Select the graph that correctly represents ƒ(x) = 1∕2(x – 3)^2 – 6.

Answer:

(0.3)^x has the slowest growth rate.

Step-by-step explanation:

Out of the given functions, the function with the slowest growth rate is (0.3)^x.

To see why, we can compare the growth rates of each function as x increases.

For 5(e)^x, the base e is approximately equal to 2.718, which means that the function grows very quickly as x increases.

For 6(1.03)^x, the base is slightly greater than 1, which means that the function grows at a moderate rate as x increases.

For 2(3)^x, the base is greater than 1, which means that the function grows more quickly than (1.03)^x, but less quickly than (e)^x.

For (0.3)^x, the base is less than 1, which means that the function decreases as x increases. Therefore, this function has the slowest growth rate out of the given functions.

So, (0.3)^x has the slowest growth rate.

Answer:

Step-by-step explanation:

A hands-on experience is the only modality that would give visitors a complete understanding of the plight of modern enslaved people.

According to the given information, it takes 0.5 hours to travel 16.5 highway miles and 7 city miles and 1.6 hours to travel 44 highway miles and 28 city miles.

What is the average rate?

The average rate is a measure of the average speed at which something occurs over a period of time. It is calculated by dividing the total amount of something by the total time it took to occur.

The formula for the average rate or average speed is:

Average speed = Total distance traveled / Total time taken

a. To find the time it takes to travel 16.5 highway miles and 7 city miles, we can use the formula:

time = distance/speed

The time it takes to travel 16.5 highway miles is:

time on highway = 16.5 / 55 = 0.3 hours

The time it takes to travel 7 city miles is:

time in city = 7 / 35 = 0.2 hours

The total time it takes to travel both distances is:

total time = time on highway + time in city = 0.3 + 0.2 = 0.5 hours

Therefore, it takes 0.5 hours (or 30 minutes) to travel 16.5 highway miles and 7 city miles.

b. To find the time it takes to travel 44 highway miles and 28 city miles, we can use the same formula:

time on highway = 44 / 55 = 0.8 hours

time in city = 28 / 35 = 0.8 hours

total time = time on highway + time in city = 0.8 + 0.8 = 1.6 hours

Therefore, it takes 1.6 hours (or 96 minutes) to travel 44 highway miles and 28 city miles.

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Approximately 342,857 empty cans would weigh 200 pounds.

A ratio is an ordered pair of numbers a and b, written a / b where b does not equal 0. A proportion is an equation in which two ratios are set equal to each other. For example, if there is 1 boy and 3 girls you could write the ratio as 1 : 3 (for every one boy there are 3 girls).

To solve this problem, we need to use proportional reasoning. We know that six empty cans weigh 3.5 ounces, so we can set up a proportion to find out how many cans are needed to weigh 200 pounds:

6 cans : 3.5 ounces = x cans : 200 pounds

To solve for x, we need to cross-multiply and simplify:

6 cans * 200 pounds = 3.5 ounces * x cans

1200 pounds = 3.5 ounces * x cans

x cans = 1200 pounds / 3.5 ounces

x cans = 342857.14 cans (rounded to two decimal places)

Therefore, approximately 342,857 empty cans would weigh 200 pounds.

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Therefore, we have proven that cos (0) - sin (0) = √2 sin (0), given that sin (0) + cos (0) = √2 cos (0).

Trigonometry is a branch of mathematics that deals with the study of the relationships between the angles and sides of triangles. It explores the properties and functions of angles, as well as the relationships between them in various geometrical shapes.

Given by the question.

We have:

sin (0) + cos (0) = √2 cos (0)

Simplifying the left-hand side:

sin (0) + cos (0) = 1

Substituting back into the original equation:

1 = √2 cos (0)

Dividing both sides by √2:

1/√2 = cos (0)

Now we need to prove that:

cos (0) - sin (0) = √2 sin (0)

Substituting our value for cos (0):

1/√2 - sin (0) = √2 sin (0)

Multiplying both sides by √2:

√2/2 - √2 sin (0) = 2 sin (0)

Adding √2 sin (0) to both sides:

√2/2 = 3 sin (0)

Dividing both sides by 3:

sin (0) = √2/6

Substituting back into the equation we were trying to prove:

cos (0) - sin (0) = cos (0) - √2/6

We can now use our earlier equation to substitute for cos (0):

cos (0) - √2/6 = √2/2 - √2/6

Simplifying:

cos (0) - sin (0) = √2 sin (0)

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By the given definition of matrix A, the value of the matrix A⁻¹ is [tex]\left[\begin{array}{ccc}-1&0&1&2&-1&-2&0&0&-1\end{array}\right][/tex] and it is true that AA⁻¹ = I₃.

