8s + 8 < s + 9 solve the following inequality for a( simplest form)

Work Shown:

[tex]\displaystyle \int \sin(x) dx = -\cos(x)+C\\\\\displaystyle \int f(x)dx = g(x)+C\\\\\displaystyle \int_{a}^{b} f(x)dx = g(b)-g(a)\\\\\displaystyle \int_{0}^{\pi} f(x)dx = g(\pi)-g(0)\\\\\displaystyle \int_{0}^{\pi} \sin(x) dx = -\cos(\pi)-(-\cos(0))\\\\[/tex]

[tex]\displaystyle \int_{0}^{\pi} \sin(x) dx = -\cos(\pi)+\cos(0)\\\\\displaystyle \int_{0}^{\pi} \sin(x) dx = -(-1)+1\\\\\displaystyle \int_{0}^{\pi} \sin(x) dx = 1+1\\\\\displaystyle \int_{0}^{\pi} \sin(x) dx = 2\\\\[/tex]

The area is 2 square units. You can use WolframAlpha or GeoGebra to confirm.

Considering the expression of a line, the equation of the line that passes through the pair of points (-3,5) and (1,-1) is y=-1.5x +0.5.

A linear equation o line can be expressed in the form y = mx + b

where

Knowing two points (x₁, y₁) and (x₂, y₂) of a line, the slope of the line can be calculated using:

m= (y₂ - y₁)÷ (x₂ - x₁)

Substituting the value of the slope and the value of one of the points, the value of the ordinate to the origin can be obtained.

In this case, being (x₁, y₁)= (-3, 5) and (x₂, y₂)= (1, -1), the slope m can be calculated as:

m= ( -1 -5)÷ (1 - (-3))

m= ( -1 -5)÷ (1 + 3)

m= (-6)÷ 4

m= -1.5

Considering point 2 and the slope m, you obtain:

-1= -1.5×1 + b

-1= -1.5 +b

-1 + 1.5= b

0.5= b

Finally, the equation of the line is y=-1.5x +0.5.

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If a random sample of 20 gas stations was selected to estimate the gas prices. The type of error that could have occurred is a type I error, which is the rejection of a true null hypothesis. The probability of making a type I error is equal to the level of significance (α), which is 10% in this case. So, there is a 10% chance of making a type I error in this test.

To test if the true average gas price is under $3.50, we can use a one-tailed t-test with a null hypothesis of:

H0: μ >= $3.50

And an alternative hypothesis of:

Ha: μ < $3.50

where μ is the true population mean gas price.

Using a t-test with a sample size of 20, we can calculate the t-value as:

t = (x -bar - μ) / (s / sqrt(n))

where x-bar is the sample mean, μ is the hypothesized population mean, s is the sample standard deviation, and n is the sample size.

Plugging in the given values, we get:

t = (3.40 - 3.50) / (0.22 / sqrt(20))

t = -2.03

Looking up the t-distribution table with degrees of freedom (df) = 19 and a 10% alpha level (α = 0.10), we find the critical t-value as -1.325.

Since our calculated t-value (-2.03) is less than the critical t-value (-1.325), we reject the null hypothesis in favor of the alternative hypothesis. Therefore, we can conclude that there is sufficient evidence to suggest that the true average gas price is under $3.50.

The type of error that could have occurred is a Type I error, which is rejecting a true null hypothesis. In this case, it means that we conclude that the true average gas price is under $3.50 when it's actually not. The probability of making a Type I error is equal to the alpha level, which is 10% in this case. So there's a 10% chance that we falsely reject the null hypothesis.

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The ice cream factory could make 720 quarts of ice cream in 36 hours.

We can use a proportion to solve this problem:

quarts / hours = constant rate of production

Let x be the number of quarts that could be made in 36 hours. Then we have:

200 quarts / 10 hours = x quarts / 36 hours

Cross-multiplying, we get:

10x = 200 * 36

Simplifying, we get:

10x = 7200

Dividing both sides by 10, we get:

x = 720

Therefore, the ice cream factory could make 720 quarts of ice cream in 36 hours.

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The mean, median, mode and range are 42.2, 30, 87 and 95 respectively.

The mean (average) of a data set is found by adding all numbers in the data set and then dividing by the number of values in the set.

