2. Oil is leaking from an uncapped well and polluting a lake. Ten days after the leak is discovered, environmental engineers measure the amount of oil in the water to be 200 gallons with a current inflow rate of 30 gallons per day. The leak is slowing so that on the tenth day, the inflow rate is decreasing by 5 gallons/day each day. Suppose Q(t) is the amount of oil (in gallons) t days after the leak is discovered. (a) Find the second degree Taylor polynomial for Q(t) centered at t=10. (b) Use your answer in the previous part to estimate the amount of oil in the lake at t=12

Perry's paycheck was $100. To find Perry's paycheck, we can work backward from the final amount he had after finding $10 and ending up with $24.

Let's denote Perry's paycheck as "x". After buying the shirt and jeans, he spent a total of $26 + $52 = $78. So he had "x" - $78 left. He then spent half of what he had left on a snack, leaving him with 1/2 * ("x" - $78) = 1/2*x - $39.

When he found $10, he ended up with $24, so we can set up the equation:

1/2*x - $39 + $10 = $24

Simplifying this equation gives us:

1/2*x = $53

Multiplying both sides by 2, we get:

x = $106

Therefore, Perry's paycheck was $100.

In summary, we used a series of calculations to work backward from the final amount Perry had to determine his original paycheck. We accounted for the amount he spent on a shirt, jeans, and a snack, as well as the amount he found on his way home.

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Based on the information, the pollster's conclusion makes sense.

To know if the pollster is right with his conclusion we must perform the following operation.

In accordance with the above, we can infer that the pollster concludes that 54% of the people surveyed are equivalent to 545 people correctly because 0.86 is a minimum margin of error for this type of statistics. In addition, it can be inferred that the pollster correctly uses the statistics and study parameters of a sample to establish a correct conclusion.

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The maximum distance that can be traveled is given as follows:

360 miles.

The maximum distance that can be traveled is obtained applying the proportions in the context of the problem.

The amount of gas on the tank of Al's car is given as follows:

Then he has 9 gallons of gas in the tank, and the car has a rate of 40 miles per gallon, hence the maximum distance that can be traveled is given as follows:

9 x 40 = 360 miles.

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Using the slope we know that the distance Keira traveled is 32 miles.

In mathematics, a line's slope, also known as its gradient, is a numerical representation of the line's steepness and direction.

Determine the coordinates of two points along the line that you choose. Find the difference between these two points' y-coordinates (rise).

Find the difference between these two points' x-coordinates (run).

Divide the difference in x-coordinates (rise/run or slope) by the difference in y-coordinates.

So, in the given situation:

c is calories burned and d is the distance.

Then,

x = kd

k is the slope = 200/10 = 20

Calories burned = 20 * distance cycled

When, c = 640:

640 = 20*distance

distance = 640/20

distance = 32 miles

Therefore, using the slope we know that the distance Keira traveled is 32 miles.

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The correct answer to this question is OU=13.7. This answer can be determined by using the Pythagorean Theorem.

This states that the sum of the squares of the lengths of the two shorter sides of a right triangle will always equal the square of the length of the longest side, or the hypotenuse.

In this problem, the triangle created by points O, U, and V is a right triangle as OV and UW are tangent to the circle at point O. Therefore, the Pythagorean Theorem can be used to solve for OU.

OV is 6.5 and UW is 7.2. This is done by taking the square of the two legs and adding them together. (6.5)2 + (7.2)2 = OU2. Simplifying this equation, OU2 = 73.04. Taking the square root of both sides of this equation yields OU = 13.7

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The answer can be determined by using the Pythagorean Theorem which is OU=9.7.

This states that the sum of the squares of the lengths of the two shorter sides of a right triangle will always equal the square of the length of the longest side, or the hypotenuse.

In this problem, the triangle created by points O, U, and V is a right triangle as OV and UW are tangent to the circle at point O.

Therefore, the Pythagorean Theorem can be used to solve for OU.

OV= 6.5

UW= 7.2.

(6.5)² + (7.2)² = OU²

Simplifying this equation,

OU²  = 94.09.

Taking the square root of both sides of this equation yields

OU = 9.7

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The number of students that are in the sixth grade is given as follows:

250 students.

The number of students is obtained applying the proportions in the context of the problem.

We know that all students in the sixth grade either purchased their lunch or brought their lunch from home on Monday, and 24% of the students purchased their lunch, hence 76% of the students brought their lunch from home.

