P1
A fruit punch that contains 70% truit juice is combined with 100% fruit juice. How many ounces of each should be used to make 54 oz of a mixture that is 90%
Part: 0/2

If a and b are both positive integers, then (2a − 1) mod (2b − 1) = 2a mod b − 1, because the left side can be rewritten as (2a mod (2b − 1)) - 1, which equals the right side.

To show that (2a − 1) mod (2b − 1) = 2a mod b − 1, let's break it down step-by-step:

1. Consider (2a − 1) mod (2b − 1). Apply the property of modular arithmetic, which states that (A mod N) = (A mod N) mod N.

2. This gives us (2a mod (2b − 1)) - 1.

3. Observe that 2a mod (2b − 1) can also be written as 2a mod (2(b − 1) + 1), which equals 2a mod 2(b - 1) + 2a mod 1.

4. Since 2a mod 1 = 0, we have 2a mod 2(b - 1) + 0 = 2a mod 2(b - 1).

5. Apply the distributive property of modular arithmetic to get 2(a mod (b - 1)) = 2a mod b.

6. Substitute this back into the expression from step 2: (2a mod b) - 1.

7. Therefore, (2a − 1) mod (2b − 1) = 2a mod b − 1.

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The data set is a Population, because it is a collection of salaries for all baseball players in the league.Therefore option D is correct.

To determine whether the data set is a population or a sample:

Population, because it is a collection of salaries for all baseball players in the league.

The data set includes every single baseball player's salary in the league, which makes it a population.

It is not a sample because it includes every individual in the group being studied, rather than just a subset of them.

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The solution set for the inequality 2 - 4x > 6 is x < -1, expressed in interval notation as (-∞, -1).

To solve the inequality 2 - 4x > 6, we need to isolate the variable x on one side of the inequality.

2 - 4x > 6

Subtract 2 from both sides:

-4x > 4

Divide both sides by -4, remembering to flip the inequality since we are dividing by a negative number:

x < -1

Therefore, the solution set for the inequality 2 - 4x > 6 is x < -1, expressed in interval notation as (-∞, -1).

To graph this solution set, we can draw a number line and shade everything to the left of -1.

<=================|----------->

                 -1    

The shaded part of the number line represents the solution set (-∞, -1).

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The required answer is the probability of getting the sequence BD is 1/30.

a) To find the probability that one chip will be A and one will be E, we need to first determine the total number of possible outcomes. Since we are selecting two chips without replacement, there are 6 ways to choose the first chip and 5 ways to choose the second chip. Therefore, there are 6 x 5 = 30 possible outcomes.

Next, we need to determine the number of outcomes where one chip is A and one chip is E. There are two ways this can happen: A can be the first chip and E can be the second, or E can be the first chip and A can be the second. Therefore, there are 2 possible outcomes where one chip is A and one chip is E.

The probability of getting one chip that is A and one that is E is therefore 2/30, or 1/15.

b) To find the probability that the first chip will be F, we again need to determine the total number of possible outcomes. Since there are 6 chips, there are 6 ways to choose the first chip.

Out of those 6 possible outcomes, only 1 of them results in the first chip being F. Therefore, the probability of the first chip being F is 1/6.

c) To find the probability that the first chip will be B and the second will be D, we again need to determine the total number of possible outcomes. There are 6 ways to choose the first chip and 5 ways to choose the second chip, giving us 6 x 5 = 30 possible outcomes.

Out of those 30 possible outcomes, only 1 of them results in the first chip being B and the second chip being D (BD).

Therefore, the probability of getting the sequence BD is 1/30.

a) To find the probability that one chip will be A and one will be E, you first need to determine the total number of possible outcomes when selecting two chips without replacement. There are 6 choices for the first chip and 5 choices for the second chip, so there are 6 x 5 = 30 possible outcomes.

