A scientist is studying bacterial growth over time. Her research is conducted by placing 10 strep bacteria in a petri dish containing enough food for the bacteria to live and thrive for the course of the research. She records the number of bacteria present every two hours for 6 hours. The collected data is shown in the table below and on the given graph.
a. What function, linear or exponential, do you think would be the best choice to model this data? Explain why you think your choice is the best-fit.
b. Explain how to determine which function is the best choice mathematically, without using a graph.
Demonstrate your method.
c. Think of an example of at least 4 data points that would be best modeled by the function you did NOT choose in part a. Explain how you set up the data so that it worked with this type of function.

The maximum-volume of open box which could made from a "10-inch" by "20-inch" piece of cardboard by cutting out squares of equal sides from 4 corners is 192.5 inch³.

We know that "Volume" of box is ⇒  V = L × W × H.

The dimensions of a piece of cardboard is 10 inches by 20 inches.

Let x = height of box, x will also be side of square taken out of each corner.

The length of box = 20 - 2x; width of box = 10 - 2x and

So, equation of Volume is ⇒ V = (20-2x) × (10-2x) × (x),

⇒ V = 4x³ - 60x² + 200x

On differentiating both sides, to get "dv/dx" :

We get,

⇒ dv/dx = 12x² - 120x + 200

Using general formula for finding roots of equation,

We get,  x = 7.886 and x = 2.113,

Since 7.886 × 2 = 15.772 inches, this is greater than the 10 inch side,

So,  x = 2.113 inches.

We have cut "2.113 inches" from each-corner to get maximum volume.

The sizes of cubes are :

⇒ Length = 20 - (2 × 2.113) = 15.774,

⇒ width = 10 - (2 × 2.113) = 5.774

⇒ Height = 2.113

So, volume = 15.774 × 5.774 × 2.113 = 192.5 cubic inches.

Therefore, the volume of cube is 192.5 inch³.

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Gas company charges are typically not linear due to the use of fixed fees, variable charges, and tiered pricing structures. While it is possible for a company to adopt a linear rate structure, it is not common in practice.

The charges levied by gas companies can vary depending on a variety of factors, including the amount of gas used, the location, the season, and other factors. However, when considering a particular rate structure, we can analyze whether the charges are linear or not.

A linear rate structure implies that the cost of gas usage is directly proportional to the amount of gas consumed. In other words, if you use twice as much gas, you would be charged exactly twice as much.

In practice, most gas companies use non-linear pricing structures that incorporate fixed charges, variable charges, and tiered pricing structures. For instance, a gas company may charge a fixed monthly fee for the gas supply, regardless of the amount of gas used. Additionally, there may be variable charges that depend on the amount of gas consumed, with different rates applying to different usage levels. Finally, some companies may apply tiered pricing structures, where the price per unit of gas increases as consumption levels increase.

For example, suppose a gas company charges a fixed fee of $10 per month for the gas supply, and then $0.10 per cubic foot of gas for the first 500 cubic feet used in a month, and then $0.15 per cubic foot for all additional usage. In this case, the pricing structure is not linear, as the rate per cubic foot of gas changes depending on the amount consumed.

It is also important to note that gas companies may adjust their pricing structures from time to time in response to market conditions, regulatory changes, or other factors. Therefore, it is important for consumers to review their bills regularly to ensure that they understand the charges and are not being overcharged.

In conclusion, gas company charges are typically not linear due to the use of fixed fees, variable charges, and tiered pricing structures. While it is possible for a company to adopt a linear rate structure, it is not common in practice.

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To factor the polynomial 10x³ + 35x² − 4x − 14 by grouping, the greatest common factor is factored out from the first two terms and the last two terms separately. Both terms have a common factor of (2x + 7), which is the missing factor in the parentheses, and the correct answer is (B) 2x + 7.

To factor the polynomial by grouping, we need to factor out the greatest common factor from the first two terms and the last two terms separately:

10x³ + 35x² − 4x − 14

5x2(2x + 7) − 2(2x + 7)

We can see that both terms have a common factor of (2x + 7), so the missing factor in the parentheses is (2x + 7). Therefore, the answer is (B) 2x + 7.

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Answer:

V = 75[tex]\pi[/tex] or 235.619449... cm / 235.62 cm

Hope this helps!

Step-by-step explanation:

The formula for the volume of a cone is  [tex]V=\frac{1}{3} *h*\pi r^{2}[/tex].

