How to get the equation

a) The exponential function that represents the population of the bacteria after t hours is given as follows:

y = a(2)^(t/12). (a is the initial amount).

b) The initial amount is given as follows: a = 2.

The standard definition of an exponential function is given as follows:

y = a(b)^(x/n).

In which:

Considering that the colony doubles in 12 hours, the parameters b and n are given as follows:

b = 2, n = 12.

Hence the function is defined as follows:

y = a(2)^(t/12).

After 4 days = 96 hours, there are 400 bacteria, hence the initial amount is obtained as follows:

400 = a(2)^(96/12)

256a = 400

a = 400/256

a = 2. (rounding).

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The average rate of change from x = 4 to x = 9 is given as follows:

-1.

The average rate of change of a function is given by the change in the output divided by the change in the input.

The numeric values are given as follows:

f(4) = 25, f(9) = 20.

Hence the change in the output is of:

20 - 25 = -5.

The change in the input is of:

9 - 4 = 5.

Hence the rate is given as follows:

r = -5/5

r = -1.

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Answer:

2/3, 8/12

Step-by-step explanation:

(4 divided by 2)/(6 divided by 2) = 2/3

(4 times 2)/(6 times 2)= 8/12

Multiply/Divide the numerator and the denominator by the same number

There are infinte equivalent fractions to 4/6! 4000/6000 is one!

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The probability of observing at least one head in 10 tosses of a fair coin is approximately 0.999 or 99.9%.

When you flip a fair coin, there are two possible outcomes: heads or tails. Since the coin is fair, the probability of observing heads is the same as the probability of observing tails, and each individual coin toss is independent of all the others.

To find the probability of observing at least one head in 10 coin tosses, we can use the complement rule. The complement of the event "observing at least one head" is the event "observing no heads" or "observing all tails."

The probability of observing all tails in 10 coin tosses is (1/2)^10, since the probability of observing tails on any given toss is 1/2, and the tosses are independent.

Therefore, the probability of observing at least one head in 10 coin tosses is:

1 - (1/2)^10

This is the probability of the complement of the event "observing all tails," which is the event "observing at least one head."

We can calculate this probability using a calculator or by simplifying the expression:

1 - (1/2)^10 = 1 - 1/1024

= 1023/1024

≈ 0.999

So the probability of observing at least one head in 10 coin tosses is approximately 0.999, or 99.9%. This means that it is very likely (but not guaranteed) that you will observe at least one head if you flip a fair coin 10 times.

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The volume after rotation about y- axis is V = π ∫[a,b] (R² - r²) dx, about line y= 3 is V = 9π ∫[a,b] (b - a² - 2b + 2a - 1) dx.

A) If we rotate the region bounded by the curves y = 3√x and y = 3x about the y-axis, the resulting solid is a volume is:

V = π ∫[a,b] (R² - r²) dx

Where R is the outer radius, r is the inner radius, and [a, b] is the interval over which the region is being rotated.

For the region being rotated, the outer radius R is given by the maximum value of y = 3√x, which is 3√b. The inner radius r is given by the minimum value of y = 3x, which is 3a. So we have:

R = 3√b and r = 3a

We can substitute these expressions into the formula to get:

V = π ∫[a,b] (3² * b - 3 * a²) dx

V = 9π ∫[a,b] (b - a²) dx

B) If we rotate the region bounded by the curves y = 3√x and y = 3x about the line y = 3, the resulting solid is a volume obtained is:

V = π ∫[a,b] (R² - r²) dx

Where R is the outer radius, r is the inner radius, and [a, b] is the interval over which the region is being rotated.

For the region being rotated, the outer radius R is given by the maximum value of y = 3√x minus the line y = 3, which is 3√b - 3. The inner radius r is given by the minimum value of y = 3x minus the line y = 3, which is 3a - 3. So we have:

R = 3√b - 3 and r = 3a - 3

We can substitute these expressions into the formula to get:

V = π ∫[a,b] (3² * b - 3² * a² - 6 * 3 * b + 6 * 3 * a - 9) dx

V = 9π ∫[a,b] (b - a² - 2b + 2a - 1) dx

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_____ The given question is incorrect, the correct question is given below:

Set up the integral that uses the method of disks/washers to find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified lines.  y = 3√x, y = 3x

A) About the y-axis

B) About the line y=3

Answer:

64°

Step-by-step explanation:

the sum of the 3 angles in a triangle = 180°

let the third angle be x , then

x + 90° + 26° = 180°

x + 116° = 180° ( subtract 116° from both sides )

x = 64°

the other angle is 64°

The Coordinate of 3×A, A"" will be (- 24, 12) is a scale factor of 3 is used.

