The function y=56.35x can be used to determine the
cost of taking a group of students to a water park. If at
least 15 but no more than 25 students go to the water
park, what is the domain of the function?

A) the limit of the difference quotient at x0 = 3 for f(x) is  lim(h → 0) (f(3+h) - f(3) )/h = 3² + 3(3) - 6 = 12

B) the expression of a line passing through the same point that f(x) passes through at x = 3 with a slope equal to the limit you found in part is y = 12x - 29.

C) See graph attached.

Part A ) The difference quotient of a function f(x) at x0 is given by:

(f(x0+h) - f(x0)) /h

To calculate the limit of the difference quotient at x0 = 3 for f(x) we need to find the limit of the above expression as h approaches 0.

f(x) = x³ - 4x² + 2

(f(3+h) - f(3))/h

= ( (3+h)³ - 4(3+h)² + 2 - (3³ - 4(3)² + 2) )/h

 = (27 + 27h +   9h² + h³ - 4(9 + 6h + h²) + 2 - 19)/h

= (h³ + 3h² - 6h)/h

= h² + 3h - 6

Therefore, the limit of the difference quotient at x0 = 3 for f(x) is:

lim(h → 0) (f(3+h) - f(3))/h

= 3² + 3(3) - 6 = 12

Part B)  To find the equation of a line passing through the point (3, f(3)) with a slope equal to the limit found in part A, we can use the point-slope form of the equation of a line...

y - y1 = m (x - x1)

where m is the slope and (x1, y 1) is the given point.

Substituting the values

y - f(3) = 12(x - 3)

Expanding and simplifying

y = 12x - 29

so this means that the equation of the line passing through the point (3, f(3)) with a slope equal to the limit found in part A is y = 12x - 29.


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Full Question:

Part A

Consider the function f(x) = x3 − 4x2 + 2. Calculate the limit of the difference quotient at x0 = 3 for f(x).

Part B

Find the equation of a line passing through the same point that f(x) passes through at x = 3 with a slope equal to the limit you found in part A.

Part C

Graph the function f(x) and the line you identified in part B. How do the graphs relate to each other? What does the slope of the line represent about the function at the given point?

Answer:

part a) will be pie r +2r

i m not sure about others

Answer:

a

Step-by-step explanation:

i hope this help

The graph represents a functional relationship between two variables X and Y. We get y=3 for the value x=2

Domain is defined as the set of values of independent variables.

In the question, all x coordinates are domain.

In a function, for each domain we get a possible output from the dependent variables which is termed as range. In this graph, the y coordinates are range because y value is dependent on x value.

The graph shows five points are plotted on XY coordinate system that represents a relationship between independent variable X and dependent variable Y.

hence, this graph is plotted from a function.

from the graph we get the domain as {-4, -2, 0, 2, 4}

the ranges are = {-3, -2, -1, 1, 3}

we get, y=3 for the value of x= 2  

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A. The system of equations in standard form is: 4x + 5y = 540 and 3x + 5y = 440.

B. The y-intercept of the equation representing the prices on Saturday night is 108, and the y-intercept of the equation representing the prices at the matinee on the second day is 88.

A) Let's define the variables:

Let x represent the number of children attending.

Let y represent the number of adults attending.

On Saturday night:

The equation for the revenue generated on Saturday night is:

4x + 5y = 540 (since children's tickets cost $4 and adults' tickets cost $5, and the total revenue is $540).

Matinee on the second day:

The equation for the revenue generated at the matinee is:

3x + 5y = 440 (since children's tickets cost $3 and adults' tickets still cost $5, and the total revenue is $440).

Therefore, the system of equations in standard form is:

4x + 5y = 540

3x + 5y = 440

B) Let's rewrite the system of equations in slope-intercept form:

On Saturday night:

4x + 5y = 540

Rearranging the equation, we get:

5y = -4x + 540

Dividing both sides by 5, we get:

y = (-4/5)x + 108

The y-intercept of this equation is 108.

Matinee on the second day:

3x + 5y = 440

Rearranging the equation, we get:

5y = -3x + 440

Dividing both sides by 5, we get:

y = (-3/5)x + 88

The y-intercept of this equation is 88.

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She should use 4 grams of fat, and 2 calories per half cup serving

This is a group of equations which may have none, one or more solutions.

From the question, we have the following system of equations

\(13x+8y=68\)

\(145x+330y=1240\)

Where:

x represents the grams of fats, and y represents the number of calories

Next, we plot the graph of both equations (see attachment)

From the attachment, the solution to the system is:

(x,y) = (4,2)

Hence, she should use 4 grams of fat, and 2 calories per half cup serving

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The number of downloads that were standard version were 560 songs.

