Points z1 and z2 are shown on the graph. (10 points)

complex plane, point z sub 1 at 4 to the right of the origin and 6 units up, point z sub 2 at 5 units to the left of the origin and 2 units up

Part A: Identify the points in standard form and find the distance between them.

Part B: Give the complex conjugate of z2 and explain how to find it geometrically.

Part C: Find z2 − z1 geometrically and explain your steps.

The flux of material past x = 0 is zero for all times.

The flux of material past x = 0 can be calculated by integrating the product of density and velocity over the spatial domain.

This is calculated as;

Φ = ∫ ρv dx

ρ = exp[cos(t − x)]

v = sin(t − x)

where;

The flux of material past x = 0 is calculated as;

Φ = ∫ exp[cos(t − x)] sin(t − x) dx

∫ exp[cos(t − x)] sin(t − x) dx, is the integration of an odd function over a symmetric interval [-π, π] which is zero.

Φ = ∫ exp[cos(t − x)] sin(t − x) dx = 0

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Part A: The total value of the loan with simple interest is $59,660.50.

Part B: The total value of the loan with quarterly compounded interest is $60,357.91.

Part C: The total value of the loan with continuously compounded interest is $60,441.82.

Part D: The best choice for the small business owner is the loan with simple interest, as it has the lowest total value and thus the lowest cost to repay.

Part A:

To calculate the total value of the loan with simple interest, we can use the formula:

[tex]Total value = Principal \times (1 + interest rate \times time)[/tex]

Where,

Principal = $50,000

Interest rate = 5.25% = 0.0525

Time = 3 years and 8 months = 3.67 years.

[tex]Total $ value = $50,000 \times (1 + 0.0525 \times 3.67) = $59,660.50[/tex]

Therefore, the total value of the loan with simple interest is $59,660.50.

Part B:

To calculate the total value of the loan with quarterly compounded interest, we can use the formula:

[tex]Total value = Principal \times (1 + (interest rate / n))^{(n \times time) }[/tex]

Where,

Principal = $50,000

Interest rate = 5.25% = 0.0525

Time = 3 years and 8 months = 3.67 years

n = 4 (since interest compounds quarterly)

[tex]Total value = $50,000 \times (1 + (0.0525 / 4))^{(4 \times3.67) } = $60,357.91[/tex]

Therefore, the total value of the loan with quarterly compounded interest is $60,357.91.

Part C:

To calculate the total value of the loan with continuously compounded interest, we can use the formula:

[tex]Total value = Principal \times e^{(interest rate \times time)}[/tex]

Where,

Principal = $50,000

Interest rate = 5.25% = 0.0525

Time = 3 years and 8 months = 3.67 years

[tex]Total value = $50,000 \times e^{(0.0525 \times 3.67) } = $60,441.82[/tex]

Therefore, the total value of the loan with continuously compounded interest is $60,441.82.

Part D:

From the calculations in Parts A, B, and C, we can see that the loan with continuously compounded interest has the highest total value of $60,441.82.

This is followed by the loan with quarterly compounded interest with a total value of $60,357.91, and finally the loan with simple interest with a total value of $59,660.50.

Therefore, the best choice for the small business owner would be to take the loan with simple interest, as it has the lowest total value and thus the lowest cost to repay.

However, if the small business owner is willing to pay a higher cost for the loan, they could consider the loan with quarterly or continuously compounded interest, which have higher total values.

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At the end of 2 years, Jackson's account balance is $700.40.

We have,

The formula for calculating simple interest is:

I = Prt

Where:

I is the interest earned

P is the principal amount

r is the interest rate

t is the time period

In this case,

Jackson invested $680.00 at an annual interest rate of 1.5% for 2 years. Therefore:

P = $680.00

r = 1.5% = 0.015

t = 2 years

Plugging these values.

I = Prt = $680.00 x 0.015 x 2 = $20.40

So,

Jackson earned $20.40 in interest over 2 years.

To find his total account balance at the end of 2 years, we need to add the interest earned to the initial principal:

Total balance

= $680.00 + $20.40

= $700.40

Therefore,

At the end of 2 years, Jackson's account balance is $700.40.