Therefore mean = (23+87+19+34+37+37+5+14+100+26)/10

= 422/10 = 43.2

The median is the middle value when a data set is ordered from least to greatest. The mode is the number that occurs most often in a data set.

median = 26+34/2 = 60/2 = 30

Mode is the number with the highest frequency

Therefore, the mode is 87

Range Is the difference between the highest and the lowest

= 100-5 = 95

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The equation of the circle is (x + 1)² + (y - 1)² = 16.

option D

The radius of the circle is 4, and the center of the circle is (-1, 1).

The equation of a circle with center (a,b) and radius r is given by:

(x - a)² + (y - b)^2 = r²

In this case, the center of the circle is (-1, 1) and the radius is 4. Plugging these values into the equation, we get:

(x - (-1))² + (y - 1)² = 4²

Simplifying the equation, we get:

(x + 1)² + (y - 1)² = 16

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Answer: (a) Let the circle roll along the x-axis with the center at the origin. Then, the parametric equations of the cycloid are given by:

x = t - sin(t)

y = 1 - cos(t)

where t is the angle in radians that the circle has rotated from its initial position.

(b) The arclength of one period of the cycloid is given by the integral:

L = ∫(0,2π) √[dx/dt]^2 + [dy/dt]^2 dt

Substituting the parametric equations from part (a), we get:

dx/dt = 1 - cos(t)

dy/dt = sin(t)

Therefore,

[dx/dt]^2 + [dy/dt]^2 = 2 - 2cos(t)

Substituting back into the arclength integral, we get:

L = ∫(0,2π) √(2 - 2cos(t)) dt

This integral can be evaluated using the half-angle formula:

cos(t) = 1 - 2sin^2(t/2)

Substituting this into the integral and simplifying, we get:

L = 8∫(0,π/2) √(sin^3(t/2)) dt

This integral can be evaluated using a substitution u = cos(t/2), du = -sin(t/2)dt:

L = 16∫(0,1) √[(1-u^2)^3] du

This integral can be further simplified by making the substitution u = sin(θ):

L = 16∫(0,π/2) cos^4(θ) dθ

This integral can be evaluated using integration by parts and trigonometric identities. After some algebraic manipulations, we get:

L = 8π

Step-by-step explanation:

The future value of the account after 25 years is $210,595.95.

We can use the formula for future value of an annuity:

FV = P * (1 + r)^n + PMT * ((1 + r)^n - 1) / r

Where

Substituting the given values, we get:

FV = 19675.13 * (1 + 0.0925)^25 + 3000 * ((1 + 0.0925)^25 - 1) / 0.0925

Simplifying and evaluating, we get:

FV = $210,595.95

Therefore, the future value of the account after 25 years is $210,595.95.

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Answer:

l = 2w - 3

2(2w - 3) + 2w = 27

4w - 6 + 2w = 27

6w - 6 = 27

6w = 33, so w = 5.5 feet and l = 8 feet

Answer:

Width: 5.5 feet

Length: 8 feet

Step-by-step explanation:

Let's begin by assigning variables to stand in for the rectangle's width and length.

Let x represent the rectangle's width.

The length is three feet shorter than double the width, according to the problem. As a result, the length may be written as:

2x - 3

The lengths of all the sides make up a rectangle's perimeter, which in this instance is:

2w + 2l

By substituting the width and length measurements we discovered:

2x + 2(2x - 3)

Simplifying:

2x + 4x - 6 = 27

6x = 33

x = 5.5

The rectangle is 5.5 feet wide as a result.

We may change x into the expression we discovered earlier to figure out the length:

2(5.5) - 3 = 8

The rectangle is 8 feet long as a result.

As a result, the rectangle's measurements are:

Width: 5.5 feet

Length: 8 feet

The first recommendation among the given options is "Buy more than one ticket at a time".

This recommendation suggests that purchasing multiple tickets in a single transaction can result in cost savings. Many airlines offer discounted fares for group bookings, which means buying multiple tickets at once can save money compared to buying them individually.

Buying tickets in bulk not only saves money, but it also provides a level of flexibility in terms of travel plans. For instance, if a group of travelers plan to visit a particular destination together, they can book their tickets together and have a better chance of securing seats on the same flight. Additionally, in case there are changes in the travel plans, such as the need to reschedule or cancel, group bookings provide more flexibility in terms of making changes or requesting refunds.

Overall, buying more than one ticket at a time can be a wise decision, especially if one is traveling with a group or planning multiple trips in the near future. By doing so, one can save money and have more flexibility in terms of travel plans, making the overall travel experience smoother and more enjoyable.

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The statement that is true of the pizza is A. About half of the pizza is left.

The answer to the subtraction of the fractions is 7/24.