The first step is to form a system of equations:

2.9s + 4.5h = 5629 equation 1

s + h = 1450  equation 2

Where:

Multiply equation 2 by 4.5:

4.5s + 4.5h = 6525 equation 3

Subtract equation 3 from equation 1:

896 = 1.6s

Divide both sides of the equation by 1.6:

s = 560

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Answer:

-235+84i

Step-by-step explanation:

I hope this is what you were looking for.

Antonio needs to score at least 111 points in his next game to reach his goal of an average of at least 100 points per game.

What are basic arithmetic operations?

Basic arithmetic operations are fundamental mathematical calculations that involve manipulating numerical values using a set of basic operations.

To find out how many points Antonio needs to score in his next game to reach an average of at least 100 points per game, we can use the following formula:

Total points needed = (number of games + 1) × average score goal - total points scored so far

In this case, Antonio has played 5 games, so the number of games he will have played after his next game is 6. His goal is an average of 100 points per game, so his average score goal is 100. The total points he has scored so far is:

105 + 95 + 97 + 82 + 110 = 489

Substituting these values into the formula, we get:

Total points needed = (6) × (100) - 489

Total points needed = 600 - 489

Total points needed = 111

Hence, Antonio needs to score at least 111 points in his next game to reach his goal of an average of at least 100 points per game.

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Answer:

  f(x) = 14(3/2)^x

Step-by-step explanation:

The function can be written ...

  f(x) = a(b^x)

where a = f(0), and b = f(1)/f(0). The table tells us ...

  a = f(0) = 14

  b = f(1)/f(0) = 21/14 = 3/2

The exponential function can be written ...

  f(x) = 14(3/2)^x

The categorical data, in this case, refers to information that can be sorted into specific categories or groups. Among the options provided, hair color would be considered categorical data.Option c is correct

Hair color falls into distinct categories such as blonde, brown, black, red, or other variations. It is a qualitative characteristic that cannot be measured on a numerical scale.On the other hand, the other options can be classified as numerical data.

The number of siblings (option A) and age (option B) are both quantitative variables that can be represented numerically. They can be counted or measured using specific numerical values. Similarly, height (option D) is a quantitative measurement that can be expressed using numbers or units of length.

In summary, while number of siblings, age, and height are numerical data, hair color is an example of categorical data as it falls into distinct categories or groups.Option c is correct.

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Answer:

(d)

Step-by-step explanation:

pie r^2 is the area

2 pie r is the circumference

Answer:

50?

Step-by-step explanation:

31: 62*100 =

(31*100): 62 =

3100: 62 = 50

Answer:3 f + 36 = 85 I think, hope this helps

Step-by-step explanation:

Answer:

36° , 54° , 90°

Step-by-step explanation:

since the triangle is right then one angle is 90°

the ratio of the smaller angles = 3 : 2 = 3x : 2x ( x is a multiplier )

the sum of the 3 angles in the triangle is 180° , that is

3x + 2x + 90 = 180

5x + 90 = 180 ( subtract 90 from both sides )

5x = 90 ( divide both sides by 5 )

x = 18

Then

3x = 3 × 18 = 54°

2x = 2 × 18 = 36°

the 3 angles measure 36° , 54° , 90°

The  coordinate of the point that lies on the line is (2,0)

Given y/x−2=3/11

In order to find the coordinates of points that lie on the line:

First, find the value of x when y = 0:

y/x−2=3/11, when y = 0:

0/x−2=3/11

3(x-2) = 0

3x -6  = 0

3x = 6

x = 6/3 = 2

Also, find the value of y when x = 0:

y/x−2 = 3/11 when x = 0:

y/0−2 = 3/11

y/-2 = 3/11

11y = -6

y = -6/11

Since the coordinates of the two points are (2,0) and  (0,-6/11 ). Therefore, option a with the coordinate (2,0) is the answer.

Therefore,  the values to be plotted on our graph are (0, 1) and (-0.2, 0). The image of the graph is attached

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The measure and shape of the angle will not change when you translate it, only its location in space.

When you translate an angle, you are moving its position in space without changing its size or shape. The angle will maintain its measure, and the relationship between its sides and vertex will remain the same.

If you translate an angle by a certain distance in a specific direction, the new location of the angle will be a specified distance away from the original location in that direction. However, the angle itself remains the same, and it will not be affected by the translation process.

In other words, the measure and shape of the angle will not change when you translate it, only its location in space.

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The arrow is at a height of 48 ft after approx. 3 - √6 s and after 3 + √6 s.

To find the time it takes for the arrow to reach a height of 48 ft, we can use the formula for the height of the arrow:

s = v0t - 16t^2

Here, s represents the height of the arrow, v0 is the initial velocity, and t is the time.