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Answer:

The median is the best measure of center, and it equals 19.

Step-by-step explanation:

The line for the median is exactly on 19

The line in the box of a box plot represents the median of the data. For this particular data set shown in the box plot, the median is 19.

In the described box plot, the line in the box that is at the number 19 represents the median of the data. This is because a box plot illustrates the five number summary of a data set: the minimum, the first quartile, the median (second quartile), the third quartile, and the maximum. The line inside the box always represents the median. Therefore, the correct choice is: The median is the best measure of center, and it equals 19.

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The solution of the function on the interval is x = 0.524 radians.

The solution of the function on the interval is calculated as follows;

cot(x)= √3

1/tan(x) = √3

Simplify the expression as follows;

1 = √3tan(x)

tan(x) = 1/√3

The solutions of this equation in the interval [-2,2], is calculated as;

x = tan⁻¹(1/√3)

x = 0.524 radians

Another solution of x;

x = tan⁻¹(1/√3) + π = 3.67 radians

This value is not in the given interval, so there is only one solution to the equation cot(x) = √3 in the interval [-2,2], which is approximately x = 0.524 radians.

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Answer:

[tex]x=(x-3)^{\frac{9}{5}}[/tex]

Step-by-step explanation:

The first thing we are going to do is change both sides of the equation to be base (x-3):

[tex](x-3)^{log_{(x-3)}(x)}=(x-3)^{\frac{9}{5}}[/tex].

Now, since the log is in base (x-3), the base (x-3) and the log cancel out:

[tex]x = (x-3)^\frac{9}{5}[/tex]

This is the final answer.

In this example, a positive relationship between the x- and y-values is demonstrated. This basically demonstrates that when one value increases, the other typically does too, and vice versa when one value decreases in the graph.

An x and y coordinate plane with values between -6 and +10 is displayed on the graph.

The points (2, 0), (4, 2), (6, 4), (negative 2, negative 4), and (negative 4, negative 6) are all connected by a line. The y values rise in proportion to the x values because the line has a positive slope.

In this example, a positive relationship between the x- and y-values is demonstrated. This basically demonstrates that when one value increases, the other typically does too, and vice versa when one value decreases.

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The complete question is-

Which statement describes the relationship between the x- and y-values shown in the graph?

A coordinate plane has x-axis and y-axis with values ranging from negative 6 to 10. A positive slope passes through the points (negative 4, negative 6), (negative 2, negative 4), (2, 0), (4, 2), and (6, 4).

Answer:

40 is the surface area.

Explanation on hiw to get the answer:

Answer: 5, 9

Step-by-step explanation:

factors of 45:

1, 3, 5, 9, 15, 45

Sums: 46, 24, 14

the factors 5 and 9 multiply to get 45 and add to get 14

voila!

Diagonals that bisect each other: Rhombus, Rectangle, Square.

Diagonals that bisect each other and are congruent: Rectangle, Square.

Diagonals that bisect each other and are perpendicular to each other: Square.

A parallelogram does not necessarily have diagonals that bisect each other.

A rhombus has diagonals that bisect each other.

This means that the diagonals intersect at their midpoints.

A rectangle has diagonals that bisect each other.

Additionally, the diagonals of a rectangle are congruent, meaning they have the same length.

A square has diagonals that bisect each other.

The diagonals of a square are congruent and perpendicular to each other.

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The determinant of the given 10 x 10 matrix is 1024.

A matrix is described as  a rectangular array or table with rows and columns and numbers, symbols, or expressions that is used to represent a mathematical object or a property of such an object.

We apply the knowledge that the determinant of a diagonal matrix is the product of its diagonal entries in order to find the determinant of a 10 x 10 matrix with 2s on the main diagonal and zeros elsewhere,

Determinant = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2

We then calculate for the product and have:

Determinant = [tex]2 ^ 1^0[/tex] = 1024

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The measure of angle CDE is 70°.

Option C) 70° is correct.

Angle BAC measures 80°.

