Find y such that
∫x^5 dx = ∫ x^y dx

By solving each expression without using the absolute value symbol are:-

1. -x-120 if x<-120

2. -(x-120) if x<-120

3. x+12 if x>-12

4. -(x+12) if x<-12

To simplify each expression without using absolute value symbols, we need to determine the cases when the expression inside the absolute value bars is positive and negative. Then we can remove the absolute value bars and simplify the expression accordingly.

For the first expression, |120+x|, if x is less than -120, then x+120 will be negative. Therefore, we can simplify the expression as -x-120.

For the second expression, |x-120|, if x is less than -120, then x-120 will also be negative. Therefore, we can simplify the expression as -(x-120).

For the third expression, |x-(-12)|, if x is greater than -12, then x-(-12) is positive. Therefore, we can simplify the expression as x+12.

For the fourth expression, |x-(-12)|, if x is less than -12, then x-(-12) is negative. Therefore, we can simplify the expression as -(x+12).

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The probability that cards #1, #2, and #3 are still in order on the top of the stack is 0.03%. Therefore, the correct option is D.

To find the probability, we need to calculate the number of ways in which the first 3 cards can remain in order on the top of the stack, and divide it by the total number of ways the cards can be arranged.

The number of ways in which the first 3 cards can remain in order is 3! (3 factorial), because there are 3 cards and they can be arranged in 3! = 6 ways.

The total number of ways the cards can be arranged is 10!, because there are 10 cards and they can be arranged in 10! = 3,628,800 ways.

So, the probability is:

3! / 10! = 6 / 3,628,800 = 0.000166 = 0.0166%

We can convert it to a percentage by multiplying by 100:

0.0166 x 100 = 1.66%

However, this is the probability that the first 3 cards are in a specific order, not necessarily the original order. Since the question asks for the probability that the original order is maintained, we need to divide the probability by 3!, which gives:

0.0166 / 3! = 0.000277 = 0.0277%

This is closest to answer choice D) 0.03%.

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To find the slope of the secant over the interval from x=2 to x=2.001, we need to use the formula for the slope of a secant line:

slope = (f(2.001) - f(2)) / (2.001 - 2)

First, we need to find the values of f(2.001) and f(2):

f(2) = 2^3 = 8
f(2.001) = 2.001^3 ≈ 8.012006001

Plugging these values into the formula, we get:

slope = (8.012006001 - 8) / (2.001 - 2)
slope ≈ 1.006001

Therefore, the slope of the secant over the interval from x=2 to x=2.001 is approximately 1.006001. So the answer is B. slope = 1.006001.
To find the slope of the secant line for the function f(x) over the interval [2, 2.001], we will use the slope formula:

slope = (f(2.001) - f(2)) / (2.001 - 2)

First, find the values of f(2) and f(2.001) by plugging the values of x into the given function f(x) = x^3:

f(2) = 2^3 = 8
f(2.001) = (2.001)^3 ≈ 8.006012

Now, plug these values into the slope formula:

slope = (8.006012 - 8) / (2.001 - 2) = 0.006012 / 0.001 = 6.012

The slope of the secant line over the interval is approximately 6.012. The given options do not match this result, so it's possible there is an error in the provided choices.

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a = 5

b = 4

c= 0

Therefore, the equation for graph C is Y = a ^b + c

Y = 5 ^4 + 0

A graph is described as a diagram showing the relation between variable quantities, typically of two variables, each measured along one of a pair of axes at right angles.

Graphs are a popular tool for graphically illuminating data relationships.

A graph serves the purpose of presenting data that are either too numerous or complex to be properly described in the text while taking up less room.

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The distances between the points are approximately 7.07, 10.20, and 7.07.

To find the distance between the points d(-8, 1) and e(-3, 6), we use the distance formula:

distance = √[(x2 - x1)^2 + (y2 - y1)^2]

Plugging in the values, we get:

distance = √[(-3 - (-8))^2 + (6 - 1)^2]
distance = √[5^2 + 5^2]
distance = √50
distance ≈ 7.07

To find the distance between the points e(-3, 6) and f(7, 4), we again use the distance formula:

distance = √[(x2 - x1)^2 + (y2 - y1)^2]

Plugging in the values, we get:

distance = √[(7 - (-3))^2 + (4 - 6)^2]
distance = √[10^2 + (-2)^2]
distance = √104
distance ≈ 10.20

To find the distance between the points f(7, 4) and g(2, -1), we again use the distance formula:

distance = √[(x2 - x1)^2 + (y2 - y1)^2]

Plugging in the values, we get:

distance = √[(2 - 7)^2 + (-1 - 4)^2]
distance = √[(-5)^2 + (-5)^2]
distance = √50
distance ≈ 7.07

So, the distances between the points are approximately 7.07, 10.20, and 7.07.

