determine the domain of the function f(x) = x^2 + 6x +5/x^2 - 25

a. (-1, 5)
b. (5, 5)
c. (-infinity, -1) U (-1,5) U (5, infinity)
d. (-infinity, -5) U (-5, 5) U (5, infinity)

Answer:

Cylindrical:

5r^2(sin(t)^2 - cos(t)^2) +r(3cos(t) + 5sin(t)) - 3z^2 + 4z + 12 = 0

Spherical:

5r^2sin(t)^2 (sin(f)^2 - cos(f)^2) - 3r^2 cos(t)^2 + r sin(t) (3cos(f) + 5sin(f)) + 4r cos(t) +12 = 0

Step-by-step explanation:

In cylindrical coordinates:

x = r cos(t)

y = r sin(t)

z = z

Let us reorganize the original equation

−5x^2+3x+5y^2+5y−3z^2+4z+12=0

5 (y^2-x^2) + 3x + 5y - 3z^2 + 4z + 12 = 0

Now, we can replace x and y:

5 (r^2 sin(t)^2 - r^2 cos(t)^2) + 3rcos(t) + 5r sin(t) - 3z^2 + 4z + 12 = 0

5r^2(sin(t)^2 - cos(t)^2) +r(3cos(t) + 5sin(t)) - 3z^2 + 4z + 12 = 0

In spherical coordinates:

x = r sin(t) cos(f)

y = r sin(t) sin(f)

z = r cos(t)

Let us reorganize the equation:

5 (y^2-x^2) - 3z^2  + 3x + 5y + 4z + 12 = 0

5r^2sin(t)^2 (sin(f)^2 - cos(f)^2) - 3r^2 cos(t)^2 + r sin(t) (3cos(f) + 5sin(f)) + 4r cos(t) +12 = 0

Answer:

Confidence interval = [ 0.4432, 0.4968]

Step-by-step explanation:

Registered voters approved of the job Barack Obama was doing as president,

p = 47% = 0.47

Total sample size, n = 2,294

Confidence level = 99%

now,

confidence interval for the proportion of all registered voters who approved of the job Barack Obama was doing as president

= p ± [tex]z\times\sqrt{\frac{p\times(1-p)}{n}}[/tex]

for 99% confidence level, z value is 2.58

= 0.47 ± [tex]2.58\times\sqrt{\frac{0.47\times(1-0.47)}{2294}}[/tex]

= 0.47 ± 0.0268

or

Confidence interval = [0.47 - 0.0268, 0.47 + 0.0268 ]

or

Confidence interval = [ 0.4432, 0.4968]

Answer:

9 days

Step-by-step explanation:

Experimental probability is the number of times an event occurs divided by the total number of trials of the event.

We know the experimental probability, based on past data, that rain will fall 20% chance.

So, one can expect, that in the next 45 days, rain will fall 20% of the days, according to our experimental probability.

First, we find decimal equivalent of 20%. We divide by 100:

20% = 20/100 = 0.2

Now we multiply this with the number of days:

0.2 * 45 = 9

Thus,

One can expect it to rain 9 days

40 minutes is 2/3 of an hour.

30 miles per hour x 2/3 hour = 20 miles.

30 minutes = 1/2 hour.

60 miles per hour x 1/2 hour = 30 miles.

Total = 20 + 30 = 50 miles.

The probability that the sum of two spins on this spinner will be five is 7/32, which can be calculated by determining individual probabilities for each combination that gives a sum of 5 and then adding these probabilities together.

The question is asking for the probability that the sum of two spins on a specific spinner will be equal to five. First, we need to look into the different ways we can achieve a sum of 5. These possible combinations include (1,4), (2,3), and (3,2), where the values in the parentheses represent the results of the first and second spins respectively.

Now we calculate the probability of each combination happening. The probability of getting a 1 on a spin is 3/8 because there are three sections numbered 1 out of eight total sections. The probability of getting a 2 on a spin is 1/8 because there's only one section numbered 2. Likewise, the probability of getting a 3 is 4/8 (simplified to 1/2) because there are four sections numbered 3.

The combined probability of each scenario would be (3/8*1/2) for (1,4), (1/8*1/2) for (2,3) and (1/2*1/8) for (3, 2). To find the total probability, we simply add these probabilities together. That gives us (3/16) + (1/32) + (1/32) = 7/32. Therefore, the probability that the sum of two spins will be five is 7/32.

