10. A culture starts with 157 bacteria cells, and the number of cells doubles every 15 minutes.
a. Write an equation for the number of cells C after t 15-minute periods.
b. Use the equation to predict the number of cells after 2 hours.

Using Algebraic expression,  It will take approximately 3.47 years for an investment of $930,000 to double at an annual interest rate of 20%, compounded continuously.

Algebraic expression can be defined as combination of variables and constants.

The formula for continuous compound interest is given by:

[tex]A = Pe^{(rt)}[/tex]

where A is the amount after time t, P is the initial principal, r is the annual interest rate (as a decimal), and e is the constant 2.71828... (also known as Euler's number).

To find the time it takes for an investment to double at a 20% annual interest rate, we need to solve for t in the equation:

[tex]2P = Pe^{(rt)}[/tex]

Dividing both sides by P, we get:

[tex]2 = e^{(rt)}[/tex]

Taking the natural logarithm of both sides, we get:

[tex]ln(2) = rt ln(e)[/tex]

Since ln(e) = 1, we can simplify to:

ln(2) = rt

Solving for t, we get:

[tex]t = ln(2) / r[/tex]

Substituting P = $930,000 and r = 0.20, we get:

[tex]t = ln(2) / 0.20t = 3.47 years[/tex]

Therefore, it will take approximately 3.47 years for an investment of $930,000 to double at an annual interest rate of 20%, compounded continuously.

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Answer:

Step-by-step explanation:

We can use the given formula to set up an equation and solve for x:

V = 20,000(0.8)x

where V is the value of the car after x years.

We want to find x when the value of the car is $8,200, so we set V equal to 8,200 and solve for x:

8,200 = 20,000(0.8)x

Dividing both sides by 20,000:

0.41 = 0.8x

Taking the natural logarithm of both sides:

ln(0.41) = x ln(0.8)

Dividing both sides by ln(0.8):

x = ln(0.41) / ln(0.8)

Using a calculator:

x ≈ 6.11

Therefore, it will take about 6.11 years for the car to be worth $8,200.

The distance between the parallel lines y = -1/3x + 3 and y = -1/3x - 7 is approximately 9.49 units.

A parallel line is a pair of lines that never cross and are always spaced at the same distance apart. The y-intercepts are different, but they have the same slope. In other terms, two lines are parallel if they have the same slope but distinct y-intercepts. Due to their role in defining and producing forms like rectangles, parallelograms, and trapezoids, parallel lines play a crucial role in geometry and trigonometry. In the actual world, parallel lines are also used in things like roads, railroad tracks, and electrical cables.

The standard equation of the line is given as:

y = mx + c

where, m is the slope.

c is the y-intercept.

Comparing the equation with the given equation we have the slope of the parallel lines as:

y = -1/3x +3 and y =-1/3x -7

slope = m = -1/3

y-intercept of 1st line = (0, 3)

y-intercept of the 2nd line = (0, -7)

The distance between two parallel lines is the perpendicular distance between them.

To calculate the perpendicular distance we find the equation of the perpendicular line.

The slope of the perpendicular line is negative reciprocal of the slope of the two given lines, which is 3.

The point-slope form is:

y - y1 = m(x - x1)

Substituting the values:

y - 3 = 3(x - 0)

y = 3x + 3

The intersection of this perpendicular line with the other line.

3x + 3 = -1/3x - 7

10/3x = -10

x = -3

Substituting x = -3 into either equation of the parallel lines gives:

y = -1/3(-3) - 7 = -6

Using the distance formula we have:

d = √((x2 - x1)² + (y2 - y1)²)

d = √((-3 - 0)² + (-6 - 3)²)

d = √(9 + 81)

d = √(90)

d ≈ 9.49

Hence, the distance between the parallel lines y = -1/3x + 3 and y = -1/3x - 7 is approximately 9.49 units.

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The linear equation on the graph is written as:

y = 20x + 10

Remember that the general linear equation can be written as:

y = a*x + b

Where a is the slope and b is the y-intercept.

By looking at the graph, we can see that the line intercepts the y-axis at y = 10, then the y-intercept is b = 10.

We can write:

y = a*x + 10

Now we can also identify the point (1, 30), replacing the values of that point in the equation above we will get:

30 = a*1 + 10

30 = a + 10

30 - 10  =a

20  =a

Then the linear equation is:

y = 20x + 10

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We can write the expression in the simplest form as -

[tex]$4x^{2} y^{2} \sqrt[3]{x^{2} y^{2} }[/tex].