Miron was 1/3 of an hour faster today than yesterday.

Initially, 2/3 of the pizza was left over. Marcy ate 2/9 of the pizza.

2/3 - 2/9

(2/3) x (3/3) = 6/9

6/9 - 2/9 = 4/9

So, 4/9 of the pizza is left. Since 4/9 is less than half but more than very little.

2/3 - 3/8

Find a common denominator, which in this case is 24:

(2/3) x (8/8) = 16/24

(3/8) x (3/3) = 9/24

Now subtract:

16/24 - 9/24 = 7/24

The answer in simplest form is 7/24

Miron finished his tasks in 3/4 of an hour yesterday and 5/12 of an hour today.

3/4 - 5/12

Find a common denominator, which in this case is 12:

(3/4) x (3/3) = 9/12

9/12 - 5/12 = 4/12

4/12 = 1/3

Miron was 1/3 of an hour faster today than yesterday.

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Answer:x+2

Step-by-step explanation:

Answer:

Greater than 7.

Step-by-step explanation:

To solve for x in the inequality, we need to isolate x on one side of the equation. First, we need to subtract

$ [tex]\large \boxed{\mathrm{1328}}[/tex]

from each side, giving $30x > $210. Then, we divide both sides by 30 to find that x > 7. Therefore, to make the inequality true, x needs to be greater than 7.

Answer:

x=0, 5 (-5±√15)/2

Step-by-step explanation:

see images for solution

5.) The measure of angle D of the given triangle would be = 59.5°

To calculate the missing angles, sine rule is used which is ;

Sine rule = a/sin A = b/sin B

Where a = 23

A = 56°

b = 24

B = X

That is;

23/sin56° = 24/sinx

Make sin X the subject formula;

Sin X = sin56×24/23

= 0.829037572×24/23

= 19.89690174/23

= 0.865082684

X = Sin-1 (0.865082684)

= 59.89214806

= 59.9° (approximately)

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Given that biology is the subject, the Probability of being on time is 32.8%.

In this case:

P(on-time arrival in class | biology) = P(on-time arrival in class and biology) / P(biology) is the result of applying the conditional probability formula.

According to the table, 82.4% of students will be on time for biology class. Since there are four topics listed, each with an equal weight, the likelihood of biology is 25%. Therefore, we have:

P(on-time arrival in class | biology) = 82.4% / 25% = 0.328 = 32.8%

Therefore it can be concluded that the probability of on-time arrival in class given that the subject is biology is 32.8%.

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All answers that describe the quadrilateral are parallelogram and rhombus.

In the diagram provided as seen from online sources, the lines, BO=OD and this equals x. Next, SO=OC which equals s. Since the diagonals bisect each other, we can refer to this as a parallelogram.

Next, angle ABO equals angle ADO and since the opposite sides are parallel and equal, we can also refer to the quadrilateral as a rhombus.

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Answer: 50

Step-by-step explanation
One method to estimate the number of fish in the pond is to use the mark-and-recapture method. This method assumes that the proportion of marked fish in the second sample is the same as the proportion of marked fish in the entire population. Using this assumption, we can estimate the total number of fish in the pond as follows:

Let N be the total number of fish in the pond.

Let M be the number of fish caught in the first sample (50 fish).

Let R be the number of marked fish caught in the second sample (8 fish).

Let r be the unknown total number of marked fish in the pond.

According to the mark-and-recapture method, we have the following proportion:

M/N = R/r

We can rearrange this equation to solve for N:

N = M * r / R

Substituting the values given in the problem, we have:

N = 50 * r / 8

Simplifying, we get:

N = 6.25 * r

This means that the estimated total number of fish in the pond is 6.25 times the number of marked fish in the second sample. Since we know that 8 fish were marked in the second sample, the estimate for the total number of fish in the pond is:

N = 6.25 * 8 = 50

Therefore, the estimated number of fish in the pond is 50

The equation is given by y = 2.5x. It means that the cost of each gallon of gas is $2.5

An exponential equation is an expression that shows how numbers and variables using mathematical operators.

Let y represent the cost in dollars for x gallons of gas she puts in the tank

The graph passes through point (0, 0) and (3, 7.5), hence:

y - 0 = [(7.5 - 0) / (3 - 0)](x - 0)

y = 2.5x

The equation is given by y = 2.5x

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The volume of the rectangular prism for the given dimensions length , width , and the height is equal to 168 cubic inches.