Now, there are 2 ways to select chips A and E: AE or EA. So the probability of selecting one A and one E is:
P(A and E) = Number of favorable outcomes (AE or EA) / Total possible outcomes = 2/30 = 1/15

b) To find the probability that the first chip will be F, you need to consider that there are 6 chips in total. Only 1 of them is F, so the probability is:
P(First chip is F) = Number of favorable outcomes (F) / Total possible outcomes = 1/6

c) To find the probability that the first chip will be B and the second chip will be D, you need to consider the possible outcomes. There is only 1 favorable outcome: selecting B first and then D. So the probability is:
P(First chip is B and second chip is D) = Number of favorable outcomes (BD) / Total possible outcomes = 1/30

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Consider a homogeneous linear system with a 4 by 9 coefficient matrix. The coefficient matrix has rank at most equal to 4 and at least 5 free variable columns.

This is because the rank of a matrix cannot exceed the number of rows or columns it has. In this case, the matrix has 4 rows, so the rank cannot exceed 4.

Additionally, the number of free variable columns can be found by subtracting the rank of the matrix from the number of columns. In this case, there are 9 columns, so subtracting the maximum rank of 4 gives us 5 free variable columns.

Therefore, The coefficient matrix has rank at most equal to 4 and at least 5 free variable columns.

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Answer:

Let’s call the width of the rectangle “w”. Since the length of the rectangle is twice its width, we can call the length “2w”. We know that the diagonal of the rectangle is equal to the square root of 45 cm. Using Pythagorean theorem, we can find that:

diagonal^2 = length^2 + width^2 45 = (2w)^2 + w^2 45 = 4w^2 + w^2 45 = 5w^2 w^2 = 9 w = 3

So the width of the rectangle is 3 cm and its length is twice that, or 6 cm.

Step-by-step explanation:

The probability that a randomly selected student spends at most 185 hours on the project is 1 (or 100%).

The mean of a gamma distribution with parameters a and ß is a/ß², so in this case, the mean is 50/3 = 16.67 hours.

The variance of a gamma distribution with parameters a and ß is a/ß², so in this case, the variance is 50/9 = 5.56 hours. Therefore, the standard deviation is the square root of the variance, which is approximately 2.36 hours.

We can convert X to a standard normal variable Z using the formula:

Z = (X - μ) / σ

where μ is the mean of X and σ is the standard deviation of X. Substituting in the values we calculated in Step 1, we get:

Z = (X - 16.67) / 2.36

To find the probability that a randomly selected student spends at most 185 hours on the project,

we need to find the corresponding Z-score for X = 185 and then find the area under the standard normal curve to the left of that Z-score.

Z = (185 - 16.67) / 2.36 = 69.53

Using a standard normal table or calculator, we can find that the area to the left of Z = 69.53 is essentially 1. Therefore, the approximate probability that a randomly selected student spends at most 185 hours on the project is 1 (or 100%).

This is because the gamma distribution with a large a is well approximated by a normal distribution, and so the probability of X being more than a few standard deviations away from the mean is extremely small.

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Answer:

(3)

Step-by-step explanation:

the limit lines are the same (and correct) in all 4 pictures.

the difference is the applicable side of the lines.

y <= x + 3

because of the "<=" the valid area is below the line. in our case to the right and below the line.

that eliminates (1) and (4).

y >= -2x - 2

because of the ">=" the valid area is above the line. in our case right and above the line.

so, (3) is correct.

Answer:

Step-by-step explanation:

You are asked for the area and perimeter of a figure comprised of a square and two sectors.

The perimeter of the figure is the sum of the lengths of the outside edges. You recognize vertical edges AD and BC as being the sides of a square that are 4 units long.

The other two sides of the square are AB and CD, but these are not part of the perimeter. The significance of those is that they are radii of the sectors ABE and CDF. The straight segments of AE and CF of those sectors have the same length (4 units) as the side of the square. Those straight segments are part of the perimeter.

In effect, the four straight segments of the perimeter are all 4 units.

The curved edges of the two sectors have a length that is found using the formula ...

  s = rθ

where r is the sector radius, and θ is the central angle in radians.

The angle is shown as 30°, which is 30°(π/180°) = π/6 radians. The radius is the square side length, 4, so each curved line has length ...

  s = (4)(π/6) = 2/3·π

The perimeter of the figure is the sum of the lengths of the straight segments and the curved arcs:

  P = 4(4 units) +2(2/3π units) = 16 +4/3π units ≈ 20.19 units

As with the perimeter, the area is composed of the area of a square and the areas of two sectors.

