Enter the number that belongs in the green box

a) The value of f(5) of the given graph is: Positive

b) The point at which f(x) = 0 is at x = 1.

c) The interval for which the function f(x) ≤ 0 is from the interval: -2 ≤ x ≤ 1

The domain of a function is defined as the set of input values that can be plugged into a function. This set is the x values in a function such as f(x). The range of a function is defined as the set of output values that the function assumes. This set is the values that the function brings out after we plug an x value in.

a) The value of f(5) of the given graph is:

f(5) = 1

Thus, f(5) is positive

b) The point at which f(x) = 0 is the x-intercept which in this case it is at x = 1.

c) The interval for which the function f(x) ≤ 0 is from the interval:

-2 ≤ x ≤ 1

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Answer:

Step-by-step explanation:

You want the equation representing the relation between Children's tickets and Senior tickets, and the solution to the problem if ...

The augmented matrix for the problem, and its solution, are shown in the attachment.

<95141404393>

Based on the information, there are 25 cars with good brakes and good tires.

Total cars = 50

Cars with faulty brakes = 10

Cars with faulty tires = 15

Cars with faulty brakes and good tires = 2

Let's calculate the number of cars with good brakes and good tires:

Cars with faulty brakes or faulty tires = Cars with faulty brakes + Cars with faulty tires - Cars with faulty brakes and good tires

= 10 + 15 - 2

= 25

Cars with good brakes and good tires = Total cars - Cars with faulty brakes or faulty tires

= 50 - 25

= 25

Therefore, there are 25 cars with good brakes and good tires.

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The die was rolled 20 times, so the total number of possible outcomes is 20.

The die landed on 5 a total of 4 times, so the probability that a die will land on 5 is 4/20 = 0.2 or 20%.

The die landed on 1 a total of 5 times, so the probability that a die will land on 1 is 5/20 = 0.25 or 25%.

If a probability is unlikely, then the probability is less than 0.5 or 50%. This is because the probability of an event ranges from 0 (impossible) to 1 (certain), so any probability less than 0.5 means that the event is less likely to happen than not.

An order-preserving function is a function that preserves the order of its inputs. In other words, if x is less than y, then f(x) will be less than f(y).

The statement "f is order-preserving if x < y implies f(x) < f(y)" means that if x is less than y, then f(x) must be less than f(y). This is a necessary condition for a function to be order-preserving. However, it is not a sufficient condition. For example, the function f(x) = x^2 is not order-preserving, because 2 < 3, but f(2) = 4 > f(3) = 9.

In summary, order-preserving functions are useful in situations where we need to preserve the order of a set of data.

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There are 32 fishes with length between 30 - 40 cm in length

To obtain the frequency value , multiply the frequency density by the upper limit of the frequency bar.

upper limit = 40

frequency density = 0.8

Fishes between 30 - 40 cm long :

Therefore, there are 32 fishes that are between 30 - 40 cm long .

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The equation of the level curve is [tex]x^2 + y^2 = 9[/tex].

The given function is [tex]f(x,y) = 64 - x ^{2}  - y ^{2}[/tex].

The equation for the level curve that passes through the point [tex](\sqrt2 ; \sqrt7)[/tex] can be obtained by equating [tex]f(x,y)[/tex] with [tex]f(\sqrt2, \sqrt7)[/tex].

Thus we have, [tex]f(x,y) = 64 - x ^{2}  - y ^{2} = 64 - 2 - 7 = 55[/tex]

This is the value of [tex]f(x,y)[/tex] at the point [tex](\sqrt2 ; \sqrt7)[/tex].

Now let us substitute [tex]f(x,y) = 55[/tex] and solve for [tex]y[/tex]. We get,[tex]64 - x ^{2}  - y ^{2} = 55 \\ \Rightarrow y^2 = 9 - x^2[/tex]

Now, substitute the value of [tex]y^2[/tex] in the point [tex](\sqrt2 ; \sqrt7)[/tex] to get the value of [tex]x[/tex]. Thus,[tex]7 = 9 - x ^{2} + 2 \\ \Rightarrow x ^{2} = 4[/tex]

Therefore, the point [tex](\sqrt2 ; \sqrt7)[/tex] lies on the curve [tex]y^2 = 9 - x^2[/tex] or [tex]x^2 + y^2 = 9[/tex].