Given that the initial velocity, v0, is 96 ft/s and the height, s, is 48 ft, we can set up the equation:

48 = 96t - 16t^2

Now, let's solve this equation to find the time it takes for the arrow to reach a height of 48 ft.

Rearranging the equation:

16t^2 - 96t + 48 = 0

Dividing the equation by 16 to simplify:

t^2 - 6*t + 3 = 0

We now have a quadratic equation in the form of at^2 + bt + c = 0, where a = 1, b = -6, and c = 3.

Using the quadratic formula:

t = (-b ± √(b^2 - 4ac)) / (2a)

Plugging in the values:

t = (6 ± √((-6)^2 - 413)) / (2*1)

t = (6 ± √(36 - 12)) / 2

t = (6 ± √24) / 2

Simplifying the square root:

t = (6 ± 2√6) / 2

t = 3 ± √6

Therefore, the arrow reaches a height of 48 ft after approximately 3 + √6 seconds and 3 - √6 seconds.

In summary, the arrow takes approximately 3 + √6 seconds and 3 - √6 seconds to reach a height of 48 ft, assuming an initial velocity of 96 ft/s.

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Note the complete question is

The height of an arrow shot upward can be given by the formula s = v0*t - 16*t², where v0 is the initial velocity and t is time.How long does it take for the arrow to reach a height of 48 ft if it has an initial velocity of 96 ft/s?  

Solve (t-3)^2=6

The arrow is at a height of 48 ft after approx. ___ s and after ___ s.

The equation y = \(-0.024x^2\) + 0.0791x + 4.873 represents a quadratic function with a downward-opening parabol

The given expression is a quadratic equation in the form y = -0.024x^2 + 0.0791x + 4.873. Let's analyze its components and characteristics.

The equation represents a quadratic function, where x is the independent variable and y is the dependent variable. The coefficients in front of each term determine the shape, position, and direction of the graph.

The term with the highest power of x is -0.024x^2, which indicates a downward-opening parabola. The coefficient -0.024 determines the steepness of the curve. A negative coefficient means the parabola is concave down.

The term 0.0791x is the linear term and determines the slope of the line. A positive coefficient indicates an upward or positive slope. It affects the overall direction and position of the graph.

The constant term 4.873 is the y-intercept. It indicates the point at which the graph intersects the y-axis when x = 0.

To analyze the graph of the quadratic equation further, we can calculate the vertex. The x-coordinate of the vertex can be found using the formula x = -b/(2a), where a and b are the coefficients of x^2 and x, respectively. In this case, a = -0.024 and b = 0.0791. Substituting these values into the formula, we have x = -0.0791 / (2 * -0.024) ≈ 1.643. By substituting this x-coordinate into the equation, we can find the y-coordinate of the vertex.

Overall, the equation y =\(-0.024x^2 + 0.0791x\) + 4.873 represents a quadratic function with a downward-opening parabola. The specific properties, such as the vertex and other key points, can be determined by further calculations and analysis of the equation.

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Answer:

Step-by-step explanation:

Please, share any answer choices that were given to you.

                                       -1                 -1                 -1

-1/64 is equivalent to   -------- and ---------- and ---------

                                      4^3              2^6             8^2

Answer:

  $4125

Step-by-step explanation:

Let 'a' represent the additional income Mrs. King must earn. In order for her disposable income to remain the same, we must have ...

  (66000 +a)(1 -0.20) = 66000(1 -0.15)

  66000 +a = 66000(0.85/0.80) = 66000(1 +0.05/0.80)

  a = 66000(5/80) = 4125

Mrs King must earn $4125 more before tax to maintain her after-tax income.

Looks like the integral should be

\(\displaystyle\int\frac{x+2}{\sqrt{x+(2x+3)^2}}\,\mathrm dx\)

Expand the denominator:

\(\displaystyle\int\frac{x+2}{\sqrt{4x^2+13x+9}}\,\mathrm dx\)

Notice that

\(u=4x^2+13x+9\implies \mathrm du=(8x+13)\,\mathrm dx\)

In order to use this substitution, expand the integral as

\(\displaystyle\frac18\int\frac{8x+13}{\sqrt{4x^2+13x+9}}\,\mathrm dx+\frac38\int\frac{\mathrm dx}{\sqrt{4x^2+13x+9}}\)

In the first integral, use the previously mentioned substitution:

\(\displaystyle\frac18\int\frac{8x+13}{\sqrt{4x^2+13x+9}}\,\mathrm dx=\frac18\int\frac{\mathrm du}{\sqrt u}=\frac14\sqrt u\)

\(=\dfrac14\sqrt{4x^2+13x+9}\)