Two-dimensional figures can be transformed mathematically in order to travel about a plane or coordinate system.

Dilation: The preimage is scaled up or down to create the image.

Reflection: The picture is a preimage that has been reversed.

Rotation: Around a given point, the preimage is rotated to create the final image.

Translation: The image is translated and moved a fixed amount from the preimage.

Given, The smaller rectangle is the image of the red rectangle.

The vertices of A(- 8, - 4) and A'(- 4, 2).

As we can see it is a dilation by a scale factor of (1/2).

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Answer:

The answer to the question is Sixteen.

The probability that a randomly selected female is taller than 67 inches is 0.8133, or approximately 81%.

To calculate this probability, we need to standardize the height of 67 inches using the formula:

z = (x - mu) / sigma

where x is the height we're interested in, mu is the mean, and sigma is the standard deviation.

z = (67 - 64.5) / 2.8 = 0.8929.

Then, we can look up the area to the right of z = 0.8929 in the standard normal distribution table, which is 0.1867.

Finally, we subtract this value from 1 to get the probability that a randomly selected female is taller than 67 inches:

P(X > 67) = 1 - 0.1867

P(X > 67) = 0.8133

So, the probability that a randomly selected female is taller than 67 inches is 0.8133, or approximately 81%.

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The results of the hypothesis test H0:β0 = 0 against the alternative H1:β1 ≠ 0 is Reject at α = 0.10, Do not reject at α = 0.05. Hence, option D is the accurate solution.

To determine the results of the hypothesis test H0:β0 = 0 against the alternative H1:β1 ≠ 0, we will calculate the t-statistic and compare it to the critical values from the t-distribution at the desired significance level.

The t-statistic for the null hypothesis is given by -

t = (Bo - 0) / SE(Bo)

where Bo is the estimate of the intercept, SE(Bo) is its standard error, and 0 is the hypothesized value under the null hypothesis.

Substituting the given values, we get,

t = 15.52 / 3.242 = 4.785

The degrees of freedom for this test are n - 2 = 10, where n is the number of observations.

At a significance level of 0.10, the critical values for a two-tailed test with 10 degrees of freedom are ±1.812. Since our calculated t-statistic of 4.785 is greater than the critical value of 1.812, we reject the null hypothesis at α = 0.10.

Similarly, at a significance level of 0.05, the critical values for a two-tailed test with 10 degrees of freedom are ±2.228. Since our calculated t-statistic is less than the critical value of 2.228, we do not reject the null hypothesis at α = 0.05.

Therefore, the correct answer is (D) Reject at α = 0.10, Do not reject at α = 0.05.

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The correct answer is C. $ 3,666.30 is in the savings account.

A percentage is a number or ratio that can be expressed as a fraction of 100. A percentage is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%", although the abbreviations "pct.", "pct" and sometimes "pc" are also used. A percentage is a dimensionless number; it has no unit of measurement.

here, we have,

Let's calculate how much money you have in the CD after 5 months, this way:

Gross Income = $ 4,520

Total deductions on taxes = 27.9% (11.75 + 7.65 + 8.50) * 4,520 = $ 1,261.08

Realized income = Gross income - Total deductions on taxes

Realized income = 4,520 - 1,261.08

Realized income = $ 3,258.92

Fixed expenses = 3,258.92 * 0.3 = $ 977.68

Emergency fund = 977.68 * 5 = $ 4,888.40

Certificate of Deposit = 4,888.40 * 0.75 = $ 3,666.30

The correct answer is C. $ 3,666.30

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Based on the information, it can be inferred that the graph shows a curve because it refers to exponential growth. Then, the curve shows a higher frequency of views in less time.

According to the information, it can be inferred that the visualizations in the first hours would be less than time, then they would be proportional to time and finally they would show more frequency with respect to time.

According to the information, a time association table with the visualizations would look like this:

views / time

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The circumference of the circle is 88in (nearest inches)

The circumference is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out to a line segment. More generally, the perimeter is the curve length around any closed figure.

The circumference of a circle is given as;

C = πd

d = diameter = 28in

C = 3.14 × 28

C = 88 in ( nearest inches)

therefore the circumference of the circle is 88in

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Your expected winnings from playing this game are 2 pounds.

A game is an activity or a form of play, often with a set of rules and goals, that is undertaken for enjoyment, competition, or skill development.

Let's analyze this game to see what your expected winnings are.

If the first roll is 1, 2, or 3, the game continues and you have a 3 in 6 chance (or 1/2 chance) of continuing to roll the die. Each subsequent roll has the same probabilities and outcomes as the first roll.

Let's start with the case where you win on the first roll with a probability of 1/2. In this case, your winnings are 1 pound.