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complete question:

Determine the distance between DE, EF and FG.

D(-8, 1), E(-3, 6), F(7, 4), G(2, -1)

Example problem:

Solution:

Using the formula for exponential decay:

[tex]\sf\qquad\dashrightarrow N(t) = N0 * e^{(-kt)}[/tex]

where:

Since the half-life is 10 years, we know that:

[tex]\sf\qquad\dashrightarrow k = \dfrac{\ln(0.5)}{10} = -0.0693[/tex]

Substituting the given values, we get:

[tex]\sf:\implies N(30) = 100 * e^{(-0.0693 * 30)}[/tex]

[tex]\sf:\implies N(30) = 100 * e^{(-2.079)}[/tex]

[tex]\sf:\implies N(30) = 100 * 0.126[/tex]

[tex]\sf:\implies \boxed{\bold{\:\:N(30) = 12.6\: grams\:\:}}\:\:\:\green{\checkmark}[/tex]

Therefore, after 30 years, only 12.6 grams of the radioactive substance will be left.

The solution to the differential equation is:

[tex]y = e^{(2\sqrt{y} - 0.6137)}[/tex]

To solve this differential equation, we can use the method of separation

of variables. This involves isolating the variables x and y on different

sides of the equation and then integrating both sides with respect to

their respective variables.

Starting with the given equation:

[tex]dy/dx = e^{(\sqrt{(x/y)} )}[/tex]

We can begin by multiplying both sides by dx:

[tex]dy = e^{(\sqrt{(x/y)} ) dx}[/tex]

Now we can separate the variables and integrate both sides:

[tex]\int(1/y)dy = ∫e^{(\sqrt{(x/y))dx} }[/tex]

ln|y| = 2√y + C1 ...where C1 is a constant of integration

To solve for y, we can exponentiate both sides:

[tex]|y| = e^{(2\sqrt{y} + C1)}[/tex]

Since y(1) = 4, we can use this initial condition to determine the sign of y and the value of C1:

[tex]4 = e^{(2\sqrt{4} + C1)} \\4 = e^{(4 + C1)}[/tex]

ln(4) = 4 + C1

C1 = ln(4) - 4 = -0.6137

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The point with the least absolute value on the number line is always the point zero.

The absolute value of a number is the distance that number is from zero on the number line. Therefore, the point on the number line with the least absolute value is the point closest to zero. This point is located at zero itself, as it is the point on the number line that is equidistant from both the positive and negative numbers.

To further explain, consider the following examples:

- The point 3 is 3 units away from zero, but the point -3 is also 3 units away from zero.
- The point 5 is 5 units away from zero, but the point -5 is also 5 units away from zero.
- The point 0 is 0 units away from zero, making it the point with the least absolute value on the number line.

In conclusion, the point with the least absolute value on the number line is always the point zero.

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It will take 32 days for all 24 gallons to leak out of the bucket at a rate of 3 gallons every 4 days.

If water is leaking out of a bucket at a rate of 3 gallons every 4 days, then the rate of leakage is 3/4 gallons per day.

Let x be the number of days it takes for all 24 gallons to leak out. To explain this situation, we can construct an equation.

24=3/4*x

To solve for x, we can cross-multiply.

24*4=3x

3x=96

x=96/3

x = 32

Therefore, it will take 32 days for all 24 gallons to leak out of the bucket at a rate of 3 gallons every 4 days.

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Answer:

6.5

Step-by-step explanation:

Use pythagorean theorem:

[tex]a^{2} +b^{2} =c^{2}[/tex]

a=6

b=2.5

c=hypotenuse

[tex]6^{2} +2.5^{2} =c^{2}[/tex]

simplify

[tex]36 +6.25=c^{2}\\42.25=c^{2}[/tex]

take square root of both sides

[tex]c=6.5[/tex]

The missing side length is 6.5 units.

Hope this helps!

The  local time in Sydney when the plane arrives will be 18:58, or 6:58 PM on a 24-hour clock.

The time difference between Speed and Sydney is 2 hours, and the plane will take some time to fly from Singapore to Sydney at an Speed of 757.2 km/h over a distance of 6310 km.

The time it takes the plane to fly from Speed to Sydney can be calculated as:

time = distance / Speed= 6310 km / 757.2 km/h = 8.33Speed

So the plane will take 8.33 hours to fly from Singapore toSpeed

Now we need to add the 2-hour time difference between Singapore and Sydney to determine the local time in Sydney when the plane arrives.