An expression is a term made up of both constants and variables.

Given is the expression as -

[tex]$(8x^{7} y^{4} )^{2/3} \\[/tex]

We can solve as -

[tex]$(8x^{7} y^{4} )^{2/3} \\[/tex]

[tex](8^{2/3} x^{7\times\frac{2}{3} } y^{4\times\frac{x}{y} 2/3})[/tex]

[tex]$4x^{2} y^{2} \sqrt[3]{x^{2} y^{2} }[/tex]

Therefore, we can write the expression in the simplest form as -

[tex]$4x^{2} y^{2} \sqrt[3]{x^{2} y^{2} }[/tex].

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The greatest common factor of 4x³y² and 12x²y is: 4x²y.

The GCF is a term used to refer to the largest common factor between any two or more numbers (Greatest Common Factor). The GCF of two natural integers, x and y, is the largest possible number that equally and completely divides both x and y. GCF is typically calculated using the following three techniques: division, multiplication, and prime factorization.

Now in the given question,

4x³y² can be written as = 2×2×x×x×x×y×y

12x²y can be written as = 2×2×3×x×x×y

The common factors among them are:

2×2×x×x×y = 4x²y.

Therefore, the greatest common factor of 4x³y² and 12x²y is: 4x²y.

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The coordinate point for the centroid are (8/3, 1/3).

From the question, we have the following parameters that can be used in our computation:

Coordinates = (1,2), (2, 1) and (5, -2)

To find the centroid of a triangle, you can calculate the average of the x-coordinates and the average of the y-coordinates of its three vertices.

Let (x1, y1), (x2, y2), and (x3, y3) be the coordinates of the vertices of a triangle.

Then, the coordinates of the centroid (xc, yc) are given by:

xc = (x1 + x2 + x3) / 3

yc = (y1 + y2 + y3) / 3

Substitute the known values in the above equation, so, we have the following representation

xc = (1 + 2 + 5) / 3 = 8/3

yc = (2 + 1 - 2) / 3 = 1/3

Hence, the coordinates of the centroid are (8/3, 1/3).

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Therefore , the solution of the given problem of equation comes out to be  the system of equations has an answer of x = -20 and y = 10.

The same letter is frequently used in mathematical formulas to attempt to enforce variable uniformity among two assertions. Mathematical equations, also referred to as assertions, are used to demonstrate the equality of many academic figures. Using the outcome y + 6 = 12 as an example, the normalise divides the numbers 12 and b + 6 into 2 parts. The relationship between each symbol component and the amount of lines can be found. A symbol's meaning typically runs counter to itself.

Here,

The set of linear equations can be used to find the values of x and y:

144x + 288y = 1440 \s28x + 84y = 280

By dividing both parts of the second equation by 28, we can begin by making it simpler:

x + 3y = 10

Then, by separating x, we can use this equation to solve for x in terms of y:

x = 10 - 3y

In order to simplify the first problem, we can use this expression in place of x:

=>  144(10 - 3y) + 288y = 1440

=>  1440 - 432y + 288y = 1440

=>  -144y = -1440

=>  y = 10

Given that y = 10, we can solve for x by substituting this number into the equation x + 3y = 10:

=>  x + 3(10) = 10

=>  x + 30 = 10

=> x = -20

As a result, the system of equations has an answer of x = -20 and y = 10.

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Answer:

8

Step-by-step explanation:

160×5%

[tex]\frac{160 * 5}{100}[/tex] = 8

The area of the irregular room is  82.5 ft²

It is important to note that the area of a triangle is represented with the formula

A = 1/2bh

Given that;

From the image shown, we have that;

Area of the figure = sum of the areas of the two triangle

This represented as;

Area = 1/2 × 14(6) + 1/2 × 9(9)

expand the brackets

Area = 1/2 (84) + 1/2(81)

divide the values

Area = 42 + 40. 5

Add the values

Area = 82.5 ft²

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50

Step-by-step explanation:

plot it on the number line

Answer:

D

y=5x-3

0=5x-y+3

5x-y= -3

If the base triangle has a base of 8 meters and a height of 6.9 meters. the surface area of the pyramid is 139.6 square meters.