Point D is located on side BC such that BD = CD.

Angle ADE measures 30°.

To find: Measure of angle CDE.

Since angle BAC measures 80° and angle ADE measures 30°, we can determine the measure of angle CDE by subtracting the sum of these two angles from 180° (the total measure of angles in a triangle).

Measure of angle CDE = 180° - (80° + 30°)

= 180° - 110°

= 70°

Therefore, the measure of angle CDE is 70°.

The correct answer is:

C) 70°

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The  complete question may be like;

In the diagram below, triangle ABC is shown. Angle BAC measures 80°. Point D is located on side BC such that BD = CD. Angle ADE measures 30°. What is the measure of angle CDE?

A) 50°

B) 60°

C) 70°

D) 80°

Please let me know if you have any specific requirements or if there's anything else I can assist you with.

W do not have sufficient evidence at the 0.01 level of significance to reject the claim made by the marketing executive

We'll perform a one-sample z-test for proportions.

From the survey, we know:

Sample size (n) = 1000

Sample proportion) = 0.748

Under the null hypothesis:

Proportion (p0) = 0.78

Now we can calculate the z-score:

z = (0.748 - 0.78) / ✓(0.78 * (1 - 0.78)) / 1000]

= -0.032 / ✓(0.1716 / 1000]

= -0.032 / 0.0131

≈ -2.44

From the z-table, the P-value for -2.44 is approximately 0.015.

However, we need to double this value to get the two-tailed P-value: 2 * 0.015 = 0.03.

If the P-value is less than the significance level (α = 0.01), we reject the null hypothesis. In this case, the P-value (0.03) is greater than α, so we do not reject the null hypothesis.

Based on the data from the Leichtman Research Group survey, we do not have sufficient evidence at the 0.01 level of significance to reject the claim made by the marketing executive.

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Answer:

Area_shaded_region = 2 * (√(1 - cos^2(4π/9))) - (16/9)π m^2

Step-by-step explanation:

To find the area of the shaded region, we need to subtract the area of the sector from the area of the triangle formed by the radius and the two radii connecting to the endpoints of the central angle.

First, let's find the area of the sector:

The formula for the area of a sector is (θ/360) * π * r^2, where θ is the central angle and r is the radius.

Given that the radius is 2 m and the central angle is 160°, we have:

θ = 160°

r = 2 m

Converting the angle to radians:

θ_radians = (160° * π) / 180° = (8π/9) radians

Now, we can calculate the area of the sector:

Area_sector = (θ_radians / (2π)) * π * r^2

= (8π/9) / (2π) * π * 2^2

= (4/9) * π * 4

= (16/9)π m^2

Next, let's find the area of the triangle:

The formula for the area of a triangle is (1/2) * base * height.

The base of the triangle is equal to the length of the radius, which is 2 m.

The height of the triangle can be found using the formula h = r * sin(θ/2).

θ = 160°

r = 2 m

Converting the angle to radians:

θ_radians = (160° * π) / 180° = (8π/9) radians

Calculating the height:

h = 2 * sin(θ_radians/2)

= 2 * sin((8π/9)/2)

= 2 * sin(4π/9)

= 2 * (√(1 - cos(4π/9)^2))

= 2 * (√(1 - cos^2(4π/9)))

Now, we can calculate the area of the triangle:

Area_triangle = (1/2) * base * height

= (1/2) * 2 * 2 * (√(1 - cos^2(4π/9)))

= 2 * (√(1 - cos^2(4π/9)))

Finally, we can find the area of the shaded region by subtracting the area of the sector from the area of the triangle:

Area_shaded_region = Area_triangle - Area_sector

= 2 * (√(1 - cos^2(4π/9))) - (16/9)π m^2

This is the exact answer in terms of π, with the correct unit of measurement (m^2).

The graph of the function is added as an attachment

The asymptote: y = 7 and the points are (-5, 6) and (-7, -18)

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

f(x) = -(1/5)ˣ ⁺ ⁵+ 7

The above function is an exponential function that has been transformed as follows

Next, we plot the graph using a graphing tool by taking not of the above transformations rules

The graph of the function is added as an attachment, where we have the following points

Asymptote: y = 7

(-5, 6) and (-7, -18)

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Answer:

See below for proof.