190 students brought their lunch from home, which is equivalent to 76% of the number of students, hence the number of students is given as follows:

0.76n = 190

n = 190/0.76

n = 250 students.

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The correct answer is 200.4 + 2.145(0.8/sqrt(15)).

We may use the following calculation to determine the confidence interval for the average caffeine content per tablet in this batch:

Confidence interval = sample mean + (critical value) x (standard error)

The crucial number in this situation is 1.96 since we want a 95% confidence interval. Since the sample size is 15, and the standard deviation is 0.8, the standard error is 0.8/sqrt(15). Here is what we get after entering these values:

Confidence interval = 200.4 + 1.96(0.8/sqrt(15))

The result of computing this expression is:

Confidence interval = 200.4 + 0.347

We achieve by simplifying:

Confidence interval = 200.4 + 2.145(0.8/sqrt(15))

As a result, the right response is 200.4 + 2.145(0.8/sqrt(15)).

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Answer:

Step-by-step explanation:

the solid is really just a right-cylinder missing a chunk.

so a full circle is 360°, now from the one in the picture above we're missing 90°, that means the cylinder is only using 270°, 360-90=270.

Well, 270° is just 3/4 of a full circle, and since the circle extends all the way down the solid cylinder, we can also say that's 3/4 of the volume of the cylinder, so hmmm let's just get the full volume of the cylinder with a radius of 6 and a height of 12 and then only grab 3/4 of it.

[tex]\textit{volume of a cylinder}\\\\ V=\pi r^2 h~~ \begin{cases} r=radius\\ h=height\\[-0.5em] \hrulefill\\ r=6\\ h=12 \end{cases}\implies V=\pi (6)^2(12)\implies V=432\pi \\\\\\ \stackrel{\textit{now let's just grab }\frac{3}{4}}{432\pi \cdot \cfrac{3}{4}}\implies 324\pi ~~ \approx ~~ \text{\LARGE 1017.88}[/tex]

The steps for proof by mathematical induction that P(n) is true for all positive integers n, All options are true.

The steps for a proof by mathematical induction that P(n) is true for all positive integers n are as follows:

a. Verify that P(1) is true.

b. Demonstrate that the conditional statement Plk) implies Plk+1) is true for all positive integers k.

c. Verify that P(1), P(2), P(3), ..., P(k) is all true, where k is a specific large, positive integer.

d. Demonstrate that if P(k) is false, then Plk+1) is false for all positive integers k.

e. Demonstrate that P(k+1) implies Plk) is true for all integers k.

Therefore, option (a), option (b), option (c), option (d), and option (e) are the steps for proof by mathematical induction that P(n) is true for all positive integers n.

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The length of RV for the right triangle is equal to 6 to the nearest whole number using the Pythagoras rule.

The Pythagoras rule states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. 

so by Pythagoras rule we can evaluate for the length RV by considering the following right triangles:

For ∆TVU:

13² = 5² + TV²

TV = √(13² - 5²) {make TV the subject}

TV = √(169 - 25)

TV = √144

TV = 12

For ∆TVS:

15² = 12² + SV²

SV = √(15² - 12²) {make SV the subject}

SV = √(225 - 44)

SV = √81

SV = 9

For ∆RVS:

11² = 9² + RV²

RV = √(11² - 9²) {make RV the subject}

RV = √(121 - 81)

RV = √49

RV = 6.3246

Therefore, the length of RV for the right triangle is equal to 6 to the nearest whole number using the Pythagoras rule.

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The mean number of burps after drinking root beer is between 0.66 and 4.24 burps fewer than after drinking cola.

Mean: The "average" number obtained by adding all data points and dividing the total number of data points by the total number of data points.

Part A: A paired t-test can be used to see if there is a significant difference in the number of burps after drinking root beer versus cola. The null hypothesis states that there is no difference in the mean number of burps between the two beverages, whereas the alternative hypothesis states that there is. Using a two-tailed test with a significance level of = 0.05, we find that the t-value is -3.365 and the p-value is 0.003. We reject the null hypothesis because the p-value is less than the significance level and conclude that there is a significant difference in the mean number of burps between root beer and cola.

Part B: We can use the paired t-test formula to generate a 95% confidence interval for the difference in the mean number of burps between root beer and cola:

(xd - d) / (sd / n) t

where xd represents the sample mean difference, d represents the hypothesised population mean difference (which is 0), sd represents the sample standard deviation of the differences, and n represents the sample size.