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Answer:

The probability that the maximum number of draws is required is 0.2286

Step-by-step explanation:

The probability that the maximum number of draws happens when you pick different colors in the first four pick.

Assume you picked one sock in the first draw. Its probability is 1, since you can draw any sock.

In the second draw, 7 socks left and you can draw all but the one which is the pair of the first draw. Then the probability is [tex]\frac{6}{7}[/tex]

In the third draw, 6 socks left and you can draw one of the two pair colors which are not drawn yet. Its probability is [tex]\frac{4}{6}[/tex]

In the forth draw, 5 socks left and only one pair color, which is not drawn. The probability of drawing one of this pair is [tex]\frac{2}{5}[/tex]

In the fifth draw, whatever you draw, you would have one matching pair.

The probability combined is 1×[tex]\frac{6}{7}[/tex] ×[tex]\frac{4}{6}[/tex]× [tex]\frac{2}{5}[/tex] ≈ 0.2286

Final answer:

The probability that the maximum number of draws is required to get a matching pair of socks from four pairs is 1/7, or about 14.29%.

Explanation:

The question is asking for the probability that the maximum number of draws (7) from 8 socks (4 pairs) is needed before obtaining a matching pair. To calculate this probability, consider that with each draw, you are less likely to pick a sock that matches the ones already drawn, until the last pair remains.

The first sock can be any of the 8, so there's no chance of failure here. For the second sock, to avoid a match, we have 6 out of 7 chance, since one would match the first. For the third sock, to continue avoiding a match, there is a 4 out of 6 chance. This pattern continues until we are left with just two socks, and we must draw the one that matches the last sock drawn.

Therefore, the probability that the maximum number of draws is required is:

(6/7) * (4/6) * (2/4) * (1/2) = 1/7 or about 14.29%.

The probability it will be heads on the next flip is 0.24.

From the information given, a bag comes up heads 40% of the time, and the other will come up heads 60%.

The probability it will be heads on the next flip will be:

= 40% × 60%

= 0.4 × 0.6

= 0.24

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Answer:

x=-4 is a vertical asymptote

Step-by-step explanation:

A vertical asymptote of the graph of a rational function f(x) is a line x=a, such that one of of these statements is fulfilled

[tex]\displaystyle \lim _{x\to a^{+}}f(x)=\pm \infty[/tex]

[tex]\displaystyle \lim _{x\to a^{+}}f(x)=\pm \infty[/tex]

Our function is

[tex]\frac{x^2+7x+10}{x^2+9x+20}[/tex]

To find the candidate values of a, we set the denominator to zero

[tex]x^2+9x+20=0[/tex]

Factoring

[tex](x+4)(x+5)=0[/tex]

Which gives us two possible vertical asymptotes: x=-4 or x=-5

We now must confirm if one of the two conditions are true for each value of a

[tex]\displaystyle \lim _{x\to -4^{-}}\frac{x^2+7x+10}{x^2+9x+20}[/tex]

The numerator can be factored as

[tex]x^2+7x+10=(x+2)(x+5)[/tex]

So our limit is

[tex]\displaystyle \lim _{x\to -4^{-}}\frac{(x+2)(x+5)}{(x+4)(x+5)}[/tex]

Simplifying:

[tex]=\displaystyle \lim _{x\to -4^{-}}\frac{(x+2)}{(x+4)}=+\infty[/tex]

We can see x=-4 is a vertical asymptote

Checking with x=-5, and using the simplified limit:

[tex]\displaystyle \lim _{x\to -5^{-}}\frac{(x+2)}{(x+4)}=3[/tex]

[tex]\displaystyle \lim _{x\to -5^{+}}\frac{(x+2)}{(x+4)}=3[/tex]

The limit exists and is 3, so x=-5 is NOT a vertical asymptote

The only vertical asymptote of the function is x=4

Answer:

20/7 days (just less than 3 days)

Step-by-step explanation:

Recall that (1 job) = (rate)(time), so time = (1 job) / (rate).

Set up and solve the following equation:

  1 job

------------------------------- = time required for 2 pumps working together

1 job         1 job

---------- + -------------

4 days      10 days

This comes out to:

              1 job

------------------------------------------- = time required

10 job-days      4 job-days

------------------ + -----------------

   40 days           40 days

or:

    1 job

-------------------- = (40/14) days, or 20/7 days (just less than 3 days)

14 job·days

-----------------

   40 days  

The number of days required to complete the work together by the pump is 20 / 7 days.

The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.

Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.