Step-by-step explanation:

[tex]\boxed{\textsf{Prove that}\;\;\sin^3A + \sin^3(60^{\circ} + A) + \sin^3(240^{\circ}+ A) = \sin^3A}[/tex]

Rewrite 240° as (180° + 60°):

[tex]\sin^3A + \sin^3(60^{\circ} + A) + \sin^3(180^{\circ}+60^{\circ}+ A)[/tex]

As sin(180° + x) = -sin(x), we can rewrite sin³(180° + 60° + A) as:

[tex]\sin^3(180^{\circ}+60^{\circ}+ A)=-\sin^3(60^{\circ}+ A)[/tex]

Substitute this into the expression:

[tex]\sin^3A + \sin^3(60^{\circ} + A) -\sin^3(60^{\circ}+ A)[/tex]

As the last two terms cancel each other, we have:

[tex]\sin^3A[/tex]

Hence proving that:

[tex]\sin^3A + \sin^3(60^{\circ} + A) + \sin^3(240^{\circ}+ A) = \sin^3A[/tex]

As one calculation:

    [tex]\sin^3A + \sin^3(60^{\circ} + A) + \sin^3(240^{\circ}+ A)[/tex]

[tex]=\sin^3A + \sin^3(60^{\circ} + A) + \sin^3(180^{\circ}+60^{\circ}+ A)[/tex]

[tex]=\sin^3A + \sin^3(60^{\circ} + A) -\sin^3(60^{\circ}+ A)[/tex]

[tex]=\sin^3A[/tex]

[tex]\hrulefill[/tex]

[tex]\boxed{\textsf{Prove that}\;\;\sin^3A + \sin^3(120^{\circ} + A) + \sin^3(240^{\circ}+ A) = -\dfrac{3}{4}\sin 3A}[/tex]

Use the sine and cos double angle identities to rewrite sin(3x) in terms of sin(x):

[tex]\begin{aligned}\sin(3x)&=\sin(2x+x)\\&=\sin2 (x)\cos (x)+\sin (x)\cos2 (x)\\&=(2\sin (x)\cos (x))\cos (x)+\sin (x)(1-2\sin^2 (x))\\&=2\sin (x)\cos^2 (x)+\sin (x)-2\sin^3 (x)\\&=2\sin (x)(1-\sin^2 (x))+\sin (x)-2\sin^3 (x)\\&=2\sin (x)-2\sin^3 (x)+\sin (x)-2\sin^3 (x)\\&=3\sin (x)-4\sin^3 (x)\end{aligned}[/tex]

Rearrange to isolate sin³x:

[tex]\begin{aligned}\sin(3x)&=3\sin (x)-4\sin^3 (x)\\\\4\sin^3 (x)&=3\sin (x)-\sin (3x)\\\\\sin^3 (x)&=\dfrac{3\sin (x)-\sin (3x)}{4}\end{aligned}[/tex]

Use this expression to rewrite the terms in sin³A on the left side of the equation:

   [tex]\sin^3A + \sin^3(120^{\circ} + A) + \sin^3(240^{\circ}+ A)[/tex]

[tex]=\dfrac{3\sin A-\sin3A}{4}+ \dfrac{3\sin (120^{\circ} + A)-\sin (3(120^{\circ} + A))}{4}+\dfrac{3\sin (240^{\circ}+ A)-\sin (3(240^{\circ}+ A))}{4}[/tex]

[tex]=\dfrac{3\sin A-\sin3A+3\sin (120^{\circ} + A)-\sin (360^{\circ} + 3A)+3\sin (240^{\circ}+ A)-\sin (720^{\circ}+ 3A)}{4}[/tex]

As sin(x ± 360°n) = sin(x), we can simplify:

[tex]\sin(360^{\circ}+3A) = \sin (3A)[/tex]

[tex]\sin(720^{\circ}+3A) = \sin (3A)[/tex]