First, let's find the area of the base triangle:

Area of base triangle = (1/2) * base * height

Area of base triangle = (1/2) * 8 * 6.9

Area of base triangle = 27.6 square meters

Now, let's find the area of one of the triangular faces:

Area of triangular face = (1/2) * base * height

Area of triangular face = (1/2) * 8 * 7

Area of triangular face = 28 square meters

Since the pyramid has four triangular faces, the total area of the triangular faces is 4 times the area of one triangular face, which is:

4 * 28 = 112 square meters

Total surface area = area of base triangle + area of triangular faces

Total surface area = 27.6 + 112

Total surface area = 139.6 square meters

Therefore, the surface area of the pyramid is 139.6 square meters.

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Answer:

9.4ft²

Step-by-step explanation:

The formula for the area of a rectangle is base*height, or b*h.

Since we know the final area of the room and one of the side lengths, we can plug those into the formula I just stated so that we get

15*h=141

All that's left to do is solve for h, which is the other side that we are solving for. To do this, we will divide 15 on both sides:

15h/15=141/15

The 15s on the left side cancel out, and we simply divide 141 by 15 to get our final answer of 9.4ft².

a. The critical points are (0,0), (0,k), (k,0), and (k,k).

b. x³ and y³ are always non-negative, it can see that this critical point is a local minimum.

c. The function has a local minimum for all k > 0

The function is defined as a mathematical expression that defines a relationship between one variable and another variable.

(a) To find the critical points of f(x, y) = x³ + y³ - 3xy, we need to find where the partial derivatives with respect to x and y are equal to zero:

∂f/∂x = 3x² - 3ky = 0

∂f/∂y = 3y² - 3kx = 0

Solving these equations simultaneously, we get:

x = y

3x² - 3kx = 0

Substituting the first equation into the second, we get:

3x² - 3kx = 0

3x(x - k) = 0

So x = 0 or x = k. Since x = y, we also have y = 0 or y = k.

Therefore, the critical points are:

(0,0), (0,k), (k,0), and (k,k).

To determine the nature of these critical points, we need to compute the second partial derivatives of f(x):

∂²f/∂y² = 6y

∂²f/∂x∂y = -3k

At the point (0,0), we have:

∂²f/∂x² = 0

∂²f/∂y² = 0

∂²f/∂x∂y = 0

So we cannot use the second derivative test to determine the nature of this point.

At the point (0,k), we have:

∂²f/∂x² = 0

∂²f/∂y² = 6k

∂²f/∂x∂y = -3k

The determinant of the Hessian matrix is:

∂²f/∂x² × ∂²f/∂y² - (∂²f/∂x∂y)² = 0 - (-3k)² = 9k² > 0

Since ∂²f/∂y² is positive, we have a local minimum at the point (0,k).

Similarly, we can show that the points (k,0) and (k,k) are also local minima when k > 0.

(b) When k = 0, the partial derivatives are:

∂f/∂x = 3x² = 0

∂f/∂y = 3y² = 0

So x = y = 0 is the only critical point.

The second partial derivatives are:

∂²f/∂x² = 6x = 0

∂²f/∂y² = 6y = 0

∂²f/∂x∂y = 0

So we cannot use the second derivative test to determine the nature of this point. However, since the function is symmetric about the x and y axes, and x³ and y³ are always non-negative, we can see that this critical point is a local minimum.

(c) To find the values of k for which f(x) has a local minimum or maximum, we need to consider the discriminant of the Hessian matrix at each critical point:

For the point (0,k):

∂²f/∂x² × ∂²f/∂y² - (∂²f/∂x∂y)² = 0 - (-3k)² = 9k² > 0

So function has a local minimum for all k > 0

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The triangle AEC is similar to triangle BDC.

Those triangles look the same but are different in size.

And in similar triangles,

the corresponding sides are in proportion to each other and the corresponding angles are equal to each other.

Given:

5. ΔAEC is a triangle.

Assuming BD is parallel to AE.

The ratio of the corresponding sides,

10/5 = 2

8/4 = 2

The ratio is constant.

A similar triangle is made by drawing a line parallel to one of the triangle's sides.

Therefore, triangle AEC ~ triangle BDC.

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Answer:

To solve this problem, we need to use vector addition. We can start by breaking down each force into its x and y components.