Given that one pump can empty a pool in 4 days, whereas a second pump can empty the pool in 10 days.

The number of days will be calculated as,
1 / N  = ( 1 / 10 ) + ( 1 / 4 )

1 / N = (10 + 4 ) / 40

1 / N = 14 / 40

1 / N = 7 / 20

N = 20 / 7 days

Therefore, the number of days required to complete the work together by the pump is 20 / 7 days.

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Answer:

  see below

Step-by-step explanation:

We presume the damping constant is the opposite of the multiplier of time in the exponential term. Then the equations are ...

(a)  y = a·e^(-ct)·sin(ωt)

(b)  y = a·e^(-ct)·cos(ωt)

__

These are the standard equations for simple harmonic motion assuming there is no driving function.

  a = initial amplitude*

  c = damping constant**

  ω = frequency of oscillation in radians per second

  t = time in seconds

_____

* Of course, when y(0) = 0, the motion never actually reaches this amplitude because it is subject to decay before it can.

__

** In electrical engineering, damping is often specified in terms of a time constant, the time it takes for amplitude to decay to 1/e (≈36.8%) of the original amplitude. If that time is represented by τ, then the exponential factor is e^(-t/τ).

In a damped harmonic motion, the equation for displacement y at time t would be y = ae-ct/2mcos(ωt+φ). At t = 0, if y = 0, the phase constant φ is π/2; if y = a, then φ = 0.

In the context of damped harmonic motion, the displacement y at time t of an object with damping constant c, period 2π/ω, and initial amplitude a generally would follow the equation: y = ae-ct/2mcos(ωt+φ) , where m is the mass of the object and φ is the phase constant.

(a) If y = 0 at time t = 0, then the phase constant φ is π/2.

(b) If y= a at time t= 0, then φ=0 because all the displacement is in the stretch or compression, not the oscillation.

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Answer:

Step-by-step explanation:

Multiply the budget total by the percentage for each category:

__

Find "the rest" by subtracting the total of percentages from 100%, or by subtracting the total of budget categories from $2400. Here, everything (not including savings) adds to 90% × $2400 = $2160, so the amount to savings is 10%, or $240.

In a family's monthly budget of $2,400, $1,056 is spent on housing, $552 on food, $144 on clothing, $408 on transportation, and the remaining $240 is put into savings.

To calculate how much money is spent on each category for a family with a monthly budget of $2,400, we need to multiply each percentage by the total budget. For example:

  Transportation: 17% of $2,400 is $408.00.  

The rest of the money, which is 10% of the total budget (the remaining percentage when subtracting the other categories from 100%), is put into savings. Therefore, the amount put into savings is 10% of $2,400, which equals $240.00.

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Answer:

Self selection sampling.

Step-by-step explanation:

A poll that does not attempt to generate a random sample, but instead invites people to volunteer to participate is called - self selection sampling.

Self-selection sampling is a sampling method where researchers allow the people or individuals, to choose to take part in research on their own accord.

Answer:

Step-by-step explanation:

number 2 is false

Answer:

23.12

Step-by-step explanation:

Jordan has entered 3.4 x 6.8 on his calculator.

3.4 x 6.8 = (34/10) x (68/10) = (34 x 68) / (100) = 23.12

Or an easier way would be, observing the two numbers.

In 3.4 , there is one digit after decimal point.

In 6.8 there is one digit after decimal point.

So in their product there should be two digits after the decimal point.

So the answer is 23.12

Answer:

Step-by-step explanation:

Describing the function rule means that you are going to write the equation of the parabola using that roots.  If x = 6 + 4i, then the factor for that is

(x - 6 - 4i).

If x = 6 - 4i, then the factor for that is

(x - 6 + 4i).

FOILing that together gives you a long string of x- and i-terms with a constant or 2 thrown in:

[tex]x^2-6x+4ix-6x+36-24i-4ix+24i-16i^2[/tex]

What's nice here is that 4ix and -4ix cancel each other out; likewise 24i and -24i.  Once that is all canceled away, we are left with

[tex]x^2-12x+36-16i^2[/tex]

The i-squared is what makes this complex.  i-squared = -1, so

[tex]x^2-12x+36-16(-1)[/tex] and

[tex]x^2-12x+36+16[/tex] and

[tex]x^2-12x+52=y[/tex]

Answer:The length of the missing side can be calculated by the following formula;

Step-by-step explanation:

So the scale used was

1 inch = 6 feet

Step-by-step explanation:

Given

Length of library on drawing = 13 inches

Actual length of library = 78 feet

In order to find the scale we divide the actual length by the length of library on drawing.