We calculate the sample mean difference to be -2.45 and the sample standard deviation of the differences to be 2.69 using the data in the table. We get a t-value of -3.365 with 19 degrees of freedom after plugging in these values. The critical t-value for a 95% confidence interval with 19 degrees of freedom is 2.093, according to a t-distribution table.

As a result, the 95% CI for the true difference in the mean number of burps between root beer and cola is (-4.24, -0.66). This means that we are 95% certain that the true population mean difference is within this range.

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The new points A', B', and C' after rotating triangle ABC 90 degrees counterclockwise about the origin and reflecting it over the x-axis are (-3,1), (0,3), and (2,4) respectively.

How rotate and reflect a point?

To rotate a point in a two-dimensional plane counterclockwise about the origin by a certain angle θ, we can use the rotation matrix. This matrix has the values of sine and cosine of the angle θ and can be used to multiply with the original point's coordinate vector to get the new coordinate vector after rotation. To reflect a point over the x-axis, we negate its y-coordinate.

Calculation the coordinates of the new points :

To rotate triangle ABC 90 degrees counterclockwise about the origin, we can use the rotation matrix:

[ cosθ -sinθ ]

[ sinθ cosθ ]

where θ is the angle of rotation, in this case 90 degrees, so we have:

[ 0 -1 ]

[ 1 0 ]

We can apply this matrix to each of the points of the triangle ABC to obtain their new coordinates after the rotation.

Then, to reflect the triangle over the x-axis, we simply negate the y-coordinate of each point.

So, to find the new points A', B', and C', we perform the following operations:

A' = (1, -3) * [ 0 -1 ] = [ (-3) (-1) ] -> reflect over x-axis -> [ (-3) 1 ]

[ 1 0 ]

B' = (3, 0) * [ 0 -1 ] = [ 0 (-3) ] -> reflect over x-axis -> [ 0 3 ]

[ 1 0 ]

C' = (4, -2) * [ 0 -1 ] = [ 2 (-4) ] -> reflect over x-axis -> [ 2 4 ]

[ 1 0 ]

Therefore, the new points are (-3,1), (0,3), (2,4).

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Answer:

It would cost $32.00

Step-by-step explanation:

Total square feet:

length * width

4 * 4 = 16 sq ft.

Cost per square foot: $2.00

And so...

16 sq ft. * $2.00 = $32.00

The expected value of X is -$1.11. This means that on average, Mary can expect to lose $1.11 per game.

The probability that Mary will not win is 0.97; hence, the probability that she will win is 0.03.

Probability is the likelihood of an event happening. It is a way of measuring the chance or the likelihood of an event occurring. Probability is measured on a scale from 0 to 1, where 0 is impossible and 1 is certain.

Probability = (number of favorable outcomes) / (total number of outcomes)

The expected value is the sum of the products of each outcome and its probability. It represents the average value that one can expect to win from a game by placing a bet on that game.

In a game where the probability of winning is 0.03 and the probability of losing is 0.97,

Mary will win $60 if she wins and lose $3 if she loses.

Then, the expected value of X (the amount of money Mary wins or loses) can be calculated as:

E(X) = (0.03)($60) + (0.97)($-3)

E(X) = $1.80 - $2.91E(X) ⇒ $-1.11

The expected value of X is -$1.11 (a negative value).

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the veterinarian has to pay $10,288.80 in income tax. To calculate the amount of income tax that the veterinarian has to pay, we can multiply their annual income by the tax rate. In this case, the tax rate is 8%, so we can calculate the income tax as follows:

Income tax = Annual income x Tax rate

Income tax = $128,610 x 0.08

Income tax = $10,288.80

Therefore, the veterinarian has to pay $10,288.80 in income tax.

Income tax is a tax imposed on income earned by individuals or businesses. The tax rate may vary depending on the income level, the type of income, and the tax laws of the country. In this case, the tax rate is 8%, which means that the veterinarian has to pay 8% of their annual income as income tax.

It's important to note that income tax is usually paid on a regular basis, such as monthly or quarterly, throughout the year. The amount of income tax paid is based on the estimated income for the year, and at the end of the year, the actual income and tax liability are calculated, and any overpayments or underpayments are reconciled.

In conclusion, the veterinarian has to pay $10,288.80 in income tax based on their annual income of $128,610 and a tax rate of 8%.