The area of the square is the square of its side length:

  A = s²

  A = (4 units)² = 16 units²

The area of each sector is effectively the area of a triangle with base equal to the arc length (2/3π) and height equal to the radius of the arc (4 units). The sector area is ...

  A = 1/2rs

  A = 1/2(4 units)(2/3π units) = 4/3π units²

The area of the whole figure is the sum of the area of the square and the areas of the two sectors:

  A = square area + 2×(sector area)

  A = 16 units² + 2×(4/3π units²) = (16 +8/3π) units² ≈ 24.38 units²

__

Additional comment

In general, you find the perimeter and/or area of a strange figure by decomposing it into parts whose perimeter and area you can compute. (When you get to calculus, those parts will be infinitesimally small and there will be an infinite number of them.) At this point, you will generally be making use of formulas that should be familiar.

The formula for the area of a sector is usually written ...

  A = 1/2r²θ

Here, we have made use of our previous computation of s=rθ to write the area formula as A = 1/2rs. The similarity to the triangle area formula is not accidental.

(a)  The estimated value of f(61) is 384.

(b) The estimated value of f''(58) is 0.

(a) Using linear approximation, we have:

f(61) ≈ f(60) + f'(60)(61 - 60)

Substituting the given values, we get:

f(61) ≈ 378 + 6(1)

≈ 384

Therefore, the estimated value of f(61) is 384.

(b) Since f is a differentiable function, we can use the second derivative test to estimate f''(58) as follows:

f''(58) ≈ lim h → 0 [tex](f(58 + h) - 2f(58) + f(58 - h)) / h^2[/tex]

Using linear approximation, we have:

f(58 + h) ≈[tex]f(58) + f'(58)h + f''(58)h^2/2[/tex]

f(58 - h) ≈ [tex]f(58) - f'(58)h + f''(58)h^2/2[/tex]

Substituting these values, we get:

f''(58) ≈ lim h → 0[tex][ (f(58) + f'(58)h + f''(58)h^2/2) - 2f(58) + (f(58) - f'(58)h + f''(58)h^2/2) ] / h^2[/tex]

Simplifying and rearranging terms, we get:

f''(58) ≈ lim h → 0[tex][ (f(58 + h) - 2f(58) + f(58 - h)) /[/tex][tex]h^2 - f''(58)h^2][/tex]

Taking the limit as h approaches 0, we get:

f''(58) ≈ f''(58)(0) = 0

Therefore, the estimated value of f''(58) is 0.

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The formula to find the total number of trees, t, in r rows is [tex]t = 10 \times r[/tex]

Sequence and series - A sequence is a list of items/objects which have been arranged in a sequential way. A series can be highly generalized as the sum of all the terms in a sequence. However, there has to be a definite relationship between all the terms of the sequence.

If there are 10 trees planted in each row, then the total number of trees in one row is 10. To find the total number of trees in r rows, we need to multiply the number of trees in one row (10) by the number of rows (r):

[tex]t = 10 \times r[/tex]

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If the trees are planted in rows of 10, then the total number of trees in r rows can be found using the formula: t = 10r

A sequence is an ordered list of numbers or other mathematical objects, such as functions or geometric figures, that follow a certain pattern or rule. A sequence can be finite or infinite and can be specified using a formula or a recursive rule.

A series is a sum of the terms in a sequence, typically written using the sigma notation Σ

where t is the total number of trees and r is the number of rows. This formula simply multiplies the number of trees in each row (which is 10) by the number of rows, to get the total number of trees.

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The Set corresponding to (AUB)nc is Option C. (3, 4, 5, 7, 9).

(AUB)nc represents the complement of the union of sets A and B. To find this set, we first need to find the union of sets A and B, which is the set of all elements that are in either A or B or both.

Set A contains all prime numbers, so A = (2, 3, 5, 7, 11, ...). Set B contains (4, 7, 9, 11, 13, 14). Taking the union of sets A and B gives us:

AUB = (2, 3, 4, 5, 7, 9, 11, 13, 14)

The complement of this set (denoted by nc) contains all elements that are not in this set. Therefore, (AUB)nc contains all elements that are not in the union of sets A and B.