So,

Scale = [tex]Scale = \frac{Actual\ length}{Length\ on\ drawing}\\=\frac{78}{13}\\=6[/tex]

So the scale used was

1 inch = 6 feet

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Answer:

(x + 6, y + 0), 180° rotation, reflection over the x‐axis

Step-by-step explanation:

The answer can be found out simply , a trapezoid has its horizontal sides usually parallel meanwhile the vertical sides are not parallel.

The horizontal parallel sides are on the x-axis.

Reflection over y- axis would leave the trapezoid in a vertical position such that the trapezoid ABCD won't be carried on the transformed trapezoid as shown in figure.

So option 1 and 2 are removed.

Now, a 90 degree rotation would leave the trapezoid in a vertical position again so its not suitable again.

In,The final option (x + 6, y + 0), 180° rotation, reflection over the x‐axis, x+6 would allow the parallel sides to increase in value hence the trapezoid would increase in size,

180 degree rotation would leave the trapezoid in an opposite position and reflection over x-axis would bring it below the Original trapezoid. Hence, transformed trapezoid A`B`C`D` would carry original trapezoid ABCD onto itself

Answer:

X is less sign but is greater than sign -2

Step-by-step explanation:

Answer:

  C. ks/(60c)

Step-by-step explanation:

Units analysis is helpful here.

  A. (60 sec/min)(k calc)(s sec)/(c calc) = 60ks/c sec²/min ≠ min

  B. (k calc)(s sec)/(c calc) = ks/c sec ≠ min

  C. (k calc)(s sec)/((60 sec/min)(c calc)) = ks/(60c) min . . . correct units

  D. (60 sec/min)(c calc)/((k calc)(s sec)) = 60c/(ks) /min ≠ min

  E. (k calc)/((60 sec/min)(c calc)(s sec)) = k/(60cs) min/sec² ≠ min

The only choice in which the units work out correctly is choice C.

Answer: he has 65/20 meters of string left

Step-by-step explanation:

Montel bought a spool of string for making kites. It contained 10 3/20 meters of string. Let us first convert this length of string to improper fraction, it becomes 203/20 meters

He used 6 9/10 meters of string for kites. Changing 6 9/10 to improper fraction, it becomes 69/10 meters

To determine how many meters of string that Montel have left, we will subtract the length of string used from the total length of string. It becomes

203/20 - 69/10 = 203 - 138/20 = 65/20 meters of string left

Answer:

0 < t < [tex]\frac{5}{3}[/tex]

After 1.67 days the stocks would be sold out.

Step-by-step explanation:

The price of a certain computer stock after t days is modeled by

p(t) = 100 + 20t - 6t²

Now we will take the derivative of the given function and equate it to zero to find the critical points,

p'(t) = 20 - 12t = 0

t = [tex]\frac{20}{12}[/tex]

t = [tex]\frac{5}{3}[/tex] days

Therefore, there are two intervals in which the given function is defined

(0, [tex]\frac{5}{3}[/tex]) and ([tex]\frac{5}{3}[/tex], ∞)

For the interval (0, [tex]\frac{5}{3}[/tex]),

p'(1) = 20 - 12(1) = 20

For the interval ([tex]\frac{5}{3}[/tex], ∞),

p'(2) = 20 - 12(2) = -4

Positive value of p'(t) in the interval (0, [tex]\frac{5}{3}[/tex]) indicates that the function is increasing.

0 < t < [tex]\frac{5}{3}[/tex]

Since at the point t = 1.67 days curve is showing the maximum, so the stocks should be sold after 1.67 days.

The price of the computer stock rises for the first 5/3 days after it is issued. Ideally, if you owned the stock, you should sell it after 5/3 days to maximize your profit.

The price of the stock becomes maximum when the rate of change of the price is 0. To find when that is, we need to take the derivative of the price function, which is p'(t) = 20 - 12t, and set it equal to 0. Solving for t, we get t = 20/12 = 5/3. So, the price of the stock rises for t in (0, 5/3) days. Therefore, if you owned the stock, you should sell it after 5/3 days to maximize your profit.