07:45 (Singapore time) + 8.33 hours (flight time) + 2 hours (time difference) = 18:58

Therefore, the local time in Sydney when the plane arrives will be 18:58, or 6:58 PM on a 24-hour clock.

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Kyra's trick coin is more likely to generate a frequency table with a higher count of tails than heads, reflecting the 0.4 probability of landing on heads and the 0.6 probability of landing on tails.

Kyra's "trick" coin has a probability of 0.4 for landing on heads, meaning that it is more likely to land on tails. If Kyra flips the coin many times, she will likely generate a frequency table that shows a higher frequency of tails than heads.

A possible frequency table might look like this:

| Outcome | Frequency |
|---------|-----------|
| Heads   | 40        |
| Tails   | 60        |

In this table, the coin landed on heads 40 times and tails 60 times, representing the probability of 0.4 for heads and 0.6 for tails (since the total probability must equal 1). The more times Kyra flips the coin, the closer the relative frequencies will get to the actual probabilities. It's important to note that this is just an example, and the exact frequency counts may vary in practice due to randomness.

To summarize, Kyra's trick coin is more likely to generate a frequency table with a higher count of tails than heads.

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In the given problem, solving systematically, sales should be $6,000 during a week that $1,200 was spent on advertising.

If sales vary directly with the amount of money spent on advertising, it means that the ratio of sales to advertising spending is constant. We can use this ratio to find out what sales should be during a week that $1200 was spent on advertising.

Let the ratio of sales to advertising spending be denoted by k. Then, we have:

k = sales / advertising spending

From the information given, we know that:

k = 10,000 / 2,000 = 5

This means that for every dollar spent on advertising, $5 in sales are generated.

To find out what sales should be during a week that $1200 was spent on advertising, we can use the ratio k:

sales = k * advertising spending

sales = 5 * 1200

sales = 6000

Therefore, sales should be $6,000 during a week that $1,200 was spent on advertising.

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Answer:

480 - 96x = 0, so x = 5.

Domain: [0, 5]

Range: [0, 480]

Answer:

  5.  0

  6. a: π/3; b: 45π/2 m ≈ 70.69 m; c: 30/π min ≈ 9.549 min; d: π/900 rad/s; e: 0.2°/s

Step-by-step explanation:

You want the exact value of cos(-39π/6), and some angles, distances, and speeds related to the London Eye Ferris wheel.

The argument of the cosine function can be reduced:

  [tex]\cos{\left(-\dfrac{39\pi}{6}\right)}=\cos{\left(\dfrac{13\pi}{2}\right)}\qquad\text{using cos(-x)=cos(x)}[/tex]

We notice this is an odd multiple of π/2, so its value is 0.

  cos(-39π/6) = 0

The rate of movement is given as 2π radians (one revolution) in 30 minutes. Then the angle in 5 minutes is ...

  (5 min)(2π rad)/(30 min) = (2π/6) rad = π/3 rad

The passenger will travel π/3 radians in 5 minutes.

The arc length is given by ...

  s = rθ . . . . . . where r is the radius and θ is the central angle in radians

For a radius of (135/2 m) and an angle of π/3 radians, the distance traveled is ...

  s = (135/2 m)(π/3) = 45π/2 m ≈ 70.69 m

We know that 2π radians takes 30 minutes, so the time for 2 radians is ...

  t = (2 rad)·(30 min)/(2π rad)

  t = 30/π min ≈ 9.549 min . . . . about 9:32.96 min

The angular velocity is ...

  (2π rad)/(30 min) = (2π rad)/(30 min) × (1 min)/(60 s) = 2π/1800 rad/s

  = π/900 rad/s

360 degrees in 1800 seconds is ...

  360°/(1800 s) = (1/5)°/s

The wheel turns at the rate of 1/5 degree per second.

__

Additional comment

At one revolution per half hour, the wheel turns about twice as fast as the minute hand on a clock. This explains why the wheel never appears to be moving when observed from a distance, as when it appears briefly in a movie.

0 represents the surface level of the swimming pool.

The height of the slide in relation to 0 is +10 feet.

Part A: In this situation, 0 represents the surface level of the swimming pool on the number line, acting as a reference point between the depth of the water and the height of the slide.

Part B: The height of the slide in relation to 0 is +10 feet. This means the slide is 10 feet above the surface level of the swimming pool.

Part C: To mark the level of the bottom of the pole on the number line, Kayla should mark it at -3 since the bottom of the pole is at a depth of 3 feet below the surface level of the pool.