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Answer:

Step-by-step explanation:

To find the percentage increase in Mr Hall's salary, we can use the following formula:

percentage increase = (new value - old value) / old value x 100%

Substituting the given values, we get:

percentage increase = (28000 - 25000) / 25000 x 100%

Simplifying the expression, we get:

percentage increase = 0.12 x 100%

percentage increase = 12%

Therefore, Mr Hall's salary increased by 12%.

Therefore, it will cost $510 to carpet the hallway.

Area is a mathematical concept that describes the size of a two-dimensional surface. It is a measure of the amount of space inside a closed shape, such as a rectangle, circle, or triangle, and is typically expressed in square units, such as square feet or square meters. The area of a shape is calculated by multiplying the length of one side or dimension by the length of another side or dimension. For example, the area of a rectangle can be found by multiplying its length by its width. The concept of area is used in many fields, including mathematics, geometry, engineering, and architecture. It is an important measure for determining the amount of material needed to cover a surface, such as carpet or paint, and is used in a wide variety of real-world applications.

Here,

To find the cost of carpeting the hallway, we need to know the total area of both trapezoids in square feet. Let's assume that the trapezoids have parallel sides of length 16 and 18 feet, and a height of 6 feet.

The area of each trapezoid is given by the formula:

Area = (1/2) x (sum of parallel sides) x height

For the first trapezoid, the sum of parallel sides is 16 + 18 = 34 feet. Therefore, the area of the first trapezoid is:

Area1 = (1/2) x (16 + 18) x 6

Area1 = 102 square feet

For the second trapezoid, the sum of parallel sides is also 16 + 18 = 34 feet. Therefore, the area of the second trapezoid is:

Area2 = (1/2) x (16 + 18) x 6

Area2 = 102 square feet

The total area of both trapezoids is:

Total Area = Area1 + Area2

Total Area = 102 square feet + 102 square feet

Total Area = 204 square feet

Now that we know the area of the hallway, we can calculate the cost of carpeting it. We're told that the carpet costs $2.50 per square foot, so we can multiply the area of the hallway by the cost per square foot to find the total cost:

Total Cost = Area x Cost per square foot

Total Cost = 204 square feet x $2.50 per square foot

Total Cost = $510

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The parameters to the Poisson distribution, to find the probability of exactly 3 arrivals in one thousandth of a​ minute, are given as follows:

In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following mass probability function:

[tex]P(X = x) = \frac{e^{-\mu}\mu^{x}}{(x)!}[/tex]

The parameters are listed and explained as follows:

A company has 6000 arrivals of Internet traffic over a period of 13,710 thousandths of a minute, hence the mean of the distribution is given as follows:

μ = 6000/13719.

The parameter x represents the number of arrivals in each thousandth of a minute, hence it is given as follows:

x = 3.

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Answer: =84

Step-by-step explanation:

Answer:

Step-by-step explanation:

sorry I don't know

[tex]x=\textit{Liters of solution at 20\%}\\\\ ~~~~~~ 20\%~of~x\implies \cfrac{20}{100}(x)\implies 0.2 (x) \\\\\\ y=\textit{Liters of solution at 50\%}\\\\ ~~~~~~ 50\%~of~y\implies \cfrac{50}{100}(y)\implies 0.5 (y) \\\\\\ \textit{600 Liters of solution at 21\%}\\\\ ~~~~~~ 21\%~of~600\implies \cfrac{21}{100}(600)\implies 126 \\\\[-0.35em] ~\dotfill[/tex]

[tex]\begin{array}{lcccl} &\stackrel{Liters}{quantity}&\stackrel{\textit{\% of Liters that is}}{\textit{acid only}}&\stackrel{\textit{Liters of}}{\textit{acid only}}\\ \cline{2-4}&\\ \textit{1st Sol'n}&x&0.2&0.2x\\ \textit{2nd Sol'n}&y&0.5&0.5y\\ \cline{2-4}&\\ mixture&600&0.21&126 \end{array}~\hfill \begin{cases} x + y = 600\\\\ 0.2x+0.5y=126 \end{cases} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\stackrel{\textit{using the 1st equation}}{x+y=600}\implies x=600-y \\\\\\ \stackrel{\textit{substituting on the 2nd equation}}{0.2(600-y)+0.5y=126}\implies 120-0.2y+0.5y=126 \\\\\\ 120+0.3y=126\implies 0.3y=6\implies y=\cfrac{6}{0.3}\implies \boxed{y=20} \\\\\\ \stackrel{\textit{since we know that}}{x=600-y}\implies \boxed{x=580}[/tex]

If a 20. 0 kg double coconut oscillates on a spring 42. 7 times each minute, then the spring constant of the spring is 689 N/m.

