Please help I don’t understand:(

The area of the interior above the polar axis is -0.858 square units

The area bounded by a polar curve between θ = θ₁ and θ = θ₂ is given by

[tex]A = \int\limits^{\theta_{2} }_{\theta_{1} } {\frac{1}{2}r^{2} } \, d\theta[/tex]

Now, since we have the curve r = 1 - sinθ and we want to find the area of the interior above the polar axis, we integrate from θ = 0 to θ = π, since this is the region above the polar axis.

So, [tex]A = \int\limits^{\theta_{2} }_{\theta_{1} } {\frac{1}{2}r^{2} } \, d\theta\\= \int\limits^{\pi}_{0} {\frac{1}{2}(1 - sin\theta)^{2} } \, d\theta\\= \int\limits^{\pi}_{0} {\frac{1}{2}[1 - 2sin\theta + (sin\theta)^{2}] } \, d\theta\\= \int\limits^{\pi}_{0} {\frac{1}{2}\, d\theta - \int\limits^{\pi}_{0} 2sin\theta \, d\theta+ \int\limits^{\pi}_{0} (sin\theta)^{2} } \, d\theta\\[/tex]

[tex]A = \int\limits^{\pi}_{0} {\frac{1}{2}\, d\theta - 2\int\limits^{\pi}_{0} sin\theta \, d\theta+ \int\limits^{\pi}_{0} \frac{(1 - cos2\theta)}{2} } \, d\theta\\= \int\limits^{\pi}_{0} {\frac{1}{2}\, d\theta - 2\int\limits^{\pi}_{0} sin\theta \, d\theta+ \int\limits^{\pi}_{0} \frac{1}{2} } \, d\theta -\int\limits^{\pi}_{0} \frac{cos2\theta}{2} } \, d\theta\\[/tex]

[tex]A = \int\limits^{\pi}_{0} \, d\theta - 2\int\limits^{\pi}_{0} sin\theta \, d\theta -\int\limits^{\pi}_{0} \frac{cos2\theta}{2} } \, d\theta\\= [\theta]_{0}^{\pi} - 2[-cos\theta]^{\pi} _{0} - [\frac{sin2\theta}{4}]_{0}^{\pi} \\= [\theta]_{0}^{\pi} + 2[cos\theta]^{\pi} _{0} - [\frac{sin2\theta}{4}]_{0}^{\pi}\\= [\pi - 0] + 2[cos\pi - cos0] - \frac{ [sin2\pi - sin0]}{4}\\= \pi + 2[-1 - 1] - \frac{ [0 - 0]}{4}\\= \pi + 2[-2] - \frac{ [0]}{4}\\= \pi - 4 - 0\\= \pi - 4\\= 3.142 - 4\\= -0.858[/tex]

So, the area of the interior above the polar axis is -0.858 square units

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It is to be noted that the determination of whether or not there is a significant difference between the two groups will be done using a t test.

A t-test is a statistical test that juxtaposes two samples' means. It is used in hypothesis testing, using a null hypothesis that the variance in group means is zero and an alternative hypothesis that the difference is not zero.

The conditions to use the t test are:

The degrees of freedom (k) to be utilized for this text will be derived using the conservative method given below:

df = [(s₁²/n₁) + (s₂²/n²)/[((s₁²/n₁)²/((n₁-1)) + (s₂²/n₂)²/((n₂-1))]

= [(3.0952/15) + (6.4095/15)]² / [((3.0952/15)²/14) + ((6.4095/15)²/14)]

= 24.965

Hence,

df ≈ 24 (if approximated to the floor)

Recall the the standard deviation of the population are unequal and unknown. This thus requires that we utilize the two-sample unpooled t-test.