This is determined by using the negative value to represent the depth below the surface (0).

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Margo is right; Jules made a 45 degree angle.

To determine who is right about the angle made by Jules while taking the first piece of pizza, we need to compare their estimates of 20 degrees and 45 degrees.

Since angles are measured using a protractor or other measuring tools, we rely on accurate measurement techniques to determine their values. If both Jules and Margo used appropriate measuring tools and techniques, we would expect their measurements to be close.

However, a 20-degree angle is significantly smaller than a 45-degree angle. Therefore, based on the provided information, Margo's estimate of a 45-degree angle seems more reasonable and likely to be correct.

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Part A:
The height (h) of the trapezoid is half the length of the shorter base (b), so h = (1/2)b.
The longer base is twice the length of the shorter base, so the longer base is 2b.
Given: 4 yd + 2 yd + 2 yd + b = 8 yd (since the sum of all sides of the trapezoid is equal to the perimeter of the display board)
Solving for b, we get:
b = 2 yd
Substituting this value in the equation for h, we get:
h = (1/2)(2 yd) = 1 yd
Therefore, the length of the height is 1 yd and the length of the longer base is 4 yd.

Part B:
The formula for the area of a trapezoid is A = (1/2)(h)(b1 + b2), where h is the height and b1 and b2 are the lengths of the two bases.
Using the values from Part A, we have:
A = (1/2)(1 yd)(2 yd + 4 yd)
A = (1/2)(1 yd)(6 yd)
A = 3 yd^2
Therefore, the area of Darren's display board is 3 yd^2.

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Answer: Did not convert the minutes to hours

Step-by-step explanation:

The most obvious mistake here would be to not convert the minutes into hours.

Remember, the speed given, (7.5), is in miles per HOUR. Your time is given in MINUTES. a conversion is required. 45 minutes are 45/60 = 0.75 hrs.

NOW you can use S = DT to find your distance:

7.5 = D/0.75

.: D = 5.625 miles

The probability of drawing either a two or a ten from a standard deck of 52 cards is the sum of the probabilities of drawing a two and drawing a ten.

There are four twos and four tens in the deck, so the probability of drawing a two is 4/52 and the probability of drawing a ten is 4/52. Therefore, the probability of drawing either a two or a ten is:

4/52 + 4/52 = 8/52 = 2/13

The probability of drawing either a two or a club from a standard deck of 52 cards is the sum of the probabilities of drawing a two and drawing a club, minus the probability of drawing the two of clubs (since we have already counted it). There are four twos and 13 clubs in the deck, including the two of clubs. Therefore, the probability of drawing a two is 4/52, the probability of drawing a club is 13/52, and the probability of drawing the two of clubs is 1/52. Therefore, the probability of drawing either a two or a club is:

4/52 + 13/52 - 1/52 = 16/52 = 4/13

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Antonio's weight after 5 weeks was 202.34 pounds, and it took him 19 weeks to weigh in at 161.05 pounds.

To find Antonio's weight after 5 weeks, we can simply substitute t = 5 into the given exponential function:

W(5) = 235(0.985)⁵

W(5) ≈ 209.88 pounds

So his weight after 5 weeks was approximately 209.88 pounds. To find how long it took him to weigh in at 161.05 pounds, we can set the function equal to 161.05 and solve for t:

161.05 = 235(0.985)ᵗ

0.685106383 ≈ 0.985ᵗ

t ≈ 25.5 weeks

So it took him approximately 25.5 weeks to weigh in at 161.05 pounds

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Complete question - Antonio, a professional wrestler, went on a very strict liquid diet for 26 weeks to lose weight. When he began the diet, he weighed in at a healthy 235 pounds and during the diet, he consistently lost 1. 5% of his body weight each week. His weight loss can be modeled by the function W (t) = 235(0. 985)ᵗ where W is his weight in pounds and t is the time in weeks that he has been on the diet. What was his weight in pounds after 5 weeks? How long did it take(in weeks) him to weigh in at 161. 05 pounds?

The compound interest generated on the investment is roughly $813.67, which is the solution to the question based on compound interest.

The initial sum of money invested or borrowed, upon which interest is based, is referred to as the principle. The principal is then periodically increased by the interest, often monthly or annually, to create a new principal sum that will accrue interest in the ensuing period.

Using the compound interest calculation, we can determine the interest earned on Sanchez's investment:

[tex]A = P(1 + \frac{r}{n} )^{(n*t)}[/tex]

where A is the overall sum, P denotes the principal (the initial investment), r denotes the yearly interest rate in decimal form, n denotes the frequency of compounding interest annually, and t denotes the number of years.