Hence, the equation of the level curve is [tex]x^2 + y^2 = 9[/tex].

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The masses between the Earth and the Moon found using the Newton's law of universal gravitation, whereby the gravitational force between them is 1.682 × 10²⁶ N, and the mass of the Moon is about 7.1 × 10²² kg, is about 411,144.67 meters'

The Newton's law of universal gravitation indicates that formula for finding the gravitational force F between two objects, can be expressed as follows;

F = G·(m₁·m₂)/d²

Where;

F = The gravitational force

G = The gravitational constant

m₁ and m₂ are the mass of the two objects

d = The distance between them

Where

F = 1.682 × 10²⁶ N·m²·kg⁻²

G = 6.67430 × 10⁻¹¹ N·m²·kg⁻²

m₁ = 6 × 10²⁴ kg

m₂ = 7.1 × 10²² kg

Plugging in the above values into the Newton's law of universal gravitation, we get;

1.682 × 10²⁶ = 6.67430 × 10⁻¹¹ × (6 × 10²⁴ × 7.1 × 10²²)/d²

d² = 6.67430 × 10⁻¹¹ × (6 × 10²⁴ × 7.1 × 10²²)/1.682 × 10²⁶ ≈ 169039940547

d ≈ √(169039940547) ≈ 411144.67

The distance between the Earth and the Moon, found using the specified masses is about 411144.67 meters.

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The median is a measure of central tendency in a data set. It represents the middle value when the data is arranged in ascending or descending order.

For an odd number of data points, the median is simply the value that falls exactly in the middle of the ordered list. For example, if we have the data set [1, 3, 5, 7, 9], the median is 5 because it is the value at the center.

However, when the number of data points is even, there is no single middle value. In this case, the median is calculated by taking the average of the two middle values. For example, if we have the data set [2, 4, 6, 8], the two middle values are 4 and 6. The median is then (4 + 6) / 2 = 5.

So, in summary:

For an odd number of data points, the median is the value in the middle of the ordered list.

For an even number of data points, the median is the average of the two middle values.

The new vertices after multiplying by 3 are:

A'(0, 0), B'(-12, 0), and C'(-6, 12).

To find the new coordinates of each vertex after multiplying by 3, we simply multiply the x-coordinate and y-coordinate of each vertex by 3.

Given the original vertices:

A(0, 0), B(-4, 0), and C(-2, 4)

After multiplying each coordinate by 3, we get:

[tex]A'(0 \times 3, 0 \times 3) = A'(0, 0)\\B'(-4 \times 3, 0 \times 3) = B'(-12, 0)\\C'(-2 \times 3, 4 \times 3) = C'(-6, 12)[/tex]

This means that each vertex has been scaled by a factor of 3. The x-coordinates are multiplied by 3, resulting in a horizontal stretching or compression, while the y-coordinates are also multiplied by 3, resulting in a vertical stretching or compression.

In this case, the points have been scaled up by a factor of 3, making the triangle larger.

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Answer:

385 envelopes will be needed.

Step-by-step explanation:

y = the total number of envelopes needed.

x =  the number of months

y = 5x + 325

y = 5(12) + 325

y = 60 + 325

y = 385

Helping in the name of Jesus.

Answer:

The correct answer is B.

When x increases, y increases, and when x decreases, y decreases.

answer/explanation:

hope this helps

Here is your answer.

A tank contains 5832 litres of water.

Total amount of water in tank = 5832 litres.

Each day one-third of the water in the tank is removed and not replaced.

Since one third of the water is removed for six days.