Here, H₀ is given as;

[tex]t = \frac{\bar{x_{1} -\bar{x_{2}}}}{\sqrt{\frac{s_{1}^{2} }{n_{1} } + \frac{S_{2}^{2} }{n_{2}} } } \sim t_{df}[/tex]

t = [(3.33333 - 4.13333)]/√[(3.0952/15) + (6.4095/15)]

= - 0.8/√0.6337

t = - 1.005

The 95% confidence interval is computed using the following formula:

[tex](\bar x_{2} - \bar x_{1}) \pm t_\alpha_/_2_,_df \left({\sqrt{\frac{S_{1}^{2} }{n_{1} } + \frac{S_{2}^{2} }{n_{2}} } } \right)[/tex]

= - 0.8 ± t₀.₀₂₅,₂₄ (√0.6337)

= - 0.8 ± 2.064 (√0.6337)

= -2.4429, 0.8429

To derive the 90% interval, we state:

[tex](\bar x_{2} - \bar x_{1}) \pm t_\alpha_/_2_,_df \left({\sqrt{\frac{S_{1}^{2} }{n_{1} } + \frac{S_{2}^{2} }{n_{2}} } } \right)[/tex]

= - 0.8 ± t₀.₀₅₀,₂₄ (√0.6337)

= - 0.8 ± 0.685 (√0.6337)

= -2.162, 0.562

From the intervals computed, we must fail to reject H₀

H₀ : μ₁ = μ₂

It is clear from the above intervals computed from that the differences between the mean of both groups is significant. This is because, zero is included on the two intervals.

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The monthly payments for this add on interest loan are of $164.71.

Given Information and Formula Used

It is given that for an add on interest loan,

Principal Amount, p = $8,276.17

Annual Interest Rate, r = 5.7%

Term of the loan, T = 5.5 years

The formula for simple interest is given as follows,

I = (p)(r)(t)/100 ............... (1)

The formula for total amount of add on interest is given by,

A = p + I ....................... (2)

Computing the Interest

Substitute the given values of p, r, and t in the formula (1) of interest to get,

I = (8276.17)(5.7)(5.5)/100

I = 259457.9295/100

I = $2,594.58

Computing the Monthly Payment for Add-on Interest Loan

Substituting the values of p and I in the formula (2), we obtain the total amount as,

A = $ (8276.17 + 2594.58)

A = $ 10,870.75

Monthly payment for the add on interest loan = A/t(in months)

=  $ (10,870.75/66)

= $164.71

Therefore, monthly payments of $164.71 are to be made for the add on interest loan.

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The rate if he went three years without an accident based on the information about the insurance is $552.15.

From the information given, his basic insurance cost is $613.50 per year and his insurance company offers a typical "safe-driver discount.

The amount will be:

= 613.50 - (20.45 × 3)

= 552.15

In conclusion, the rate is $552.15.

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The current yield of the bond is 9.27%.

What is current yield?

Current yield is the bond rate of return computed as the annual coupon divided by the current price of the bond

The annual coupon is the coupon rate of 10% multiplied by the bond's par value of 100

annual coupon=10%*100

annual coupon=10

current bond price=107.87

current yield=annual coupon/current bond price

current yield=10/107.87

current yield=9.27%

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12 inches

A regular polygon is any 2-D shape where all the sides and angles are congruent.

Regular Hexagon

As denoted by the prefix "hexa-", hexagons have 6 sides and internal angles. The perimeter is equal to all 6 sides added together. However, we know that these sides must be congruent because it is a regular hexagon.

Solving for Perimeter

We can create an equation that describes the perimeter of this shape, with "x" representing the length of one side.

Since all the sides are equal we can use multiplication instead of addition. To solve this equation, divide both sides by 6.

This means that all sides must be 12 inches.

The graphed piecewise function is the one in option B.

On the graph, we can see that on the lowest part we have a closed dot at x = 1

And the above part has a open dot at x = 1.

Then the piecewise function is of the form:

Such that the above part seems to be quadratic, and the part below seems to have a larger degree.

With that, we can conclude that the correct option is:

Which is option b.

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Using the Fundamental Counting Theorem, it is found that:

The 2 people can arrange themselves in 40 ways.

It is a theorem that states that if there are n things, each with [tex]n_1, n_2, \cdots, n_n[/tex] ways to be done, each thing independent of the other, the number of ways they can be done is:

[tex]N = n_1 \times n_2 \times \cdots \times n_n[/tex]

With one people in the aisle and one in the normal seats, the parameters are:

n1 = 4, n2 = 7

With both in the aisle, the parameters is:

n1 = 4, n2 = 3

Hence the number of ways is:

N = 4 x 7 + 4 x 3 = 28 + 12 = 40.

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Based on the calculations, all of the numbers belong to this arithmetic sequence.