Dimensions of the rectangular prism is equal to,

length of the rectangular prism = 12 inches

width of the rectangular prism = 8 inches

height of the rectangular prism = 1.75 inches

The formula for the volume of a rectangular prism is equal to,

Volume of the rectangular prism = length x width x height

Plugging in the given values of the dimensions , we have,

⇒ Volume of the rectangular prism  = 12 inches x 8 inches x 1.75 inches

Simplifying this expression to get the volume of rectangular prism, we get,

⇒ Volume of the rectangular prism = 168 cubic inches

Therefore, the volume of the rectangular prism is 168 cubic inches.

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The angle measure of the arc bounding the sector with area 8π square inches is 45 degrees.

The area of the sector is given by the formula:

[tex]A = (1/2) r^2[/tex]θ

where A is the sector's area,  is the sector's central angle in radians, and r is the circle's radius.

The radius of the circle is 8 inches, and the area of the sector is 8 square inches. When these values are added to the formula, we get:

8π = [tex](1/2) (8^2)[/tex] θ

Simplifying, we get:

8π = 32θ

Dividing both sides by 32, we get:

θ = π/4 radians

To convert this to degrees, we multiply by 180/π:

θ = (π/4) × (180/π) = 45 degrees

As a result, the angle of the arc that divides the 8-square-inch sector is 45 degrees..

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The  equation that shows the number of calories, c, in terms of the number servings, n  where Han wants to know how many calories he gets from the fruit snacks is  c = 90n.

Based on the provided information from the question, whereby Han wants to know how many calories he gets from the fruit snacks,  and was able to find the multilication of  number of servings, n, with calories per serving.

Therfore, we  can write the equation  as c = 90n since it is about the multiplication of the variables.

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complete question;

Each serving of a certain fruit snack contains 90 calories. Han wants to know how many calories he gets from the fruit snacks. Write an equation that shows the number of calories, c, in terms of the number servings, n. Complete the equation below c=______ Use n as your variable.

Assume that A and B are independent events. it means that the occurrence of event A does not affect the probability of event B and vice versa.

b. Assume that A and B are independent events. My response is Yes, P(A and B) can be defined as P(B) • P(A|B).

In the event that A and B are independent events, at that point the probability of A happening is one that is influenced by the event of B and vice versa. Hence, P(B) is comparable to P(B|A), showing that the probability of occasion B happening remains unaltered in any case of event A's event. Within the same vein, the likelihood of event A happening remains consistent whether event B has stay put or not, indicated as P(A) = P(A|B).

Therefore in response to question 2, The reason why independent events are not affected  by one another is that the likelihood of one event happening has no effect on the probability of the other event occurring.

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The population N(t) at any time t≥0 is given by the function N(t) = 100t^2 + 800t + 1332.

a) To find the population after 14 days, we need to integrate the given rate function from 0 to 14:

N(14) - N(0) = ∫(0 to 14) N'(t) dt

N(14) - 1500 = ∫(0 to 14) [200(t+2) - 12] dt

N(14) - 1500 = [100[tex]t^2[/tex] + 800t - 168] evaluated from t=0 to t=14

N(14) = 1500 + [100(14[tex])^2[/tex] + 800(14) - 168] - [100(0[tex])^2[/tex] + 800(0) - 168]

N(14) = 1500 + 17,512

N(14) = 19,012

So the population after 14 days is 19,012 cells.

To find the population after 34 days, we can use the same method:

N(34) - N(0) = ∫(0 to 34) N'(t) dt

N(34) - 1500 = ∫(0 to 34) [200(t+2) - 12] dt

N(34) - 1500 = [100[tex]t^2[/tex] + 800t - 168] evaluated from t=0 to t=34

N(34) = 1500 + [100(34[tex])^2[/tex]+ 800(34) - 168] - [100(0[tex])^2[/tex] + 800(0) - 168]

N(34) = 1500 + 116,768

N(34) = 118,268

So the population after 34 days is 118,268 cells.

b) To find the population N(t) at any time t≥0, we can integrate the given rate function from 0 to t:

N(t) - N(0) = ∫(0 to t) N'(x) dx

N(t) - 1500 = ∫(0 to t) [200(x+2) - 12] dx

N(t) = 1500 + ∫(0 to t) [200(x+2) - 12] dx

N(t) = 1500 + [100[tex]x^2[/tex] + 800x - 168] evaluated from x=0 to x=t

N(t) = 1500 + 100[tex]t^2[/tex] + 800t - 168

N(t) = 100t^2 + 800t + 1332

So the population N(t) at any time t≥0 is given by the function N(t) = 100[tex]t^2[/tex]+ 800t + 1332.