[tex]V = \frac{1}{3} *9*\pi 5^{2}[/tex]  ( Simplify 1/3 and 9 )

[tex]V=3*25\pi[/tex]

[tex]V=75\pi[/tex]

V = 235.619449019 or 235.62

The system of equations has two solutions: (0,2) and (-4/5, -6/5).

The substitution method is a way to solve a system of two linear equations with two variables. The method involves solving one of the equations for one of the variables, and then substituting that expression into the other equation in place of the same variable. This process reduces the system of two equations and two variables to a single equation and a single variable, which can then be solved.

We have the system of equations:

4x² + y² = 4 ........(1)

y = x + 2 ........(2)

We can solve this system of equations by substitution.

Substitute (2) into (1) to eliminate y:

4x² + (x + 2)² = 4

Simplify and solve for x:

4x² + x² + 4x + 4 = 4

5x²+ 4x = 0

x(5x + 4) = 0

x = 0 or x = -4/5

If x = 0, then from equation (2), y = 2. So one solution is (x,y) = (0,2).

If x = -4/5, then from equation (2), y = -6/5. So the other solution is (x,y) = (-4/5, -6/5).

Therefore, the system of equations has two solutions: (0,2) and (-4/5, -6/5).

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Answer:

The difference of 1.4 - 0.95 is less than one.

To understand why, we need to calculate the difference between the two numbers:

1.4 - 0.95 = 0.45

Since 0.45 is less than 1, we can conclude that the difference of 1.4 - 0.95 is less than one.

Step-by-step explanation:

Answer:

See below for proof.

Step-by-step explanation:

Given expression:

[tex]\implies \dfrac{1+\csc(x)}{\cot(x)+\cos(x)}[/tex]

[tex]\boxed{\begin{minipage}{4.3 cm}\underline{Trigonometric Identities}\\\\$\csc(x)=\dfrac{1}{\sin(x)}$\\\\\\$\cot(x)=\dfrac{\cos(x)}{\sin(x)}$\\\\\\$\sec(x)=\dfrac{1}{\cos(x)}$\\\\\end{minipage}}[/tex]

Use the trigonometric identities to rewrite csc(x) and cot(x) in terms of sin(x) and cos(x):

[tex]\implies \dfrac{1+\dfrac{1}{\sin(x)}}{\dfrac{\cos(x)}{\sin(x)}+\cos(x)}[/tex]

Simplify the numerator and denominator:

[tex]\implies \dfrac{\dfrac{\sin(x)+1}{\sin(x)}}{\dfrac{\cos(x)+\sin(x)\cos(x)}{\sin(x)}}[/tex]

[tex]\textsf{Apply the fraction rule:} \quad \dfrac{\dfrac{a}{b}}{\dfrac{c}{b}}=\dfrac{a}{c}[/tex]

[tex]\implies \dfrac{\sin(x)+1}{\cos(x)+\sin(x)\cos(x)}[/tex]

Factor out cos(x) from the denominator:

[tex]\implies \dfrac{\sin(x)+1}{\cos(x)(1+\sin(x))}[/tex]

Cancel the common factor (1 + sin(x)):

[tex]\implies \dfrac{1}{\cos(x)}[/tex]

Rewrite 1/cos(x) using the identity:

[tex]\implies \sec(x)[/tex]

Therefore, the equation has been verified.

Let's see

LHS

[tex]\\ \sf\longmapsto \dfrac{1+csc(x)}{cot(x)+cos(x)}[/tex]

[tex]\\ \sf\longmapsto \dfrac{1+\dfrac{1}{sin(x)}}{\dfrac{cos(x)}{sin(x)}+cos(x)}[/tex]

[tex]\\ \sf\longmapsto \dfrac{\dfrac{1+sin(x)}{\cancel{sin(x)}}}{\dfrac{cos(x)+sin(x)cos(x)}{\cancel{sin(x)}}}[/tex]

[tex]\\ \sf\longmapsto \dfrac{1+sinx}{cosx(1+sinx)}[/tex]

[tex]\\ \sf\longmapsto \dfrac{1}{cosx}[/tex]

[tex]\\ \sf\longmapsto sec(x)[/tex]

verified

Answer:

B

Step-by-step explanation:

False. Calculating standard error is indeed important in statistics, as it serves a different purpose than the regular standard deviation of a single data set.

