation you please to find the magnitude of:

[tex]|X| = \sqrt{2^2 + 1^2} = \sqrt{5}[/tex] - the magnitude.

And to find the phase/angle.

[tex]\theta = arcsin(\frac{b}{\sqrt{a^2+b^2} } ) = arcsin(\frac{1}{\sqrt{5}}) = 26.565^o[/tex]