Therefore:

[tex]=\dfrac{3\sin A-\sin3A+3\sin (120^{\circ} + A)-\sin (3A)+3\sin (240^{\circ}+ A)-\sin (3A)}{4}[/tex]

[tex]=\dfrac{3\sin A-3\sin3A+3\sin (120^{\circ} + A)+3\sin (240^{\circ}+ A)}{4}[/tex]

Factor out the 3 in the numerator:

[tex]=\dfrac{3\left(\sin A-\sin3A+\sin (120^{\circ} + A)+\sin (240^{\circ}+ A)\right)}{4}[/tex]

Rewrite 240° = 180° + 60°:

[tex]\sin(240^{\circ} + A) = \sin(180^{\circ} + 60^{\circ} + A)[/tex]

As sin(180° + x) = -sin(x), we can rewrite sin(180° + 60° + A) as:

[tex]- \sin(60^{\circ} + A)[/tex]

Therefore:

[tex]=\dfrac{3\left(\sin A-\sin3A+\sin (120^{\circ} + A)-\sin (60^{\circ}+ A)\right)}{4}[/tex]

As sin(120° + x) = sin(60° - x) then:

[tex]=\dfrac{3\left(\sin A-\sin3A+\sin (60^{\circ} -A)-\sin (60^{\circ}+ A)\right)}{4}[/tex]

As sin(60° - x) - sin(60° + x) = -sin(x), then:

[tex]=\dfrac{3\left(\sin A-\sin3A-\sin A\right)}{4}[/tex]

Simplify:

[tex]=\dfrac{-3\sin3A}{4}[/tex]

[tex]=-\dfrac{3}{4}\sin3A[/tex]

Hence proving that:

[tex]\sin^3A + \sin^3(120^{\circ} + A) + \sin^3(240^{\circ}+ A) = -\dfrac{3}{4}\sin 3A[/tex]

Answer:

The answer is 2,000,

Step-by-step explanation:

Probability :- Probability is a way to gauge how likely something is to happen. It is represented by a number between [tex]0[/tex] and [tex]1[/tex], with [tex]0[/tex] denoting an impossibility and [tex]1[/tex] denoting a certainty. By dividing the number of favorable outcomes by the total number of possible outcomes, the probability of an event is determined.

A total of  [tex]32 + 40 + 300 = 372[/tex]  children either take bus number seven, bus number ten, or walk to school.

The proportion of students that ride bus [tex]7[/tex] or bus [tex]10[/tex] to the total number of students determines the likelihood that a student will board either bus [tex]7[/tex]or bus [tex]10[/tex].

[tex]P(taking bus number seven or ten) = (32 + 40) / 1000[/tex]

[tex]P(taking bus number seven or ten) = 72/1000[/tex]

[tex]P(using bus number 7 or ten) = 0.072[/tex]

Therefore, the probability that the student either rides on bus 7 or rides on bus 10 is 0.072 or 7.2%.

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Applying the angle of intersecting secant-tangent theorem, the measure of arc VT is calculated as: m(VT) = 116°.

Given that the lines appear tangent, the measure of the arc VT indicated above can be calculated using the angle of intersecting secant-tangent theorem which states that:

m<U = 1/2(m(WT) - m(VT))

Given the following:

measure of angle U = 37 degrees.

measure of arc WT = 190 degrees.

Plug in the values:

37 = 1/2(190 - m(VT))

74 = 190 - m(VT)

m(VT) = 190 - 74

m(VT) = 116°

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Answer:

-3,-2,-1 is less than -4 and not greater than

Step-by-step explanation:

Answer:

your answer is c=y=1/6x

Answer:

15 feet

Step-by-step explanation:

a^2+b^2=c^2

9^2+12^2=c^2

81+144=c^2

225=c^2

15=c

distance across floor =15 feet

The solution of the exponential equation is x = -16

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

5^-x/16=5

Take the logarithm of both sides of the equation

So, we have the following representation

-x/16 log(5) = log(5)

Divide both sides of the equation by log(5)