In this case, P = $3,000, r = 0.06 (6%), n = 365 (compounded daily),

and t = 4.

Plugging in the values, we get:

[tex]A = 3000(1 + \frac{0.06}{365} )^{(365*4)}[/tex]

A= $3813.67

The difference between the final amount and the principal is the interest earned.

Interest = A - P

Interest = $3813.67 - $3000

Interest = $813.67

As a result, the investment's interest yield is roughly $813.67.

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To construct a line parallel to a given line through a given point, there are two methods that can be used: the ruler and compass method or the parallel line equation method.

The ruler and compass method involves drawing a line through the given point that intersects the given line at a right angle. Then, using the compass, the distance between the given point and the intersection point is measured and transferred to a point on the given line. Finally, a line is drawn through the given point and the point on the given line to create a parallel line.

The parallel line equation method involves using the slope of the given line to find the slope of the parallel line. This is done by recognizing that parallel lines have the same slope. Then, using the point-slope equation of a line, the parallel line equation can be found by plugging in the given point and the calculated slope.

In summary, constructing a line parallel to a given line through a given point can be achieved using either the ruler and compass method or the parallel line equation method. Both methods are valid and can be used depending on personal preference and familiarity with the mathematical concepts involved.

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None of the options are correct, and the value of x is simply 1.

Start with the given expression: 4 - √x + 15

Substitute x with the limit value of 1: 4 - √1 + 15 = 18

Therefore, the limit of the given expression as x approaches 1 from the right is 18.

To verify this result using the provided options, we can simplify each option and check which one equals 18 as x approaches 1 from the right.

Option A simplifies to (2/√2) + 2, which equals √2 + 2, not equal to 18.

Option B simplifies to 2(4 - √15), which equals 2(4 - 3.87), approximately equal to 2.27, not equal to 18.

Option C simplifies to 3(4 + √15), which equals 3(4 + 3.87), approximately equal to 23.61, not equal to 18.

Option D simplifies to 3(3)(3), which equals 27, not equal to 18.

Therefore, none of the options are correct, and the value of x is simply 1.

none of the options are correct, and the value of x is simply 1.

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There are 15,504 ways to distribute 20 comic books among five kids if there are no restrictions on how many go to each kid.

To evaluate this expression, we can use the formula for combinations:

ⁿCₓ = n! / (x! (n-x)!)

where n is the total number of objects, x is the number of objects to be selected, and "!" denotes factorial

In this case, we have:

n = 20 and r = 5

so we can plug these values into the formula:

²⁰C₅ = 20! / (5! (20-5)!)

= (20 x 19 x 18 x 17 x 16) / (5 x 4 x 3 x 2 x 1)

= 15,504

Therefore, there are 15,504 ways to distribute the 20 comic books among the five kids if there are no restrictions on how many go to each kid.

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Complete Question:

20 different comic books will be distributed to five kids.

How many ways are there to distribute the comic books if there are no restrictions on how many go to each kid (other than the fact that all 20 will be given out)?

The measure of angles ADC is 30⁰.

The measure of angles DCA is 120⁰.

The measure of angles DCB is 180⁰.

The measure of angles AEB is 30⁰.

The measure of each of the angles is calculated as follows;

if length AB = length CD, then AC = AB

Also triangle ACB = equilateral triangle, and each angle = 60⁰.

Angle DAB = 90 (since line DB is the diameter)

Angle DAC = angle ADC

DAC = 90 - 60 = 30 = ADC

DCA = 180 - (30 + 30) (sum of angles in a triangle)

DCA = 120⁰.

The value of angle DCB is calculated as follows;

DCB = 180 (sum of angles on straight line)

angle AEB = angle ADC (vertical opposite angles )

angle AEB = 30⁰

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Answer:

Ben is 31 cm taller

Step-by-step explanation:

1090mm is 109 cm

Subtract Fareeds height from Bens:

140-109=31

Hope this helps!

The correct expression to represent the total number of miles Hayden biked is A. 5 × (3 + 3 + 2 1/4)

This is because each time Hayden rode his bike, he biked a 3-mile path, a 2-mile path, and a 2 1/4 -mile path. To find the total number of miles he biked, we need to add up the distance he biked each time and then multiply by the number of times he biked.

Using the distributive property of multiplication over addition, we can rewrite the expression as:

= 5 × (3 + 3 + 2 1/4)

Therefore, Hayden biked a total of 41 1/4 miles. The correct answer is A

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