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Answer: slope=-1, slope is -1/6, line a has a greater slope

Step-by-step explanation:

slope of line a is "m" in y=mx+b form

so slope=-1

slope for line b

first find two point, slope is change in y divided by change in x

1 down for every 6 units forward, so slope is -1/6

line a has a greater slope

Answer:

A) 24

Step-by-step explanation:

The given geometric series is {12, 6, 3, 3 /2 , 3 /4 , 3/ 8 ,...}

Each term can be represented as a product of its previous term and \[\frac{1}{2}\]

The generic term of the series can be represented as the product of the first term 12 and \[\frac{1}{2}^{n-1}\] where n is the index of the term in the series.

The sum to infinity of such a series is given by the following formula:

\[\frac{term1}{1-ratio}\]

Substituting and calculating:

\[\frac{12}{1-\frac{1}{2}}\]

=\[\frac{12}{\frac{1}{2}}\]

=12*2 = 24

Answer:

The price per t-shirts is $8.5.

Step-by-step explanation:

Let us assume the price per t- shirt = $m

The cost of 75 t-shirts  =   $637.50

Now, the total cost for 75 t-shirts = 75 x ( cost of 1 t-shirt)

⇒  $637.50 = 75 x ( m)

 ⇒ m =  $637.50 / 75

   or, m = 8.5

Hence, the price per t-shirts is $8.5.

Answer:

Step-by-step explanation:

Marcy wants to buy packages of hot dogs and hamburgers for her booth at the carnival and the total cost must not exceed her budget of $150

she found packages of hot dogs that cost $6 per one and packages of hamburger is that cost $20 per one .

Let x = number of packages of hot dogs that she can buy.

Let y = number of packages of hamburgers that she can buy.

The cost of x packages of hot dogs = 6×x = $6x

The cost of y packages of hamburgers = 20×y = $20y

The equation will be

Since 6x + 20y must not exceed 150

The equation will be

6x + 20y lesser than or equal to 150

Answer: 5:42:2 am

Step-by-step explanation:

He bought a 16 1/2 pound turkey. Converting to improper fraction, it becomes 33/2 pounds of turkey.

The cookbook says to cook the turkey for 20 minutes per pound and 25 minutes per pound if the turkey is stuffed. Since he wants to stuff the Turkey, he would have to cook for 25 minutes per pound. The total amount of time that he would have to cook the stuffed turkey will be 25 × 33/2 = 412.5 minutes

Let us converted to hours

If 60 minutes = 1 hour,

412.5 minutes = 412.5/60 = 6.875

He would spend 6.875 = 55/8 hours

in cooking the stuffed turkey.

The cookbook also says to let the turkey cool for 10 minutes after coming out the oven. Let us convert the 10 minutes to hours

60 minutes = 1 hour

10 minutes = 10/60 = 1/6

Total time it will take to get the Turkey ready for dinner will be

1/6 + 55/8 = 169/24 = 7.042 hours which means 7 hours, 2 minutes and 58 seconds

If he has to have dinner ready for 12:45 pm, he needs to put the turkey in the oven at 5:42:2 am at most

Answer:She earns $9 per hour

Step-by-step explanation:

Shelia works 8 hours per day Monday, Wednesday, and Friday. This means she works a total of 8×3 = 24 hours for Monday, Wednesday and Friday

She also works 6 hours a day on Tuesday and Thursday. This means that she works for a total of 6×2 = 12 hours on Tuesday and Thursday.

Total hours worked for the week will be sum of hours worked on Monday, Tuesday, Wednesday, Thursday and Friday. It becomes

24 + 12 = 36 hours

She earns $324 per week. The amount that she earns in a week will be

Total amount earned in a week/total number of hours worked in the week. It becomes

324/36 = 9

She earns $9 per hour

Answer:

0.8997, 0.9999

Step-by-step explanation:

Given that among 46​- to 51​-year-olds, 28​% say they have called a talk show while under the influence of peer pressure.

i.e. X no of people who say they have called a talk show while under the influence of peer pressure is binomial with p = 0.28

Each person is independent of the other and there are only two outcomes

n =7

a) the probability that at least one has not called a talk showcalled a talk show while under the influence of peer pressure

=[tex]P(Y\geq 1)[/tex] where Y is binomial with p = 0.72

= [tex]1-(1-0.72)^7\\=0.9999[/tex]

b) the probability that at least one has  called a talk showcalled a talk show while under the influence of peer pressurepeer pressurethe probability that at least one has not called a talk showcalled a talk show while under the influence of peer pressure

=[tex]P(x\geq 1) =1-P(0)\\= 1-(1-0.28)^7\\=0.8997[/tex]