For muscle A:

Force = 100 lbs

Angle = 15 degrees

X-component = 100 * cos(15) = 96.93 lbs

Y-component = 100 * sin(15) = 25.04 lbs

For muscle B:

Force = 150 lbs

Angle = 30 degrees

X-component = 150 * cos(30) = 129.90 lbs

Y-component = 150 * sin(30) = 75.00 lbs

For muscle C:

Force = 75 lbs

Angle = 10 degrees

X-component = 75 * cos(10) = 73.83 lbs

Y-component = 75 * sin(10) = 13.03 lbs

Next, we can add up the x and y components separately:

X-component = 96.93 + 129.90 + 73.83 = 300.66 lbs

Y-component = 25.04 + 75.00 + 13.03 = 113.07 lbs

We can use these components to find the magnitude and direction of the resultant force:

Magnitude = sqrt(X^2 + Y^2) = sqrt((300.66)^2 + (113.07)^2) = 321.23 lbs

Direction = tan^-1(Y/X) = tan^-1(113.07/300.66) = 21.65 degrees (measured counterclockwise from the positive x-axis)

Therefore, the resultant force of the muscles is 321.23 lbs at an angle of 21.65 degrees. Here is a diagram to illustrate the vectors:

Step-by-step explanation:

      25.04 lbs    73.83 lbs    129.90 lbs

         /\           /\           /\

        /   \        /   \        /   \

       /      \     /      \     /      \

 100 lbs     15°  75 lbs    10°  150 lbs  30°

      \         /  \         /  \         /

       \      /     \      /     \      /

        \   /        \   /        \   /

      96.93 lbs   13.03 lbs   75.00 lbs

The value of c is given as follows:

c = -1.235.

The z-score of a measure X of a variable that has mean symbolized by [tex]\mu[/tex] and standard deviation symbolized by [tex]\sigma[/tex] is obtained by the rule presented as follows:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

For this problem, we have that P(z > c) = 0.8914, meaning that Z when X = c has a p-value of 1 - 0.8914 = 0.1086, hence the value of c is given as follows:

c = -1.235.

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The weight of the lead in the beaker is 0.21 pounds.

A cone is a three dimensional shape with slant sides and a circular surface. Its volume can be defined as the amount of substance that it can contain.

volume of a cone = 1/3 [tex]\pi[/tex]r^2*h

where r is the radius of its circular surface, and h is its height.

In the given question, we have;

r = d/2 = 1/2

  = 0.5 inches

h = 2 inches

Thus,

volume of the cone-shaped beaker = 1/3 [tex]\pi[/tex]r^2*h

                                 = 1/3 *22/7*(0.5)^2*2

                                 = 0.524

The volume of the lead in the cone-shaped beaker is 0.524 cubic inch.

Weight of the lead in the beaker = 0.524*0.41

                                                     = 0.2148

The weight of the lead in the beaker is 0.21 pounds.

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Answer:

2. x-10=24

Step-by-step explanation:

x is the unknown amount of candy that Charlie has.

He gives ten away (-10).

Now he has 24 pieces.

x-10 = 24

This means that when function f(x) = 4x²- 1 and g(x) = 4x + 8, f(2) - g(2) = 15, and by using the equation as a substitute.

what is function ?

A function in mathematics is a relationship between a set of inputs (also referred to as the domain) and a set of outputs (also referred to as the range), where each input has a unique outcome. Typically, the variable x stands for the input values, the variable y for the output values, and the variable f for the function itself (x). Tables, graphs, equations, and verbal descriptions can all be used to represent functions. In a wide variety of disciplines, including science, engineering, economics, and social studies, they are employed to model relationships between quantities. Functions are frequently used to explain the relationship between two quantities.

given

We must first determine the values of f(2) and g(2) independently using the provided functions in order to evaluate f(2) - g(2):

f(2) = 4(2)² - 1 = 15

g(2) = -4(2) + 8 = 0

These values can now be added to the expression f(2) - g(2) as a substitute:

f(2) - g(2) = 15 - 0 = 15

This means that when f(x) = 4x²- 1 and g(x) = 4x + 8, f(2) - g(2) = 15, and by using the equation as a substitute.

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The value of the median of the data is 35.

Algebraic expression can be defined as combination of variables and constants.

A box plot is a graphical representation of a data set that shows the distribution of the data, as well as the central tendency and the variability of the data. In this case, the box plot displays the number of push-ups completed by 9 students in a PE class.