While standard deviation measures the variability or dispersion within a data set, standard error is used to measure the accuracy of a sample mean as an estimate of the true population mean.
1. Standard deviation measures the dispersion or spread of data points in a single data set. It helps you understand how much the individual data points deviate from the mean (average) of that data set.
2. Standard error, on the other hand, is a measure of how accurate a sample mean is in estimating the true population mean.

It is calculated by dividing the standard deviation of the sample by the square root of the sample size (n).
Standard Error (SE) = Standard Deviation (SD) / √n
3. As the sample size increases, the standard error decreases, indicating that the sample mean becomes a more accurate estimate of the population mean.

This is because larger sample sizes tend to have a smaller sampling error.
4. In hypothesis testing and confidence interval estimation, standard error plays a crucial role.

It helps to determine the margin of error and to assess the significance of differences between sample means.
5. While standard deviation is a useful measure for understanding variability within a single data set, standard error is essential for making inferences about the population based on the sample data.
In conclusion, both standard deviation and standard error are important in statistics as they serve different purposes. Ignoring standard error would lead to inaccurate conclusions about the population mean based on sample data.

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88 is the measure of arc BC.​

What is arc ?

In general, an arc is any smooth wind joining two points. The length of an arc is known as its arc length. In a graph, a graph arc is an ordered brace of conterminous vertices. In particular, an arc is any portion( other than the entire wind) of the circumference of a circle.

                       An arc is the term used to describe the convex area of a circle that curves away from an angle like this, between two line segments or rays. A circle's circumference, or outside border, is known as the arc's circumference, and it is this area that is measured when an arc is drawn along it.

DE ≅ BC

  3x + 22 = 4x

      4x - 3x = 22

          x = 22

BC = 4X = 4 * 22

              =    88

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Using the Pythagorean Rule we know that player A is 19.2 ft away from us.

In a right-angled triangle, the square of the hypotenuse side is equal to the sum of the squares of the other two sides, according to Pythagoras's Theorem.

These triangle's three sides are known as the Perpendicular, Base, and Hypotenuse.

The Pythagorean Rule can be used to determine the distance between you and Player A since the distances between you, Player A, and Player B are in the shape of a right-angled triangle.

c² = a² + b²

Where;

c = hypotenuse = Distance between you and player A

a = Length of the triangle

b = Height of triangle

Then, calculate as follows:

c² = 12² + 15²

c² = 144 + 225

c = √369

c = 19.2 ft

Therefore, using the Pythagorean Rule we know that player A is 19.2 ft away from us.

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Correct question:

During a basketball game, you want to pass the ball to either Player A or Player B. You estimate that Player B is about 15 feet from you, as shown. How far away from you is Player A?

The correct answer is ± 5÷2

From the given function, the domain of the function is (-∞, ∞) and range of the function is (-∞, -5].

The given function is:

[tex]f(x) = (2/3)(x - 4)^2 - 5[/tex]

The domain of a function is the set of all possible input values (x) for which the function is defined. In this case, there are no restrictions on the input, so the domain is all real numbers:

Domain: (-∞, ∞)

The range of a function is the set of all possible output values (y) that the function can produce. To find the range of the given function, we can complete the square:

[tex]f(x) = (2/3)(x - 4)^2 - 5 \\ = (2/3)(x^2 - 8x + 16) - 5\\ = (2/3)x^2 - (16/3)x + 7.33\\ = (2/3)(x - 4)^2 - 4.67\\[/tex]

Since the term [tex](x - 4)^2[/tex] is always non-negative, the smallest value of the function occurs when [tex](x - 4)^2 = 0[/tex], which happens when x = 4. Substituting x = 4 into the equation gives:

[tex]f(4) = (2/3)(4 - 4)^2 - 5 \\ = -5[/tex]

Therefore, the range of the function is all real numbers less than or equal to -5:

Range: (-∞, -5]

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Therefore, the expression that Tesha wrote is most likely option. [tex]C: (5 *6 + 2 *3)/(3*2*3).[/tex]

An equation is a mathematical statement that shows the equality of two expressions or values. It contains an equal's sign (=) which separates the two expressions that are equal to each other. Equations can be simple or complex and can involve one or more variables, which are typically represented by letters. Equations are used extensively in mathematics, science, engineering, and many other fields to model and solve problems. Solving an equation involves finding the values of the variables that make the equation true.