In the second integral, complete the square in the denominator:

\(4x^2+13x+9=4\left(x+\dfrac{13}8\right)^2-\dfrac{25}{16}\)

Now substitute

\(x=\dfrac58\sec t-\dfrac{13}8\implies\mathrm dx=\dfrac58\sec t\tan t\,\mathrm dt\)

so that

\(\sec t=\dfrac{8x+13}5\)

Then

\(4x^2+13x+9=\dfrac{25}{16}(\sec^2t-1)=\dfrac{25}{16}\tan^2t\)

and the integral becomes

\(\displaystyle\frac{\frac{15}{64}}{\frac54}\int\frac{\sec t\tan t}{\sqrt{\tan^2t}}\,\mathrm dt=\frac3{16}\int\sec t\,\mathrm dt\)

\(=\dfrac3{16}\ln|\sec t+\tan t|\)

\(=\dfrac3{16}\ln\left|\dfrac{8x+13}5+\dfrac{\sqrt{(8x+13)^2-25}}5\right|\)

So, the integral is

\(\dfrac14\sqrt{4x^2+13x+9}+\dfrac3{16}\ln\left|\dfrac{8x+13}5+\dfrac{\sqrt{(8x+13)^2-25}}5\right|+C\)

Answer: \(35x < 4-5x+3x^2\) is \(x < \frac{1}{3}\) or \(x > \frac{4}{3}\)

Step-by-step explanation:

A) To generate the table of values, we can substitute different values of x into the equation and solve for y.

x             y

-2        26

-1        12

0         4

1         2

2        10

To plot the graph of the function, we can use the table of values and plot the corresponding points on a graph with a scale of 4 for the y-axis and 2cm to 1 unit along the x-axis. The graph should look like a parabola opening upwards.

c) To solve for the root of the equation 4-5x+3x^2=0, we can use the graph and find the x-coordinate of the point where the graph intersects the x-axis. This point is also known as the x-intercept or the root of the equation. From the graph, we can see that the root is approximately x=0.83.

To solve the equation 3x^2 -5x=2, we can rearrange it to the form 3x^2 -5x-2=0 and factor it as (3x+1)(x-2)=0. This gives us two solutions: x=-1/3 and x=2.

To determine the range of values of x for which 35x​<4-5x+3x^2, we can rearrange the inequality to the form 3x^2-40x+4>0 and factor it as (3x-1)(x-4/3)>0. This means that either (3x-1) and (x-4/3) are both positive or both negative.

When (3x-1)>0 and (x-4/3)>0, we get x>1/3 and x>4/3, which means x>4/3.

When (3x-1)<0 and (x-4/3)<0, we get x<1/3 and x<4/3, which means x<1/3.

Therefore, the range of values of x for which 35x<4-5x+3x^2 is x<1/3 or x>4/3.

Answer:

its D

Step-by-step explanation:

hope this helps

Answer:

2/3 feet.

Step-by-step explanation:

The formula for finding the area of a triangle is  A = (B x H)/2. Or, area is equal to base times height, all divided by two. In your case, we don't know the base. If we substitute in values, we get 2/5 = (B x 6/5)/2.

Next, we have to multiply both sides by two, to cancel the "/2." That results in 4/5 = B x 6/5. Then, to isolate the variable, we have to divide both sides by 6/5. 4/5 ÷ 6/5 is 4/5 x 5/6. The fives cancel out, and we are left with 4/6.

Finally, we simplify! 4/6 simplifies to 2/3.

Answer:

7250%

Step-by-step explanation:

To express 72 1/2 as a percentage, you need to convert it to a decimal first and then multiply by 100.

72 1/2 is equivalent to 72.5 as a decimal.

To convert it to a percentage:

72.5 * 100 = 7250

Therefore, 72 1/2 is equal to 7250% when expressed as a percentage.

let me know it is helpful for you or not

By performing the division, the jogging track is 0.45 miles long.

Division is a mathematical operation that involves the splitting of a quantity into equal parts or groups.

The problem asks us to convert a length of 792 yards to miles. To do this, we need to use a conversion factor to relate yards to miles. We know that there are 1760 yards in a mile, so we can use this relationship to convert the length in yards to miles.

To convert yards to miles, we divide the length in yards by the number of yards in a mile. This is because we want to cancel out the units of yards and be left with the corresponding number of miles.

So, 792 yards / 1760 yards/mile gives us the length of the jogging track in miles. We can simplify this expression by performing the division to get the result of 0.45 miles. Therefore, the jogging track is 0.45 miles long.

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Answer:

x = 4

Step-by-step explanation:

<C = <A

25° = 8x - 7

32 = 8x

x = 4