The box plot has a number line from 9 to 56 with tick marks every 2 units. The box extends from 20 to 42.5 on the number line, which means that the middle 50% of the data (i.e., the interquartile range) lies between 20 and 42.5. The line in the box represents the median of the data, which is the value that separates the data set into two equal halves.

In this case, the line in the box is at 35, which means that half of the students completed more than 35 push-ups, while the other half completed fewer than 35 push-ups.

Therefore, the value of the median of the data is 35.

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Answer: Without seeing the actual scatter plot, it is difficult to make a definitive conclusion about the relationship between the number of swings and the number of hits. However, there are a few things that the coach can look for in the scatter plot to gain some insights:

Positive correlation: If the scatter plot shows a generally upward trend, with more hits occurring as the number of swings increases, this suggests a positive correlation between the two variables. In other words, players who take more swings tend to have more hits during the tournament.

Negative correlation: On the other hand, if the scatter plot shows a generally downward trend, with fewer hits occurring as the number of swings increases, this suggests a negative correlation between the two variables. In this case, players who take fewer swings tend to have more hits during the tournament.

No correlation: If the scatter plot shows no clear trend or pattern, with hits occurring randomly across different numbers of swings, this suggests no correlation between the two variables. In this case, there is no apparent relationship between the number of swings taken and the number of hits during the tournament.

Overall, the coach can use the scatter plot to gain some insights into the relationship between the number of swings and the number of hits, but more advanced statistical analysis may be necessary to make definitive conclusions.

Step-by-step explanation:

The ball with the greater maximum height is the second ball

From the question, we have the following parameters that can be used in our computation:

f(t) - 4t^2 + 16t

g(t) = -16t^2 + 64t + 192

Differentiate and set them to 0

-8t + 16 = 0

-32t + 64 = 0

Solve for t

t = 2 in both cases

So, we have

f(2) = -4(2)^2 + 16(2) = 16

g(2) = -16(2)^2 + 64(2) + 192 = 256

Hence, the ball with the greater maximum height is the second ball

To do this, we set the function to 0 and calculate t

So, we have

f(t) = -4t^2 + 16t = 0

g(t) = -16t^2 + 64t + 192 = 0

Using a graphing tool, we have

t = 4

t = 6

This means that the first ball landed first

To change the term, we set the constant term in g(t) = -16t^2 + 64t + 192 to a negative number > -192

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Answer:

0.0166666666666666666666666666666666666666666667

Step-by-step explanation:

USE CALCULATORRRRRR

The word or phrase that describes the probability that Charlotte will win a prize, with a probability of 12%, is D. likely.

Probability refers to the likelihood of an expected outcome occurring, given the many possible chances.

Probability lies between zero and one, depending on the level of certainty or otherwise of the successful outcomes.

In this situation, Charlotte's chances of winning cannot be described as impossible, zero percent, or unlikely.

Charlotte does not have an equal chance or 50-50 odds of winning because the probability is not up to 50%.

Thus, we can conclude that Charlotte has a D. likely opportunity of winning.

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Answer:

$585 dollars will be used for cleaning supplies out of the total of $900 that was collected over the six days.

Step-by-step explanation:

$150.00 (money each day) * 6 (number of days) = $ (total)

[tex]150*6=900[/tex]

$900 in 6 days

100% (total amount) - 35% (pet food) = % (cleaning supplies)

[tex]100-35=65[/tex]

65% of his money will be used for cleaning supplied

Change the percentage into a decimal

65% = [tex]0.65[/tex]

Multiply the total amount of money by the decimal

[tex]900*0.65=585[/tex]

Out of the total $900 dollars he collected over 6 days, $585 dollars will be used for cleaning supplies and the rest ($315) will be used on pet food.

Answer:Juan has already run a total of 2.3 + 5.6 = 7.9 miles on Monday and Wednesday.

To reach his goal of running 12 miles by Saturday, he still needs to run an additional 12 - 7.9 = 4.1 miles.

Therefore, Juan needs to run 4.1 more miles to meet his goal of running 12 miles by Saturday.

Step-by-step explanation:

Answer:

2/5

Step-by-step explanation:

So if you were to half that the bottom stays the same but the top you divide by two which 4 divided by 2 is 2 so you would need 2/5 cup of flour for your new recipe.