To solve this problem, we need to simplify the expression.[tex](20 + 4) /12[/tex]:

[tex](20 + 4) / 12 = 24 / 12 = 2[/tex]

So, the expression that Tesha wrote must also simplify to 2. Let's check each of the options:

A. [tex]36 / 3 *5 = 12 * 5 = 60[/tex], which is not equivalent to 2. So, option A is not the correct one.

B. [tex](3 * 3 * 3) / 3 * 1 = 27 / 3 *1 = 9 * 1 = 9[/tex], which is not equivalent to 2. So option B is not the correct one.

C. [tex](5 * 6 + 2 * 3) / (3 * 2 * 3) = (30 + 6) / 18 = 36 / 18 = 2[/tex], which is equivalent to 2. So, option C is the correct one.

D. [tex](8 * 5 *2) / (10 * 2) = 80 /20 = 4[/tex], which is not equivalent to 2. So option D is not the correct one.

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The expression that we need to add to 5x + 7 to make it equivalent to 9(x + 1) is -2x + 2.

To make the expressions 5x + 7 and 9(x + 1) equivalent, we need to find the expression that, when added to 5x + 7, will give us 9(x + 1).

First, we can simplify 9(x + 1) by distributing the 9:

9(x + 1) = 9x + 9

Now we can see that we need to find an expression that, when added to 5x + 7, will give us 9x + 9:

5x + 7 + ? = 9x + 9

To find the missing expression, we need to isolate the variable on one side of the equation and the constant terms on the other side:

5x - 9x = 9 - 7

Simplifying this expression, we get:

-4x = 2

Dividing both sides by -4, we get:

x = -1/2

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Question:-

What expression would be added to 5x + 7 to make it equivalent to the expression 9(x + 1)?

The student can improve the investigation by using balls that are the same mass. This is because the investigation aims to study the effect of gravity on objects on Earth, and the gravitational force acting on an object is directly proportional to its mass. Therefore, in order to compare the effect of gravity on different objects, they should all have the same mass.

Using balls of different masses will make it difficult to draw any valid conclusion about how the gravitational force operates on different materials. For measuring the circumference of the balls, we would not be able to draw any conclusive results on the effect of gravity. The other option of dropping the balls from the same height is already part of the criteria mentioned and is used to ensure a consistent experimental setup.

Mariya's mistake in solving the inequality and plotting the number line is in step 3.

In mathematics, an inequality is a statement that compares two values or expressions, indicating whether they are equal, not equal, greater than, less than, greater than or equal to, or less than or equal to each other.

Inequalities can be solved by manipulating them in much the same way as equations, but with one important difference: when an inequality is multiplied or divided by a negative number, the direction of the inequality is reversed. For example, if we have the inequality -3x > 12, we can solve for x by dividing both sides by -3, but we also need to reverse the direction of the inequality, giving us x < -4.

The inequality is given as:

[tex]$\frac{a}{-13}\leq-16$[/tex]

Multiply both sides of the inequality by -13

a ≥ 208

The above means that the solution to the inequality are numbers greater than or equal to 208.

This means that we make use of a closed circle

In step 3 of Mariya's solution, the number line uses an open circle

Hence, Mariya's mistake is in step 3

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The complete question is:

Mariya was asked to solve Start Fraction a over negative 13 End Fraction less-than-or-equal-to negative 16 and then graph the solution. Her work is shown below. Step Mariya’s work Start Fraction a over negative 13 End Fraction less-than-or-equal-to negative 16 Step 1 (Negative 13) Start Fraction a over negative 13 End Fraction less-than-or-equal-to negative 16 (negative 13) Step 2 a greater-than-or-equal-to 208 Step 3 A number line going from 205 to 211. An open circle is at 208. Everything to the right of the circle is shaded. In which step, if any, did Mariya make a mistake in her work? Step 1 Step 2 Step 3 No mistake found

Answer:

1. 36

2. 28

Step-by-step explanation:

You are just counting how many sides are showing.  

The right side, you see 6 green sides.  Although you cannot see the other side, you know that it must be 6 also.  You count the orange and blue.  That is 12.  The bottom is 6.  the back is 6.

6 + 6 + 12 + 6 + 6  = 36

Helping in the name of Jesus.

The sampling interval for this account balance is $12,000.

To calculate the sampling interval, we will use the following formula:
Sampling interval = Population value (Book value) / Sample size
Student question: For an account balance with a book value of $1.5 million, the sample size is 125.

Given this information, the sampling interval will be:
Sampling interval = $1,500,000 / 125
Sampling interval = $12,000.