1st day = 1/3rd of 5832 litres is removed

[tex] = \dfrac{5832}{3} = 1944[/tex]

Amount of water left = 5832 - 1944 = 3888 litres

2nd day = 1/3rd of 3888 litres is removed

[tex] = \dfrac{3888}{3} = 1296 [/tex]

Amount of water left = 3888 - 1296 = 2592 litres

3rd day = 1/3rd of 2592 litres is removed

[tex] = \dfrac{2592}{3} = 864 [/tex]

Amount of water left = 2592 - 864 = 1728 litres

4th day = 1/3rd of 1728 litres is removed

[tex] = \dfrac{1728}{3} = 576 [/tex]

Amount of water left = 1728 - 576 = 1152 litres

5th day = 1/3rrd of 1152 litres is removed

[tex] = \dfrac{1152}{3} =384 [/tex]

Amount of water left = 1152 - 384 = 768 litres

6th day = 1/3rrd of 768 litres is removed

[tex] = \dfrac{768}{3} = 256 [/tex]

Amount of water left = 1768 - 256 = 512 litres

So, 512 litres water remains in the tank at the end of 6 days

The left-hand side of the equation is equal to the right-hand side, confirming that x = -1 is the correct solution.by using algebric operation.

To solve the equation -3x - 1/4 = 1/4x + 3, we'll follow the steps below:

1. Combine like terms:

  -3x - 1/4 = 1/4x + 3

2. Get rid of fractions by multiplying the entire equation by the least common denominator (LCD), which is 4:

  4(-3x - 1/4) = 4(1/4x + 3)

  Simplifying:

  -12x - 1 = x + 12

3. Move all the terms with x to one side and the constant terms to the other side:

  -12x - x = 12 + 1

  Simplifying:

  -13x = 13

4. Divide both sides by -13 to isolate x:

  x = 13 / -13

  Simplifying:

  x = -1

Therefore, the solution to the equation is x = -1.

In the given equation, x = -1 satisfies the equation and makes both sides equal. By substituting x = -1 back into the equation, we can verify the solution:

-3(-1) - 1/4 = 1/4(-1) + 3

3 - 1/4 = -1/4 + 3

12/4 - 1/4 = -1/4 + 12/4

11/4 = 11/4

The left-hand side of the equation is equal to the right-hand side, confirming that x = -1 is the correct solution.

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The relative frequency of the pitcher throwing exactly 4 pitches in an at-bat is given as follows:

3/20.

A relative frequency is calculated as the division of the number of desired outcomes by the number of total outcomes.

The total number of at bats in this problem is given as follows:

80.

In 12 of them, the pitcher threw exactly four pitches, hence the relative frequency of the pitcher throwing exactly 4 pitches in an at-bat is given as follows:

12/80 = 3/20.

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The solution to the system of equation is dependent and x = 20, y = -15 and z = -3

x+y+ z= 2

3x+4y- z=3

3x + 4y -2z = 6

Multiply (1) by -1 and subtract (1) from (2) to eliminate z

-x - y - z = -2

3x+4y- z=3

4x + 5y = 5

Then multiply (2) by 2 and subtract (3) from it to eliminate z

6x + 8y - 2z = 6

3x + 4y -2z = 6

3x + 4y = 0

4x + 5y = 5

3x + 4y = 0

Multiply 4x + 5y = 5 and 3x + 4y = 0 by 3 and 4 respectively

12x + 15y = 15

12x + 16y = 0

Subtract

16y - 15y = 0 - 15

y = -15

Substitute y = 15 into

12x + 16y = 0

12x + 16(-15) = 0

12x - 240 = 0

12x = 240

divide both sides by 12

x = 20

From (1)

x+y+ z= 2

20 + (-15) + z = 2

20 - 15 + z = 2

5 + z = 2

z = 2 - 5

z = -3

Therefore, x = 20, y = -15 and z = -3.

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The scale factor from shape A to shape B is determined as 0.75.

The value of the missing length x in shape B is 5.25 cm.