-x/16 = 1

So, we have

x = -16 * 1

Evaluate

x = -16

Hence, the solution to the equation is x = -16

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The volume of each figure is given below:

Figure 1: [tex]819 \ feet[/tex]³

Figure 2: [tex]1,728 \ m[/tex]³

Figure 3: [tex]10,696 \pi[/tex] yards³

Figure 4: [tex]1,079.5[/tex] m³

Figure 5: [tex]4,680[/tex] yards³

Figure 1: In the first figure we can see a Cuboid. So below is the method to find the cuboid's volume:

Given dimensions:

Length (l) = [tex]9[/tex] feet, Breadth (b) = [tex]13[/tex] feet, Width (w) = [tex]7[/tex] feet.

Formula:

[tex]\[ V = l \times b \times w \][/tex]

[tex]\[ V = 9 \, \text{feet} \times 13 \, \text{feet} \times 7 \, \text{feet} \][/tex]

Figure 2: In the given figure we see a Prism. Below is the method to find its volume:

Given dimensions:

Hypotenuse = [tex]15[/tex] m, Base = [tex]9[/tex] m, Height = [tex]12[/tex] m, Width = [tex]16[/tex] m.

Formula:

[tex]\[ V = \text{Base} \times \text{Height} \times \text{Width} \][/tex]

[tex]\[ V = 9 \, \text{m} \times 12 \, \text{m} \times 16 \, \text{m} \][/tex]

Figure 3: Here we have the method to find the volume of a Cylinder:

Given dimensions:

Length (l) = [tex]26[/tex] yards, Radius (r) = [tex]11[/tex] yards.

Formula:

[tex]\[ V = \pi \times r^2 \times l \][/tex]

[tex]\[ V = \pi \times (11 \, \text{yards})^2 \times 26 \, \text{yards} \][/tex]

Figure 4: Here, we have the method to find the volume of a Cuboid:

Given dimensions are:

Length (l) = [tex]9[/tex] m, Breadth (b) = [tex]9[/tex] m, Height (h) = [tex]13.5[/tex] m.

Formula:

[tex]\[ V = l \times b \times h \][/tex]

[tex]\[ V = 9 \, \text{m} \times 9 \, \text{m} \times 13.5 \, \text{m} \][/tex]

Figure 5: Next figure is a Prism.

The dimensions that have been given to us are:

Hypotenuse = [tex]15[/tex] yards, Base = [tex]15[/tex] yards, Height = [tex]13[/tex] yards, Width = [tex]24[/tex] yards.

Formula:

[tex]\[ V = \text{Base} \times \text{Height} \times \text{Width} \][/tex]

Now we will calculate it according to the values given in the question:

[tex]\[ V = 15 \, \text{yards} \times 13 \, \text{yards} \times 24 \, \text{yards} \][/tex]

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The mean and standard deviation are less than 88 peanuts: z = (88 - 94) / 3 = -2

between 88 and 91 peanuts: z = (91 - 94) / 3 = -1

between 91 and 94 peanuts: z = (94 - 94) / 3 = 0

between 94 and 97 peanuts: z = (97 - 94) / 3 = 1

greater than 97 peanuts: z = (97 - 94) / 3 = 1

The probability that a 16-ounce can of Nut Munchies will contain between 88 and 97 nuts is 0.8186, or approximately 82%.

The probability of the former event is 0.4772, which is greater than the probability of the latter event, which is 0.1587.

(a) The intervals below the normal distribution can be labeled using z-scores, which are calculated using the formula:

z = (x - μ) / σ

where x is the value of interest, μ is the mean, and σ is the standard deviation.

The intervals and their corresponding z-scores are:

less than 88 peanuts: z = (88 - 94) / 3 = -2

between 88 and 91 peanuts: z = (91 - 94) / 3 = -1

between 91 and 94 peanuts: z = (94 - 94) / 3 = 0

between 94 and 97 peanuts: z = (97 - 94) / 3 = 1

greater than 97 peanuts: z = (97 - 94) / 3 = 1

(b) To find the probability that a 16-ounce can of Nut Munchies will contain between 88 and 97 nuts, we need to find the area under the normal distribution curve between the corresponding z-scores. Using a standard normal distribution table or a calculator, we can find that:

P(88 ≤ X ≤ 97) = P(-2 ≤ Z ≤ 1) = 0.8186

Therefore, the probability that a 16-ounce can of Nut Munchies will contain between 88 and 97 nuts is 0.8186, or approximately 82%.