If you don't win on the first roll, the game continues with a probability of 1/2, and your expected winnings from that point on are the same as your expected winnings from the beginning of the game (since the probabilities and outcomes are the same for all rolls).

Therefore, the expected winnings from the start of the game are:

E = 1/2 * 1 + 1/2 * E

Solving for E, we get:

E = 1 + E/2

E/2 = 1

E = 2

Therefore, your expected winnings from playing this game are 2 pounds.

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The percent of the original price you end up paying will be 58%.

The amount of any product is given as though it was a proportion of a hundred. The ratio can be expressed as a quarter of 100. The phrase % translates to one hundred percent. It is symbolized by the character '%'.

The percentage is given as,

Percentage (P) = [Final value - Initial value] / Initial value x 100

A store has clearance items that have been marked down by 30% They are having a sale, advertising an additional 40% off clearance items.

Then the percent of the original price you end up paying is given as,

⇒ 1 - (1 - 0.30) x (1 - 0.40)

⇒ 1 - 0.70 x 0.60

⇒ 1 - 0.42

⇒ 0.58 or 58%

The percent of the original price you end up paying will be 58%.

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Answer:

19

Step-by-step explanation:

The position vs. time connection is best described by equations is [tex]& -(0.15 \mathrm{~m}) \cos (3.65) t[/tex] .

From the given data,

A student attaches a 3.0 kg mass to a spring with a spring constant of 40 N/m.

The student compresses the spring to the left by 0.15 m and then releases the mass allowing it to oscillate.

The position of the mass for the spring- mass oscillatory system is,

Where

The angular frequency of the system is,

The angular frequency refers to the angular displacement of any wave element per unit of time or the rate of change of the waveform phase.

Therefore, equation (1) changes as follows

[tex]x & =A \cos \left(\sqrt{\frac{k}{m}}\right) t \\[/tex]

[tex]& =-(0.15 \mathrm{~m}) \cos \left(\sqrt{\frac{40 \mathrm{~N} / \mathrm{m}}{3.0 \mathrm{~kg}}}\right) t \\[/tex]

[tex]& =-(0.15 \mathrm{~m}) \cos (3.65) t[/tex]

Therefore, the position vs. time connection is best described by equations is [tex]& -(0.15 \mathrm{~m}) \cos (3.65) t[/tex] .

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1. Assuming that interest is every month, eddie will pay 34,297.42 for his 8 month loan, every month the loan goes from 16000 *1.10 *1.10 *1.10 etc...

2. (14000*2/12*R) /100=3.5%

Tom's workplace is 0.28 miles from Fort Hood.

An expression contains one or more terms with addition, subtraction, multiplication, and division.

We always combine the like terms in an expression when we simplify.

We also keep all the like terms on one side of the expression if we are dealing with two sides of an expression.

Example:

1 + 3x + 4y = 7 is an expression.

3 + 4 is an expression.

2 x 4 + 6 x 7 – 9 is an expression.

33 + 77 – 88 is an expression.

We have,

Fort Hood from Tom's house distance = 0.8 miles

Now,

Tom's workplace is 0.35 times as far from Fort Hood as Tom's House.

This means,

Tom's workplace from Fort Hood distance.

= 0.35 x 0.8

= 0.28 miles

Thus,

Tom's workplace is 0.28 miles from Fort Hood.

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Answer:

[tex]y=0.11x+1[/tex]

Step-by-step explanation:

y = 0.11x + 1

(y is the monthly cost for x text messages)

:]

Sets of parametric equations for the rectangular equation are x=t and y=8t+7.

A parametric equation has the variables x and y expressed in terms of t which is the third independent variable.

The rectangular equation is [tex]y = 8x + 7[/tex].

If we denote the variable x by t.

Then, we can say that x=t

The parametric equation can be formed by replacing x with t in the rectangular equation [tex]y=8x+7[/tex].

Therefore, sets of parametric equations for the rectangular equation are x=t and y=8t+7.

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The equivalent ratio to the given expression 3mm : 1cm as required in the task content is; 3 : 10.

It follows from the task content that the equivalent ratio of 3mm : 1cm is to be determined .

Recall, 1cm is equivalent to 10 mm.

On this note, the given ratio can be written as; 3mm : 10 mm.

Consequently, the ratio which is equivalent to the given expression is; 3 : 10.

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Answer:

Step-by-step explanation:

The line segment joining two points P and Q can be represented by the equation of a straight line in the form y = mx + b, where m is the slope and b is the y-intercept.

To find the equation of the line, we need to find the slope, which can be calculated using the formula:

m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of the points P and Q, respectively.