The size by which the shape is enlarged or reduced is called as its scale factor.

Scale factor from shape A to shape B is calculated as;

Scale factor =   3/4 = 9/12 = 0.75

The value of the missing length x in shape B is calculated by applying the principle of similar triangles as follows;

the given lengths of shape A and shape B;

A = 4 cm and 7 cm

B = 3 cm and x cm

The value of the missing length x in shape B is calculated as;

x/3 = 7/4

x = (3 x 7/4)

x = 5.25 cm

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Answer:

To determine the formula of g in terms of f, we would need more information about the specific transformation or relationship between f and g. Unfortunately, you mentioned that there is a graph showing f as a solid blue line and g as a dotted red line, but without additional details about the graph or the transformation, it is not possible to provide the formula of g in terms of f.

If you can provide further information about the relationship between f and g or any specific points or characteristics of the graph, I would be happy to assist you in finding the formula of g in terms of f.

Step-by-step explanation:

Answer:

To find the solutions for (A) AnB, (B) Buc, and (C) AnBnc, we need to first understand the notations used.

AnB represents the intersection of sets A and B, which is the set of elements that are present in both A and B.

Buc represents the union of sets B and C, which is the set of elements that are present in either B or C or both.

AnBnc represents the intersection of sets A and B complement, also known as the relative complement of B in U. This is the set of elements that are present in A but not in B.

Now, let's find the solutions for each of the given problems:

(A) AnB:

A = {1, 2, 3, 7, 8}

B = {3, 4, 5, 7, 9}

AnB = {3, 7}

Therefore, the solution for (A) AnB is {3, 7}.

(B) Buc:

B = {3, 4, 5, 7, 9}

C = {1, 3, 5, 6, 10}

Buc = {1, 3, 4, 5, 6, 7, 9, 10}

Therefore, the solution for (B) Buc is {1, 3, 4, 5, 6, 7, 9, 10}.

(C) AnBnc:

A = {1, 2, 3, 7, 8}

B = {3, 4, 5, 7, 9}

B complement = {1, 2, 6, 8, 10}

AnBnc = {1, 2, 6, 8}

Therefore, the solution for (C) AnBnc is {1, 2, 6, 8}.

Answer:

In the situation described where we have a part-to-whole ratio of 2:5, it means that for every 2 parts of something, there are 5 parts in total. In other words, if we were to divide the whole into equal parts, the specific part we are interested in would make up 2 out of every 5 parts. This ratio provides a relative comparison between the specific part and the entire whole, indicating that the part occupies a smaller portion compared to the whole.

Step-by-step explanation:

Let's assume the original mass of the saline water (salt water) is x grams. According to the given information, the mass ratio of salt to water is initially 1:10.

This means the mass of salt in the saline water is (1/11) * x grams, and the mass of water is (10/11) * x grams.

After adding 22 grams of salt, the mass ratio of salt to water becomes 2:9.

This means the mass of salt is now (2/11) * (x + 22) grams, and the mass of water is (9/11) * (x + 22) grams.

Since the mass of salt remains the same in both scenarios, we can set up an equation:

(1/11) * x = (2/11) * (x + 22)

Simplifying the equation:

x = 2 * (x + 22)

x = 2x + 44

x - 2x = 44

-x = 44

x = -44 (disregard this negative value)

Since the value of x is negative, it is not a valid solution in this context.

It seems there may be an error or inconsistency in the given information or calculations. Please double-check the values and ratios provided to ensure accuracy.

The correct comparison between the two functions is that option A. They have the same x intercept and same end behavior.

The end behavior of a function is the statement which describes the behavior of the graph of the function towards the end of the X axis.

x-intercept of a function is the x coordinate of the point that the graph touches with the X axis.

y coordinate will be 0 there.

f touches the X axis at (1, 0).

x intercept of f = 1

g passes through the point (1, 0).