(c) To determine which event is more likely, we can compare the probabilities of each event. Using the same method as in part (b), we can find that:

P(88 ≤ X ≤ 94) = P(-2 ≤ Z ≤ 0) = 0.4772

P(X > 97) = P(Z > 1) = 0.1587

Therefore, it is more likely for a 16-ounce can of Nut Munchies to have between 88 and 94 nuts than to have over 97 nuts, since the probability of the former event is 0.4772, which is greater than the probability of the latter event, which is 0.1587.

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When not considering the order of choice, there are also 3 ways to choose 2 letters from a, b, and c.

When choosing 2 letters without replacement from the 3 letters a, b, and c, the number of ways can be calculated considering the order of choice and without considering the order of choice.

1. Considering the order of choice:

In this case, the order in which the letters are chosen matters. We can think of this as a permutation problem.

To calculate the number of ways when order matters, we use the formula for permutations:

[tex]nPr = n! / (n - r)![/tex]

where n is the total number of items and r is the number of items chosen.

In this case, we have 3 letters and we want to choose 2, so n = 3 and r = 2.

Using the formula, we get:

[tex]3P2 = 3! / (3 - 2)![/tex]

 [tex]= 3! / 1![/tex]

    = 3

Therefore, when considering the order of choice, there are 3 ways to choose 2 letters from a, b, and c.

2. Without considering the order of choice:

In this case, the order in which the letters are chosen does not matter. We can think of this as a combination problem.

To calculate the number of ways when order does not matter, we use the formula for combinations:

[tex]nCr = n! / (r!(n - r)!)[/tex]

Using the same values of n = 3 and r = 2, we get:

[tex]3C2 = 3! / (2!(3 - 2)!)[/tex]

    = [tex]3! / (2! * 1!)[/tex]

    = 3

Therefore, when not considering the order of choice, there are also 3 ways to choose 2 letters from a, b, and c.

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Answer:

The equation of the polynomial is f ( x ) = -13.5 ( x + 2 )²( x - 3 )

Given data ,

The equation of degree 3 polynomial function with real coefficients

And ,  zeros x = - 2 with multiplicity 2 and x = 3 with multiplicity 1

where function passes through the point (1, 54)

If a polynomial function has a zero x = a with multiplicity k, then the factor (x - a)^k appears in its factored form.

Therefore, a degree 3 polynomial function with zeros x = -2 with multiplicity 2 and x = 3 with multiplicity 1 can be written in factored form as:

f(x) = a(x + 2)²(x - 3)

where a is a constant factor. To find the value of a, we use the fact that the function passes through the point (1, 54):

f(1) = a(1 + 2)²(1 - 3) = 54

a(-1)²(-2) = 54

-4a = 54

Divide by -4 on both sides , we get

a = -13.5

Hence , the equation of the degree 3 polynomial function with the given zeros and passing through the point (1, 54) is f ( x ) = -13.5 ( x + 2 )²( x - 3 )

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The segment lengths for this problem are given as follows:

Suppose that we have two points of the coordinate plane, and the ordered pairs have coordinates given by [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex].

The shortest distance between them is given by the equation presented as follows, derived from the Pythagorean Theorem:

[tex]D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]

Hence the length of segment AB is given as follows:

[tex]AB = \sqrt{(2 - (-5))^2 + (4 - 4)^2} = 7[/tex]

The length of segment AE is given as follows:

[tex]AE = \sqrt{(-5 - (-5))^2 + (4 - (-5))^2} = 9[/tex]

The length of segment BC is given as follows:

[tex]BC = \sqrt{(5-4)^2 + (5 - (-5))^2} = 10.05[/tex]

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