Given that

[tex]A = \left[\begin{array}{ccc}1&0&2\\1&1&0\\1&0&1\end{array}\right][/tex]

To find the inverse of A, we can use the following formula:

A⁻¹ = (1/|A|) adj(A)

where |A| is the determinant of A and adj(A) is the adjugate matrix of A.

First, let's calculate the determinant of A:

|A| = 1(1×1 - 0×0) - 0(1×0 - 0×1) + 2(1×0 - 1×1)

|A| = -1

Next, we need to find the adjugate matrix of A.

To do this, we need to find the matrix of cofactors of A, then take its transpose:

[tex]C = \left[\begin{array}{ccc}1&-2&0&0&1&0&-1&2&1\end{array}\right]\\[/tex]

adj(A) = [tex]C^T = \left[\begin{array}{ccc}1&0&-1&-2&1&2&0&0&1\end{array}\right][/tex]

Now we can calculate A⁻¹ using the formula:

A⁻¹ = (1/|A|) adj(A)

So, we have

[tex]A^{-1} = (-1) \left[\begin{array}{ccc}1&0&-1&-2&1&2&0&0&1\end{array}\right] = \left[\begin{array}{ccc}-1&0&1&2&-1&-2&0&0&-1\end{array}\right][/tex]

Finally, we can check whether AA⁻¹ = I₃ using

[tex]A^{-1}A = \left[\begin{array}{ccc}-1&0&1&2&-1&-2&0&0&-1\end{array}\right] \left[\begin{array}{ccc}1&0&2&1&1&0&1&0&1\end{array}\right] = \left[\begin{array}{ccc}1&0&0&0&1&0&0&0&1\end{array}\right] = I_3[/tex]

Therefore, we have found the inverse of A and verified that AA⁻¹ = I₃.

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Therefore, the values of a and b are:

[tex]a = 2[/tex]

[tex]b = 9[/tex]

[tex]So, the functionf(x)=2x+9.[/tex]

In programming and mathematics, a function is a set of instructions or a block of code that performs a specific task or calculation. A function takes in one or more input values, performs a series of operations on them, and then returns a single output value. Functions are used to modularize code, making it easier to read, understand, and maintain. They can be defined and called within a program to perform repetitive tasks, manipulate data, or solve complex problems. Functions can also be used to create reusable code that can be called from multiple parts of a program or shared between different programs.

To find the values of a and b, we need to use the given information to create a system of two equations.

Let's start by using the first two rows of the table:

[tex]f(0) = a(0) + b = 7[/tex]

[tex]f(1) = a(1) + b = 9[/tex]

Simplifying these equations, we get:

[tex]b = 7\\a + b = 9[/tex]

Substituting b = 7 into the second equation, we get:

[tex]a + 7 = 9[/tex]

Solving for a, we get:

[tex]a = 2[/tex]

Now we can use this value of a to find b. Let's use the third and fourth rows of the table:

[tex]f(2) = a(2) + b = 11\\f(3) = a(3) + b = 13[/tex]

Substituting a = 2 into these equations, we get:

[tex]2 + b = 11\\3 + b = 13[/tex]

Solving for b, we get:

[tex]b = 9[/tex]

Therefore, the values of a and b are:

[tex]a = 2\\b = 9[/tex]

[tex]So, the functionf(x)=2x+9.[/tex]

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Answer:

18

Step-by-step explanation:

To determine the number of days until they both work out at the gym on the same day again, Find the lowest common multiples of 6 days and 9 days.