Option A contains 13, which is in AUB. Option B contains 4 and 7, which are also in AUB. Option D contains all elements in AUB. Therefore, the correct answer is option C, which contains (3, 4, 5, 7, 9) and does not contain any elements that are in A or B.

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The sample size increases, but the standard deviation may increase, decrease, or stay the same depending on the probability of success.

False.

The mean (or expected value) of a binomial distribution is given by the formula np, where n is the sample size and p is the probability of success. So as the sample size increases, the mean of the distribution increases proportionally, assuming the probability of success remains constant.

However, the standard deviation of a binomial distribution is given by the formula sqrt(np(1-p)). As the sample size increases, the standard deviation does not necessarily increase. In fact, it can decrease if the probability of success is small or large, and increase if the probability of success is close to 0.5. This is because the variance of the binomial distribution is given by np(1-p), which has a maximum value at p = 0.5. When the probability of success is close to 0 or 1, the variance decreases as the sample size increases because the outcome becomes more predictable. Conversely, when the probability of success is close to 0.5, the variance increases as the sample size increases because there is greater variability in the outcomes.

In summary, the mean of a binomial distribution always increases as the sample size increases, but the standard deviation may increase, decrease, or stay the same depending on the probability of success.

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Answer:

C

Step-by-step explanation:

Sin is opposite divided by hypotenuse, so that would be 21 divided by 35.

The best descriptiοn and example οf the null hypοthesis in a hypοthesis test is: "A statistical hypοthesis that there is nο difference between a parameter and a specific value, οr between twο parameters. Example: Hοu - 90"

The null hypοthesis is a statistical hypοthesis that states there is nο significant difference between twο grοups οr variables being cοmpared.

It is οften denοted as H₀ and is a statement that researchers assume tο be true until prοven οtherwise by empirical evidence.

Frοm the given οptiοns

The best descriptiοn and example οf the null hypοthesis in a hypοthesis test is: "A statistical hypοthesis that there is nο difference between a parameter and a specific value, οr between twο parameters. Example: Hοu - 90"

In a hypοthesis test, the null hypοthesis represents the default assumptiοn that there is nο significant difference between twο grοups, οr between a sample and a pοpulatiοn.

The example given, "H₀: μ = 90", represents a null hypοthesis where there is nο significant difference between a parameter (represented by the variable "Hοu") and a specific value (90).

This means that if the null hypοthesis is true, the parameter "H₀: μ" is equal tο 90 οr dοes nοt differ significantly frοm 90.

Hence,

The best descriptiοn and example οf the null hypοthesis in a hypοthesis test is: "A statistical hypοthesis that there is nο difference between a parameter and a specific value, οr between twο parameters. Example: Hοu - 90"

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A geometric shape that could be used to model a building with a quadrilateral base and triangular sides is a pyramid.

A geometric shape is a two-dimensional or three-dimensional object that can be described using mathematical formulas and properties. Examples of two-dimensional geometric shapes include squares, circles, triangles, and rectangles. Examples of three-dimensional geometric shapes include cubes, spheres, cylinders, and cones.

Geometric shapes are used in many different fields, including mathematics, science, architecture, engineering, and art. They are important for understanding spatial relationships and for solving problems related to measurement, area, volume, and other geometric properties.

Specifically, the shape would be a triangular pyramid, with the base being a quadrilateral and the sides being triangles.

A cone also has a circular base and curved sides, while a cylinder has circular bases and straight sides. A sphere is a three-dimensional shape with a curved surface, and would not be an appropriate shape to model a building with a quadrilateral base and triangular sides.

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Answer:

The answer is B. Pyramid I just took the test and got it correct.

The cylindrical coordinates of the point with rectangular coordinates (x, y, z) = (-3, 5, -1) are (ρ, θ, z) ≈ (sqrt(34), -1.03, -1).

Cylindrical coordinates are a type of coordinate system used in three-dimensional space to locate a point using three coordinates: ρ, θ, and z. The cylindrical coordinate system is based on a cylindrical surface that extends infinitely in the z-direction and has a radius of ρ in the xy-plane.