Mathematically, the nth term of an arithmetic sequence can be calculated by using this expression:

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

Where:

Next, we would determine the common difference as follows:

d = a₂ - a₁

d = 105 - 99 = 6.

d = a₃ - a₂

d = 111 - 105 = 6.

d = a₄ - a₃

d = 117 - 111 = 6.

Based on the calculations, all of the numbers belong to this arithmetic sequence.

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Complete Question:

What is the number that does not belong to this sequence 99, 105, 111, 117, 123, 129, 135, 141, 147, 153

The zeros of the polynomial function are -1.728, 0.56 and 5.167

The polynomial function is given as:

x^3 - 4x^2 - 7x + 5

Next, we plot the graph of the polynomial

From the graph, the polynomial cross the x-axis when

x = -1.728, 0.56 and 5.167

Hence, the zeros of the polynomial function are -1.728, 0.56 and 5.167

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The length and width of the rectangle are 5 inches and 1 inches respectively.

The length and width of the rectangle can be found as follows;

l = 3 + 2w

area of a rectangle = lw

where

Therefore,

5 = lw

5 = (3 + 2w)w

5 = 3w + 2w²

2w² + 3w - 5 = 0

Hence,

2w² + 3w - 5 = 0

Therefore,

w = 1  and w = - 5 / 2

width = 1 inches

length =  3 + 2(1) = 5 inches

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The solution to the inequality expression is x>5 and the correct representation is a number line from negative 3 to 3 in increments of 1. An open circle is at 5 and a bold line starts at 5 and is pointing to the right.

Inequality expressions are expression not separated by an equal sign

Given the following inequality expression

–3(2x – 5) < 5(2 – x)

Expand

-6x + 15 < 10 - 5x

Collect the like terms

-6x + 5x < 10 -15

-x < -5

x > 5

Hence the solution to the inequality expression is x>5 and the correct representation is a number line from negative 3 to 3 in increments of 1. An open circle is at 5 and a bold line starts at 5 and is pointing to the right.

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Answer:

u = -225

Step-by-step explanation:

36 - u = 261

-u = 261 - 36

-u = 225

u = -225

The information given about the landscape is illustrated below:

a. You'll need a measuring tape, two range poles, pegs, and three people.

The first person holds the zero mark between their thumbs and fingers, the second person holds the 3m mark on the tape between their thumbs and fingers, and the third person holds the 8m. When all sides of the rope are stretched, a triangle is created, and an angle near one is a right angle.

Wrap one loop of the rope around peg A with a peg through the other loop, draw a circle on the ground, place pegs B and C where the circle crosses the base line, and place peg D half way between pegs B and C, allowing pegs D and A to form lines perpendicular to the base line, forming a right angle.

B. Method 1; Take a measurement of the distance between both trees and divide by 2 to get the midpoint.

Method 2; By use of a compass to bisect the distance between both points.

Method 3; By using a protractor to mark the 90° vertical point when placed between the two endpoints.

C The three different ways he can do this.

1) by using a protractor

2) by finding a point

3) by extending the line of the angle until it reaches the other side of the field.

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Using proportions, it is found that they must win 19 of their next 30 games.

A proportion is a fraction of a total amount, and the measures are related using a rule of three.

We want a win-to-loss proportion of 2:1, which is equivalent to a win-to-total proportion of 2:3. We have that:

Hence:

[tex]\frac{5 + x}{36} = \frac{2}{3}[/tex]

3(5 + x) = 72

15 + 3x = 72

3x = 57

x = 19.

The Rangers must win 19 of their next 30 games.

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The equation of the quartic function is f(x) = x⁴ - 18 · x² + 6 · x + 72.

Herein we must construct at least a polynomial that satisfies all conditions described in the statement. According to the fundamental theorem of algebra, quartic functions may have no real roots, two real roots or four real roots, which means that one of the roots must have a multiplicity of 2.