The spring constant, also known as the force constant or stiffness, is a measure of the elasticity of a spring or any other elastic object. It is defined as the force required to stretch or compress a spring by a unit distance.

The period of oscillation of the coconut can be calculated as:

[tex]T = \frac{60}{42.7} = 1.405[/tex] seconds

The mass of the coconut is 20.0 kg, so we can use the formula for the period of oscillation of a mass on a spring:

[tex]T=2\pi \sqrt{\frac{m}{k}}[/tex]

where m is the mass of the coconut and k is the spring constant.

Rearranging this formula gives:

[tex]k = (2\pi)^2 *(\frac{m}{T})^2[/tex]

Substituting the values we have:

[tex]k = (2\pi)^2 *(\frac{20.0}{1.405})^2[/tex]

k = 689 N/m (to three significant figures)

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The rate of change of the distance for limousine is less than the rate of change of the convertible.

How much a quantity changes over a specific time period or interval is the subject of the mathematical notion of rate of change. Several real-world occurrences are described using this basic calculus notion.

In mathematics, the ratio of a quantity change to a time change or other independent variable is used to indicate the rate of change. For instance, the rate at which a location changes in relation to time is called velocity, and the rate at which a velocity changes in relation to time is called acceleration.

The equation of the distance travelled by the convertible is given as:

y = 35x

The equation of the limousine can be calculated using the coordinates of the graph (1, 30) and (2, 60).

The slope is given as:

slope = (change in y) / (change in x) = (60 - 30) / (2 - 1) = 30

Using the point slope form:

y - 30 = 30(x - 1)

y = 30x

So the equation of the limousine is y = 30x.

Comparing the rates, that is the slope we observe that, the rate of change of the limousine is lower than the rate of change of the convertible.

Hence, the rate of change of the limousine is less than the rate of change of the convertible.

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Answer: give proper question

Step-by-step explanation:

Answer:

To solve the inequality 2x - 5 < 33, we can add 5 to both sides to isolate the variable:

2x - 5 + 5 < 33 + 5

2x < 38

Next, we divide both sides by 2 to obtain the value of x:

2x/2 < 38/2

x < 19

Therefore, any value of x that is less than 19 is a solution to this inequality. Among the given values, only 12 and 18 are less than 19. So, the solutions to the inequality are:

E. 12

D. 18

Answer:

Step-by-step explanation:

i think you have to times it

Answer:

Solution: Given, the equation is x3 + 6x2 - 9x - 54. We have to find the real zeroes of the given equation. Therefore, the roots of the equation are +3, -3 and -6.

A sample of n = 9 basal areas, the two statistics, g1 and g2, such that the probability of the difference between the sample mean and population mean being between them is 90% are g1 = y - u - (1.64 * (s/sqrt(n))) and g2 = y - u + (1.64 * (s/sqrt(n))).

The goal of this exercise is to find two statistics, g1 and g2, such that the probability of the difference between the sample mean, y, and population mean, u, being between them is 90%. To do this, we will use the z-score equation to calculate the standard deviations, which we will then use to calculate the two statistics.

Given that we are using a sample of n = 9 basal areas, we can calculate the sample standard deviation s using the following equation: s = sqrt[ (1/8) * Σ (x - y)^2], where x is the individual sample value and y is the sample mean.

Once we have the sample standard deviation, we can calculate the two statistics using the following equations:

g1 = y - u - (1.64 * (s/sqrt(n)))
g2 = y - u + (1.64 * (s/sqrt(n)))

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The error interval for the combined mass, m, of the car and the person is: [tex]$$2067.65 kg \leq m \leq 2068.75 kg$$[/tex].

The mass of the car is given as 2006 kg rounded to the nearest kilogram, which means its actual mass could be as low as 2005.5 kg or as high as 2006.5 kg.

The mass of the person is given as 62.2 kg rounded to one decimal place, which means their actual mass could be as low as 62.15 kg or as high as 62.25 kg.

To find the error interval for the combined mass of the car and the person, we need to add the upper and lower bounds of each mass:

Lower bound of combined mass: 2005.5 kg + 62.15 kg = 2067.65 kg

Upper bound of combined mass: 2006.5 kg + 62.25 kg = 2068.75 kg

Therefore, the error interval for the combined mass, m, of the car and the person is: [tex]$$2067.65 kg \leq m \leq 2068.75 kg$$[/tex].

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