To convert rectangular coordinates (x, y, z) to cylindrical coordinates (ρ, θ, z), we use the following formulas:

ρ =[tex]\sqrt(x^2 + y^2)[/tex]

θ = arctan(y/x)

z = z

Substituting the given values, we get:

ρ = [tex]\sqrt((-3)^2 + 5^2)[/tex]= sqrt(34)

θ = arctan(5/-3) ≈ -1.03 radians or ≈ -58.8 degrees (measured counterclockwise from the positive x-axis)

z = -1

Therefore, the cylindrical coordinates of the point with rectangular coordinates (x, y, z) = (-3, 5, -1) are (ρ, θ, z) ≈ (sqrt(34), -1.03, -1).

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The following parts can be answered by the concept of null hypothesis.

a) The best conclusion if H0 is rejected is (ii) The scale is not in calibration.

b) If the scale is actually in calibration but the conclusion is reached that the scale is not in calibration, it would be a Type I error

a) If H0 is rejected, it means that there is enough evidence to suggest that the population mean reading on the scale is not equal to 10, which indicates that the scale is not in calibration. Therefore, the best conclusion in this case would be (ii) The scale is not in calibration.

b) If the conclusion is reached that the scale is not in calibration, but in reality, it is actually in calibration (i.e., μ=10), it would be a Type I error. This is because the null hypothesis (H0) is rejected incorrectly, leading to a false conclusion. Therefore, the type of error in this case would be a Type I error.

Therefore, the answer is:

a) The best conclusion if H0 is rejected is (ii) The scale is not in calibration.

b) If the scale is actually in calibration but the conclusion is reached that the scale is not in calibration, it would be a Type I error

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The linear equation in point-slope form is:

y + 6 = (3/4)*(x + 4)

For a linear equation whose slope is m, and we know that it passes through a point (x₁, y₁), the point slope form can be written as follows:

y - y₁ = m*(x - x₁)

Here we know that the slope of the linear equations is (3/4) and the point is (-4, -6)

Then the linear equation in the point slope form can be written as:

y + 6 = (3/4)*(x + 4)

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The confidence interval of 99% for μ₂ - μ₁ for the given mean and standard deviation is equal to (23.7377, 49.7713).

Confidence interval = 99%  

Confidence interval for μ₂ - μ₁, we need to follow these steps,

Calculate the sample mean difference and the standard error of the mean difference.

Sample mean difference

= ¯x₂ - ¯x₁

= 162.75 - 126

= 36.75

Standard error of the mean difference

= √[(s₁^2/n₁) + (s₂^2/n₂)]

= √[(8.062^2/5) + (3.5^2/4)]

= 4.0065 (rounded to four decimal places)

The t-value for a 99% confidence level with degrees of freedom

= n₁ + n₂ - 2

= 5 + 4 - 2

= 7.

Using a t-distribution table attached ,

The t-value for a 99% confidence level with 7 degrees of freedom is 3.250.

Margin of error

= t-value x standard error of the mean difference

= 3.250 x 4.0065

= 13.0213 (rounded to four decimal places)

Confidence interval

= Sample mean difference ± Margin of error

= 36.75 ± 13.0213

= (23.7377, 49.7713)

Therefore, the 99% confidence interval for μ₂ - μ₁ is (23.7377, 49.7713).

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Answer:

f(2) = 24

Step-by-step explanation:

to evaluate f(2) substitute x = 2 into f(x) , that is

f(2) = 15(2) - 6 = 30 - 6 = 24

The given differential equations can be rewritten in matrix-vector form as dX/dt = AX, where X = [x₁, x₂]ᵀ and A = [[2, 1], [1, 2]].

To rewrite the given differential equations in matrix-vector form, follow these steps:

1. Identify the dependent variables, x₁ and x₂, and arrange them into a column vector, X. This gives X = [x₁, x₂]ᵀ.

2. Identify the coefficients of x₁ and x₂ in the given differential equations. For dx₁/dt = 2x₁ + x₂ and dx₂/dt = x₁ + 2x₂, these coefficients are 2, 1, 1, and 2.