In this case, the coordinates are:

P = (-3, 2) and Q = (5, 7)

So, the slope is:

m = (7 - 2) / (5 - (-3)) = 5 / 8

Next, we can use either of the points to find the y-intercept. Let's use point P:

b = y - mx, where y and x are the y and x coordinate of the point, respectively.

In this case,

b = 2 - m * (-3) = 2 - (5/8) * (-3) = 2 + 15/8 = 89/8

So, the equation of the line joining the points P and Q is:

y = (5/8)x + 89/8

Now, to find the point where the line crosses the y-axis, we need to find the x-coordinate of the point where y = 0.

So, we have:

0 = (5/8)x + 89/8

Solving for x, we get:

x = -(89/8) / (5/8) = -89 / 5

This means that the line crosses the y-axis at the point (-89/5, 0). To find the ratio in which the line segment is divided by the y-axis, we need to find the ratio of the distance from the y-axis to point P to the distance from the y-axis to point Q.

Let's call the point of intersection with the y-axis R. The distances are then:

PR = (3, 2) and QR = (5 - (-89/5), 7)

The ratio of the distances is then:

PR / QR = (3, 2) / (5 - (-89/5), 7) = 3 / (5 + 89/5) = 3 / (94/5) = 15/47

So, the line segment joining the points P and Q is divided by the y-axis in the ratio 15:47.

Answer: 11/30

Step-by-step explanation:

First hour : 7/15

Second hour : 7/15 - 1/10

14/30 - 3/30 = 11/30

Total = 7/15 + 11/30

        =  14/30 + 11/30

        = 25/30

        = 5/6

Answer:

The name of the distribution of number of trials

X

X:

Geometric Distribution.

As the trials will be continued until the first success i.e draw of the heart is achieved.

b. The mean number of draws needed:

The probability of success or the probability of drawing a heart from the deck:

p

=

Number of hearts

Total

=

13

52

=

0.25

p=Number of heartsTotal=1352=0.25

The mean number of trials required to get the first success:

μ

=

1

p

=

1

0.25

=

4

μ=1p=10.25=4

c. The standard deviation of

X

X:

The standard deviation of geometric distribution:

σ

=

√

1

−

p

p

2

=

√

1

−

0.25

0.25

2

≈

3.464

σ=1−pp2=1−0.250.252≈3.464

d. The probability distribution table:

The probability mass function of geometric distribution:

P

(

X

=

r

)

=

(

1

−

p

)

n

⋅

p

P(X=r)=(1−p)n⋅p

where

X

∈

[

0

,

1

,

2

,

.

.

.

,

n

]

X∈[0,1,2,...,n]

As per the function f(x) = 2x + 6, the value of f(a) = 2a + 6 and f(a + h) = 2(a + h) + 6 = 2a + 2h + 6.

A function is a mathematical rule that assigns each input (or independent variable) a unique output (or dependent variable).

The function f(x) = 2x + 6 means that for any given value of x, we can compute the corresponding value of f(x) by substituting x into the expression 2x + 6.

The first expression that we need to evaluate is f(a), where a is some arbitrary value. This means that we need to substitute a into the function f(x) and simplify the expression.

Therefore, f(a) = 2a + 6.

The second expression that we need to evaluate is f(a + h), where h is also an arbitrary value. This means that we need to substitute a + h into the function f(x) and simplify the expression.

Therefore, f(a + h) = 2(a + h) + 6 = 2a + 2h + 6.

To simplify the expressions, we can collect like terms, which means combining the terms that have the same variable and exponent 2a + 2h can be written as 2(a + h).

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Mark had 480 bear stickers and 400 bird stickers at first.

A word problem in math is a math question written as one sentence or more that requires children to apply their math knowledge to a 'real-life' scenario.

Given that, Mark had 5/6 as many birds stickers as bear stickers at first. After buying another 20 stickers of each type, he has 7/8 as many bird stickers as bear stickers.

We are asked to find the number of each type of stickers did mark have at first.

Let Mark have x bear stickers at first,

Therefore,

Bird sticker = 5x/6

5x/6 + 20 = 7x/8

Multiply the equation by 48

40x + 960 = 42x

2x = 960

x = 480

5x/6 = 400

Hence, Mark had 480 bear stickers and 400 bird stickers at first.

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Answer:

128.8

Step-by-step explanation:

[tex]\frac{20}{100}[/tex] = [tex]\frac{A}{644}[/tex]

100A = 12,880

A = 128.8

The fraction of milk in each bottle is 1/15.

The unitary method is a technique for solving a problem by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value.

Given that,

Total amount of milk =1/3

Number of bottles to be filled =5

Amount of milk in one bottle = 1/3 ÷ 5/1

= 1/3 × 1/5

= 1/15

Therefore, the fraction of milk in each bottle is 1/15.

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