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a) The population after 14 days is 22,788 cells, and the population after 34 days is 126,056 cells.

b) The population at any time t≥0 is given by the equation N(t) = [tex]100t^2[/tex] + 388t + 1500, where t is measured in days and N(t) is the number of cells in the culture at time t.

a) To find the population after 14 days, we can use the given rate equation to calculate the growth rate at t=14 as follows:

N'(14) = 200(14+2) - 12 = 2808 cells/day

Then we can use the following formula to calculate the population after 14 days:

N(14) = N(0) + ∫N'(t) dt, from t=0 to t=14

N(14) = 1500 + ∫(200(t+2) - 12) dt, from t=0 to t=14

N(14) = 1500 + [tex][100t^2 + 400t - 12t][/tex]from t=0 to t=14

N(14) = 1500 + [tex][100(14)^2 + 400(14) - 12(14)] - [100(0)^2 + 400(0) - 12(0)][/tex]

N(14) = 1500 + 21288

N(14) = 22788 cells

Similarly, to find the population after 34 days, we can use the given rate equation to calculate the growth rate at t=34 as follows:

N'(34) = 200(34+2) - 12 = 6772 cells/day

Then we can use the same formula as above to calculate the population after 34 days:

N(34) = N(0) + ∫N'(t) dt, from t=0 to t=34

N(34) = 1500 + ∫(200(t+2) - 12) dt, from t=0 to t=34

N(34) = [tex]1500 + [100t^2 + 400t - 12t] from t=0 to t=34[/tex]

N(34) =[tex]1500 + [100(34)^2 + 400(34) - 12(34)] - [100(0)^2 + 400(0) - 12(0)][/tex]

N(34) = 1500 + 124556

N(34) = 126056 cells

b) To find the population N(t) at any time t≥0, we can use the same formula as above:

N(t) = N(0) + ∫N'(t) dt, from t=0 to t=t

N(t) = 1500 + ∫(200(t+2) - 12) dt, from t=0 to t=t

N(t) = [tex]1500 + [100t^2 + 400t - 12t][/tex] from t=0 to t=t

N(t) = [tex]1500 + 100t^2 + 388t[/tex]

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a) the total number of  strings of seven hexadecimal digits that do not have any repeated digits is 450,450

b) The number of strings of seven hexadecimal digits have at least one repeated digit is 267,985,006

c) the probability that a randomly chosen string of seven hexadecimal digits has at least one repeated digit is 99.8%

The total number of strings of seven hexadecimal digits is 16⁷.

For a string to not have any repeated digits, the first digit can be chosen in 16 ways, the second in 15 ways (since one digit has already been used),

the third in 14 ways, and so on, giving a total of 16 * 15 * 14 * 13 * 12 * 11 * 10 = 13,608,960 strings.

So

(16 * 15 * 14 * 13 * 12 * 11 * 10)/ (2 * 2 *2* 2* 2 *2* 2)

=  450,450

b) the number of strings with no repeated strings is 450450, so the nubmer of strings with at least one rpeated digit is

16⁷ - 450450 = 267,985,006

c) the probability that a  a randomly chosen string of seven hexadecimal digits has at least one repeated digit

= 267,985,006/ 16⁷

= 0.99832194298
= 99.8%

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P(A and B) is not equal to P(A) * P(B), we can conclude that events A and B are dependent.

Probability is a measure of the likelihood or chance of an event occurring. It is a number between 0 and 1, with 0 representing an impossible event and 1 representing a certain event. The probability of an event is calculated by dividing the number of ways the event can occur by the total number of possible outcomes.

We can determine if events A and B are independent or dependent by comparing the probability of both events occurring together with the product of their individual probabilities.

P(A and B) = 1/7

P(A) = 2/3

P(B) = 3/7

If A and B are independent events, then P(A and B) = P(A) * P(B). However, in this case, P(A and B) is not equal to P(A) * P(B):

P(A) * P(B) = (2/3) * (3/7) = 2/7

Since P(A and B) is not equal to P(A) * P(B), we can conclude that events A and B are dependent.

Events A and B are dependent since the probability of their intersection is not equal to the product of their individual probabilities. Specifically, P(A and B) = 1/7 while P(A)*P(B) = 2/7.

Therefore, P(A and B) is not equal to P(A) * P(B), we can conclude that events A and B are dependent.

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