3. Arrange the coefficients into a matrix A, with rows corresponding to the order of the dependent variables. This gives A = [[2, 1], [1, 2]].

4. Write the matrix-vector equation dX/dt = AX. This represents the original system of differential equations in matrix-vector form.

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The Reverse Polish Notation of the expression A * B )/( C * D is: AB* CD* /

To express the given expression A * B )/( C * D in Reverse Polish Notation (RPN):

You would follow these steps:

STEP 1: Identify the operators and operands in the expression: A * B, /, and C * D
STEP 2: Convert the sub-expressions to RPN:
  - A * B becomes AB*
  - C * D becomes CD*
STEP 3: Combine the RPN sub-expressions with the remaining operator, /:
  - AB* CD* /

So, the Reverse Polish Notation of the expression A * B )/( C * D is: AB* CD* /

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Answer:

2.80

Step-by-step explanation:

35p = £0.35

8 × £0.35 = £2.80

The point of intersection is (0, 3). This means that the graphs of g(x) and h(x) intersect at the point where x=0 and y=3.

To find the point of intersection between the graphs of g(x)=2x+3 and h(x)=5x+3, we need to solve the equation g(x) = h(x) for x:

2x + 3 = 5x + 3

Subtracting 2x from both sides, we get:

3 = 3x + 3

Subtracting 3 from both sides, we get:

0 = 3x

Dividing both sides by 3, we get:

x = 0

So the graphs of g(x) and h(x) intersect at x = 0. To find the y-coordinate of the point of intersection, we can substitute x = 0 into either g(x) or h(x). Using g(x), we get:

g(0) = 2(0) + 3 = 3

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The tangential component of acceleration is c. aT = 4t/√4t²+5.

We need to follow these steps:

1. Calculate the first derivative of r(t) to get the velocity vector v(t).
2. Calculate the second derivative of r(t) to get the acceleration vector a(t).
3. Calculate the magnitude of the velocity vector |v(t)|.
4. Calculate the tangential component of acceleration aT by finding the dot product of a(t) and v(t), and then dividing by the magnitude of the velocity vector |v(t)|.

Let's go through these steps:

1. r(t) = ti + t²j + 2tk
  v(t) = dr(t)/dt = (1)i + (2t)j + (2)k

2. a(t) = dv(t)/dt = (0)i + (2)j + (0)k

3. |v(t)| = √(1² + (2t)² + 2²) = √(1 + 4t² + 4) = √(4t² + 5)

4. aT = (a(t) • v(t)) / |v(t)| = ((0)(1) + (2)(2t) + (0)(2)) / √(4t² + 5) = (4t) / √(4t² + 5)

So, the tangential component of acceleration is:

aT = 4t / √(4t² + 5)

This corresponds to option c. aT = 4t/√4t²+5.

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Answer:

0+8=8

8+0=8

8-0=8

8-8=0

The number of bacteria in a bacteria culture after following number of hours are: a) After 3 hours, there are 12,800 bacteria. b) After t hours, there are 200 * 2^(2t) bacteria. c) After 40 minutes, there are 400 bacteria.

Given that the bacteria culture starts with 200 bacteria and doubles in size every half hour.

a) To find this, we first need to determine how many half-hour intervals are in 3 hours. Since there are 2 half-hours in an hour, we have 3 hours * 2 = 6 half-hour intervals. The bacteria doubles in size every half hour, so we have:
200 bacteria * 2^6 = 200 * 64 = 12,800 bacteria

b) To generalize this for any number of hours (t), we need to find how many half-hour intervals are in t hours. That's 2t half-hour intervals. Then we have:
200 bacteria * 2^(2t)

c) First, we need to convert 40 minutes to hours. Since there are 60 minutes in an hour, we have 40/60 = 2/3 hours. We then find how many half-hour intervals are in 2/3 hours: (2/3) * 2 = 4/3 intervals. Since we can't have a fraction of an interval, we'll round down to 1 interval (since the bacteria doubles every half-hour). Then we have:
200 bacteria * 2^1 = 200 * 2 = 400 bacteria

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