The root with a multiplicity of 2 is x = 3 and both x = - 4 and x = - 2 have only a multiplicity of 1, then we have the following expression by using the factor form of the definition of polynomials:

f(x) = (x - 3)² · (x + 4) · (x + 2)

Now we expand the expression to get the standard form:

f(x) = (x² - 6 · x + 9) · (x² + 6 · x + 8)

f(x) = x⁴ - 6 · x³ + 9 · x² + 6 · x³ - 36 · x² + 54 · x + 8 · x² - 48 · x + 72

f(x) = x⁴ - 18 · x² + 6 · x + 72

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Answer:4x/5-1/5 = y

Step-by-step explanation:

We need to get y by itself

4x+2=5y+3

subtract 3 from both sides

4x-1=5y

divide both sides by 5

4x/5-1/5 = y

Answer:

[tex]4x + 2 = 5y + 3[/tex]

[tex]5y + 3 = 4x + 2[/tex]

[tex]5y = 5x + 2 - 3[/tex]

[tex]5y = 4x - 1[/tex]

[tex] \frac{5y}{5} = \frac{4x - 1}{5} [/tex]

[tex]y = \frac{4x - 1}{5} [/tex]

Answer:

A) 69

Step-by-step explanation:

• First , we put the numbers of the distribution in order:

60 , 66 , 72 , 78 , 80 , 82 , 88 , 90

• Notice that the set has 8 numbers which is an even number.

Then

→ the lower quartile is the mean of the set :

60 , 66 , 72 , 78

Which is the average of the two middle numbers 66 and 72.

Then

[tex]\text{the lower quartile}=\frac{66+72}{2} = 69[/tex]

The points on the circumference are (3, 5), (3, -9), (8, -2 + √24) and (8, -2 - √24))

The circle equation is given as:

(x-3)^2 +(y+2)^2= 49

Rewrite as:

(y+2)^2= 49 -(x-3)^2

Take the square root of both sides

y+2= ±√[49 -(x-3)^2]

Subtract 2 from both sides

y = -2 ± √[49 -(x-3)^2]

Next, we determine the points

Set x = 3

y = -2 ± √[49 -(3-3)^2]

Evaluate

y = -2 ± 7

Solve

y = 5 and y = -9

So, we have

(x, y) = (3, 5) and (3, -9)

Set x = 8

y = -2 ± √[49 -(8-3)^2]

Evaluate

y = -2 ± √24

Solve

y = -2 - √24 and y = -2 + √24

So, we have

(x, y) = (8, -2 + √24) and (8, -2 - √24)

Hence, the points on the circumference are (3, 5), (3, -9), (8, -2 + √24) and (8, -2 - √24)

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None of the equation of the line is parallel because there slope are not equal.

Parallel line have the same slope.

Therefore, using slope intercept equation,

y = mx + b

where

Hence,

y = −5x − 5 , slope = -5

x + 12y = −4; y = -1 / 3 - 1 / 12 x, slope = 1 / 12

y = −2x + 4: slope = -2

y + 8 = -15x + 135; y  = -15x + 127, slope  = -15

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Answer:

305.25

Step-by-step explanation:

18.5 for one floor, multiply by 15 to get the amount for 15 floors: 18.5*15 = 277.5 sheets

We need to add 10% for waste, meaning 277.5 plus the waste: 1.1*277.5 = 305.25

The drywaller needs to order 306 core boards for the 15-story elevator shaft, considering the 10% waste.

calculate the number of core boards needed for the 15-story elevator shaft, we can follow these

Calculate the total number of sheets for all floors:

Total sheets = Number of floors × Sheets per floor

Total sheets = 15 floors × 18.5 sheets per floor

Total sheets = 277.5 sheets

Add 10% of the total for waste:

Waste percentage = 10% = 0.10 (in decimal form)

Waste sheets = Total sheets × Waste percentage

Waste sheets = 277.5 sheets × 0.10

Waste sheets = 27.75 sheets

Calculate the final number of boards needed (including waste):

Number of boards = Total sheets + Waste sheets

Number of boards = 277.5 sheets + 27.75 sheets

Number of boards = 305.25 sheets

Since you can't have a fraction of a sheet, we need to round up to the nearest whole number.

Therefore, the drywaller needs to order 306 core boards for the 15-story elevator shaft, considering the 10% waste.

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The solution to the system of inequalities is (0.667, 5.333)

The system of inequalities is given as:

y > 2x + 4

x + y ≤ 6

Next, we plot the graph of the system using a graphing tool

From the graph, both inequalities intersect at

(0.667, 5.333)

Hence, the solution to the system of inequalities is (0.667, 5.333)

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Using the z-distribution, it is found that she should take a sample of 46 students.