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Answer:

y=6x-4

Step-by-step explanation:

The slope-intercept form is y=mx+b  

The m in the equation stands for the slope.

The b in the equation stands for the y-intercept.

So you input the y-intercept given to you in the problem, -4, and the slope, 6.

So the equation should be: y=6x-4

Answer:13,595,327.65

Step-by-step explanation:

Answer:

ans:88.18cm

Step-by-step explanation:

Answer:

C) +18 ÷ +3 = +6

Step-by-step explanation:

The average weekly change in Alvin's training can be calculated by finding the total change in his training over the 3 weeks and then dividing it by the number of weeks. Since he reduced his training by 18 miles per week, his total change over the 3 weeks would be 18 miles/week * 3 weeks = 54 miles.

To represent the average weekly change in his training using an equation, we need to divide the total change by the number of weeks, which gives us:

Average weekly change = total change ÷ number of weeks

Substituting the values we found, we get:

Average weekly change = 54 miles ÷ 3 weeks = 18 miles/week

Answer:

y = 4/3x - 6

Step-by-step explanation:

The equation is y = mx + b

m = the slope

b = y-intercept

Slope = rise/run or (y2 - y1) / (x2 - x1)

Points (0, -6) and (6, 2)

We see the y increase by 8, and the x increase by 6, so the slope is

m = 8/6 = 4/3

Y-intercept is located at (0, -6)

So, the equation is y = 4/3x - 6

Hernán Cortés had around 600 men, including soldiers and sailors, under his command during the Spanish conquest of the Aztec Empire.

Hernán Cortés led the Spanish conquest of the Aztec Empire in the early 16th century. The size of his expeditionary force varied over time, but at its peak, it is estimated that he had about 600 men under his command, including soldiers and sailors.

When Cortés first set out from Cuba to explore the mainland of Mexico in 1519, he had a force of about 600 men, including 11 ships with sailors and about 100 soldiers. However, some of his soldiers were left behind to establish a settlement on the coast, reducing his fighting force to around 500 men.

As Cortés made his way inland, he was able to recruit more indigenous allies to his cause, which increased his overall fighting strength. By the time he reached the Aztec capital of Tenochtitlan in 1521, he had an army of around 80,000 men, including native allies, although the core of his fighting force was still made up of Spanish soldiers and sailors.

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The appropriate type of t-test for this study is a one-sample t-test.

A one-sample t-test is used to determine whether a sample mean is significantly different from a known or hypothesized population mean. In this case, the researcher wants to compare the obesity rate in a small community to the known obesity rate in the United States. As the population variance is unknown, the appropriate test is a one-sample t-test instead of a z-test.

The test involves calculating the t-statistic, which measures the difference between the sample mean and the hypothesized population mean, divided by the standard error of the sample mean. The t-statistic is then compared to the critical t-value from the t-distribution to determine whether the difference is statistically significant.

Therefore, a one-sample t-test is appropriate for this study.

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As the sample size increases, the variability among the sample means decreases, option A.

The number of observations used to calculate estimates for a certain population is known as the sample size. The sample size was determined by drawing from the population. Sampling is the practise of choosing a portion of the population from which to draw conclusions about the characteristics of the entire population. A subset of a population is chosen for study based on the number of entities in it.

Population data is a sizable collection of information that covers the whole population under investigation. All the components that are investigated for the research make up a population.

Yet, sample data is a component of the population. Generally, computing the entire population is rather cumbersome and challenging. In this instance, a representative sample of the population is chosen. Sample data refers to this sample.

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Answer:

To solve the equation, we need to isolate the square root term and square both sides of the equation. We have:

1 + √n + 11 = n

√n = n - 12

Squaring both sides:

n = (n - 12)^2

n = n^2 - 24n + 144

n^2 - 25n + 144 = 0

Factorizing the quadratic equation:

(n - 16)(n - 9) = 0

Therefore, n = 16 or n = 9. We need to check these solutions in the original equation to see if they are valid.

For n = 16:

1 + √16 + 11 = 16

1 + 4 + 11 = 16

16 = 16

The solution n = 16 satisfies the equation.

For n = 9:

1 + √9 + 11 = 9

1 + 3 + 11 = 9

15 = 9

The solution n = 9 does not satisfy the equation.

Therefore, the only real solution is n = 16, which corresponds to option C: -2 is an extraneous solution and 5 is a real solution.

Hope This Helps!