So x intercept of g = 1

Both have the same x intercept.

y intercept is the y coordinate where the graph touches the Y axis.

y intercept of f = -3

y intercept of g = -4

y intercept is not the same.

The graph of f is approaching to a constant value as it moves along the X axis.

Similarly, for g, the difference between the values of the function at consecutive points is decreasing and it tends to 0.

So the end behavior is same. Hence the correct option is A.

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Answer:

Approximately 116.465 seconds after the chute opens.

Step-by-step explanation:

To find the number of seconds after the chute opens when the skydiver will first be less than 100 feet in altitude, we need to set up and solve the equation H(x) < 100.

Given that H(x) = 3000 - 24.9x represents the altitude of the skydiver at time x seconds after the chute opens, we can rewrite the equation as follows:

3000 - 24.9x < 100

Now, we can solve this inequality to find the value of x:

3000 - 24.9x < 100

-24.9x < 100 - 3000

-24.9x < -2900

To isolate x, we divide both sides of the inequality by -24.9. However, we need to flip the inequality sign since we are dividing by a negative number:

x > (-2900) / (-24.9)

x > 116.465

Therefore, the skydiver will first be less than 100 feet in altitude after approximately 116.465 seconds after the chute opens.

The limit of the function as x approaches -2 is 20.To find the limit of the function (if it exists), we can substitute the given value into the function and simplify. Given function: f(x) = 4x² + 5x - 6x + 2

To find the limit as x approaches -2, we substitute -2 into the function:

f(-2) = 4(-2)² + 5(-2) - 6(-2) + 2
      = 4(4) - 10 + 12 + 2
      = 16 - 10 + 12 + 2
      = 20

Therefore, the limit of the function as x approaches -2 is 20.

To write a simpler function that agrees with the given function at all but one point, we can use a graphing utility. By plotting the given function and observing its behavior, we can create a simpler function that matches the original function except at one point.

However, without further information about the specific behavior of the given function, it is not possible to provide a more detailed explanation or a graph of the simpler function.

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To find the exact value of (7π/6) using the unit circle, locate the angle (π/6) on the unit circle in the second quadrant, determine the coordinates, and adjust the y-coordinate to be negative. The final answer is (√3/2, -1/2). This answer is obtained by understanding the reference angle and using the coordinates of the point where the angle intersects the unit circle.

To find the exact value of (7π/6) using the unit circle, follow these steps:

1. Start by understanding the reference angle. The reference angle for (7π/6) is (π/6).

2. Locate the angle (π/6) on the unit circle. This angle lies in the second quadrant.

3. Determine the coordinates of the point where the angle intersects the unit circle. For (π/6), the x-coordinate is √3/2 and the y-coordinate is 1/2.

4. Since (7π/6) lies in the second quadrant, the x-coordinate will remain the same, but the y-coordinate will be negative. Therefore, the coordinates for (7π/6) are (√3/2, -1/2).

5. The exact value of (7π/6) is (√3/2, -1/2).

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The coordinates of polygon A'B'C'D' after the rotation of 180º are given as follows:

A'(0,0), B'(-5,-2), C'(-5,5), D'(0,3).

The five more known rotation rules are given as follows:

The original coordinates for this problem are given as follows:

A(0,0), B(5,2), C(5,-5), D(0,-3).

Exchanging the sign of each of the coordinates, the coordinates after the rotation are given as follows:

A'(0,0), B'(-5,-2), C'(-5,5), D'(0,3).

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Answer:

Step-by-step explanation:

You want to show that the point (4, -4) is on the graph of x² +y² = 32, and you want the tangent and normal lines at that point.

The point is on the curve because when x = 4 and y = -4 the resulting statement is 16 +16 = 32, which is a true statement.