The confidence interval is:

[tex]\overline{x} \pm z\frac{\sigma}{\sqrt{n}}[/tex]

The margin of error is:

[tex]M = z\frac{\sigma}{\sqrt{n}}[/tex]

In which:

In this problem, we have a 95% confidence level, hence[tex]\alpha = 0.95[/tex], z is the value of Z that has a p-value of [tex]\frac{1+0.95}{2} = 0.975[/tex], so the critical value is z = 1.96.

Scores on the math portion of the SAT are believed to be normally distributed and range from 200 to 800, hence, by the Empirical Rule the standard deviation is found as follows:

[tex]6\sigma = 800 - 200[/tex]

[tex]6\sigma = 600[/tex]

[tex]\sigma = 100[/tex]

The sample size is n when M = 29, hence:

[tex]M = z\frac{\sigma}{\sqrt{n}}[/tex]

[tex]29 = 1.96\frac{100}{\sqrt{n}}[/tex]

[tex]29\sqrt{n} = 196[/tex]

[tex]\sqrt{n} = \frac{196}{29}[/tex]

[tex](\sqrt{n})^2 = \left(\frac{196}{29}\right)^2[/tex]

n = 45.67.

Rounding up, a sample of 46 students should be taken.

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The story problem that aligns with the division problem is A. A whiteboard is 2/5 meter long and has an area of 1/2 square meter. How wide is the whiteboard?

From the information given, it can be seen that 1/2 is divided by 2/5.

In this case, the story that aligns with it is that a whiteboard is 2/5 meter long and has an area of 1/2 square meter. How wide is the whiteboard?

In this case, the length is multiplied by width to get the area.

Therefore, the correct option is A.

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The explicit formula that can be used to determine the number of wolves is w = 210 (1.0692^t) .

The first step is to determine the growth rate of the wolf species.

Growth rate = [(future population / present population)^(1/number of years)] - 1

[(410 / 210)^(1/10)] - 1 = 6.92

The exponential function would have the form:

FV = P (1 + r)^t

w = 210 (1.0692^t)

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The equation that shows the relationship between the number of paragliding trips, adventure tours, and his total pay is 55a + 75p = 2725

An equation is an expression that shows the relationship between two or more numbers and variables.

Let a represent the number of adventure tours and p represent the number of paragliding trip, hence:

55a + 75p = 2725

The equation that shows the relationship between the number of paragliding trips, adventure tours, and his total pay is 55a + 75p = 2725

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The equation for a circle is:

[tex](x-a)^{2} + (y-b)^{2} + = r^{2}[/tex]

Where (a,b) is the circle's center and r is the circle's radius.

First, we can find the center point of the circle. Because the two points from the problem are the endpoints of a diameter, the midpoint of the line segment is the center point of the circle.

The formula to find the mid-point of a line segment giving the two endpoints is:

M = [tex](\frac{x_{1} + x_{2} }{2}[/tex], [tex]\frac{y_{1} + y_{2} }{y} )[/tex]

Where M is the midpoint and the given points are:

[tex](x_{1}, y_{1} )[/tex] and [tex](x_{2} , y_{2})[/tex]

Substituting the values from the two points in the problem gives:

[tex]M = (\frac{11+9}{2}[/tex],[tex]\frac{-2+4}{2} )[/tex]

[tex]M = (\frac{11+9}{2}[/tex],[tex]\frac{2-4}{2})[/tex]

[tex]M = (\frac{20}{2} , \frac{2}{2})[/tex]

[tex]M = (10,1)[/tex]

Answer: False

Step-by-step explanation:

G(x) is not a one-to-one function (since it fails the horizontal line test). Therefore, it doesn't have an inverse.

The smallest possible prime number x is 17.

An element of a whole is a fraction. The number is represented mathematically as a quotient, where the numerator and denominator are split. Both are integers in a simple fraction. A fraction appears in the numerator or denominator of a complex fraction. The numerator of a correct fraction is smaller than the denominator.

given, Two-thirds of a number x is at least 10.

So,

2/3 x ≥ 10

x ≥ 15

First prime number that is bigger than 15 is 17.

therefore, The smallest possible prime number x is 17.

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