The tangent to a circle is most easily found by first considering the normal. The normal line will be the line through the point on the circle and the center of the circle. Here, the center is (0, 0). The slope of the normal line is ...

  m = y/x = -4/4 = -1

The tangent line's slope is the opposite reciprocal of this:

  m = -1/(-1) = 1

The point-slope form of the equation for the tangent line is ...

  y -k = m(x -h) . . . . . equation for line with slope m through point (h, k)

  y -(-4) = 1(x -4) . . . . . equation for line with slope 1 through point (4, -4)

  x -y = 8 . . . . . . . . . . add 4-y to put in standard form

The equation of the tangent line is x - y = 8.

In the section above, we found the slope of the normal line is -1. We also noted that the normal line goes through the point (0, 0). Then the point-slope form of the normal line is ...

  y -0 = -1(x -0)

  y = -x

The equation of the normal line is y = -x.

__

Additional comment

The normal line can also be written as x + y = 0.

The tangent line can also be written as y = x -8.

<95141404393>

a. The object is momentarily at rest at t = 6.

b. It moves up in the interval (0, 6) and moves down in the interval (6, 12).

c. The object changes direction at t = 6.

d. It speeds up in the interval (0, 6) and slows down in the interval (6, 12).

e. The object does not have a maximum speed within the given interval, and it is moving slowest at t = 0 and t = 12.

f. The object is farthest from the axis origin at t = 6.

To solve the given problem, we start by finding the velocity and acceleration functions based on the position function provided:

Position function: [tex]s = f(t) = 192t - 16t^2[/tex]

a. Velocity function: v(t) = ds/dt = f'(t)

Taking the derivative of the position function with respect to time, we have:

[tex]v(t) = d(192t - 16t^2)/dt = 192 - 32t[/tex]

b. Acceleration function: [tex]a(t) = d^2s/dt^2 = f''(t)[/tex]

Taking the second derivative of the position function with respect to time, we have:

a(t) = d(192 - 32t)/dt = -32

Now, let's answer the given questions:

a. The object is momentarily at rest when its velocity is equal to zero.

Setting v(t) = 0 and solving for t:

192 - 32t = 0

32t = 192

t = 6

b. The object moves up when the velocity is positive (v(t) > 0) and moves down when the velocity is negative (v(t) < 0).

Interval when the object moves up: (0, 6)

Interval when the object moves down: (6, 12)

c. The object changes direction when the velocity changes sign. In this case, it changes direction at t = 6.

d. The object speeds up when the acceleration and velocity have the same sign, and it slows down when the acceleration and velocity have opposite signs.

The object speeds up in the interval (0, 6) and slows down in the interval (6, 12).

e. The object is moving fastest when its velocity is maximum. The velocity function is v(t) = 192 - 32t, and it reaches its maximum when its derivative is equal to zero.

Setting v'(t) = 0 and solving for t:

v'(t) = -32 = 0

There is no solution for t since -32 is a constant.

The object is moving slowest when its velocity is minimum, which occurs at the endpoints of the given interval.

So, the object is moving slowest at t = 0 and t = 12.

f. To find when the object is farthest from the axis origin, we look for the maximum value of the position function.

The position function is [tex]s(t) = 192t - 16t^2,[/tex] and it reaches its maximum value when its derivative is equal to zero.

Setting s'(t) = 0 and solving for t:

s'(t) = 192 - 32t = 0

32t = 192

t = 6

So, the object is farthest from the axis origin at t = 6.

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From the graph the correct option is: The graph does not represent a one-to-one function because the y-values between 0 and 2 are paired with multiple x-values.

A one-to-one function is a function where each element in the domain is paired with a unique element in the range. In other words, no two different x-values can have the same y-value.

Looking at the graph, we can see that between the y-values of 0 and 2, there are multiple x-values. For example, when y = 1, there are two corresponding x-values: x = 3 and x = 4. This violates the condition of a one-to-one function.

The graph does not represent a one-to-one function because there are multiple x-values associated with certain y-values, specifically between 0 and 2. It is important to note that a one-to-one function requires each x-value to have only one corresponding y-value, and vice versa, which is not the case in this graph.

The correct option is: The graph does not represent a one-to-one function because the y-values between 0 and 2 are paired with multiple x-values.

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