Ahmed (6-days)=6,12,18,24

Brother(9-days)=9,18,24

Answer:

x = 16

Step-by-step explanation:

∠1 + ∠2 = 90

42 + 3x = 90

3x = 90 - 42

x = 48/3 = 16

Answer: 204 ft

Step-by-step explanation: First lets find the area of the triangle, the formula for area of a triangle is base times height divided by 2. Now the height isn't given but we can find it by subtracting 18 by 10 which gives us 8. 8 times 6 is 48 divided by 2 is 24. Therefore, the area of the triangle is 24 ft. Now we find the area of the rectangle which is length times width. 18 times 10 is 180. Now we add the area of both shapes, 180 plus 24 is 204, so your answer is 204 ft.

Answer:

Step-by-step explanation:

Let the number of 50 cents coins in the container be "x".

We know that the total number of coins in the container is 600, so we can write:

[tex]x + 220 = 600[/tex]

Solving for x, we get:

[tex]x = 600 - 220[/tex]

[tex]x = 380[/tex]

Therefore, there are 380 50 cents pieces in the container.

Hope you can understand :)

The single translation that maps shape A onto shape C is given as follows:

A translation happens when either a figure or a function are moved horizontally or vertically on the coordinate plane.

The four translation rules for coordinates are defined as follows:

Translations can often be composed, that is, a combination of multiple translations can result in a single translation, as is the case for this problem.

Considering the translation of 3 units left(shape B) and then 4 units right(shape C), the horizontal translation is given as follows:

-3 + 4 = 1 -> 1 unit right.

Considering the translation of 7 units down(shape B) and then 3 units down(shape C), the vertical translation is given as follows:

-7 - 3 = 10 units down.

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Answer:

B

Step-by-step explanation:

calculate the slopes m of the 2 lines using the slope formula

m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]

with (x₁, y₁ ) = (- 6, 5 ) and (x₂, y₂ ) = (- 2, 7 )

m = [tex]\frac{7-5}{-2-(-6)}[/tex] = [tex]\frac{2}{-2+6}[/tex] = [tex]\frac{2}{4}[/tex] = [tex]\frac{1}{2}[/tex]

repeat with

(x₁, y₁ ) = (4, 2 ) and (x₂, y₂ ) = (6, 6 )

m = [tex]\frac{6-2}{6-4}[/tex] = [tex]\frac{4}{2}[/tex] = 2

• Parallel lines have equal slopes

[tex]\frac{1}{2}[/tex] ≠ 2 , then lines are not parallel

the product of the slopes of perpendicular lines equals - 1

[tex]\frac{1}{2}[/tex] × 2 = 1 ≠ - 1 , thus lines are not perpendicular

the lines are neither perpendicular nor parallel

Using Law of sines, the missing angles are:

4) ∠B = 45.74°

6) ∠C = 21.02°

8) ∠A = 60.84°

The law of sines is a rule for solving the lengths and angles of triangles and it states that:

a/sin A = b/sin B = c/sin C

4) Using law of sines, we have:

8/sin B = 11/ sin 80

(8/11) sin 80 = sin B

sin B = 0.7162

B = sin⁻¹0.7162

∠B = 45.74°

6) Using law of sines, we have:

13/sin C = 32/sin 62

sin C = (13/32) sin 62

sin C = 0.3587

C = sin⁻¹0.3587

C = 21.02°

8) Using law of sines, we have:

30/sin 104 = 27/sin A

sin A = (27/30) sin 104

sin A = 0.8733

A = sin⁻¹0.8733

A = 60.84°

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Answer:

  (a)  5.62 m/s²

  (b)  22.5 m/s

Step-by-step explanation:

You want the magnitude of the acceleration of a crate on a frictionless plane inclined at 35° from the horizontal. And you want the speed after 4 seconds.

The acceleration will be the component of the acceleration due to gravity that is in the direction down the plane.

  (9.8 m/s²)·sin(35°) ≈ 5.62 m/s²

The speed of the crate will be the product of the acceleration and time:

  v = at = (5.6 m/s²)(4 s) = 22.5 m/s

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Answer:

x = - 2 , 5

Step-by-step explanation:

to find the roots let y = 0 , that is

x² - 3x - 10 = 0 ← in standard form

(x + 2)(x - 5) = 0 ← in factored form

equate each factor to zero and solve for x

x + 2 = 0 ⇒ x = - 2

x - 5 = 0 ⇒ x = 5

The value of x is equal to: B. 12.

The measure of <BCD, in degrees is 143 degrees.

In Geometry, the vertical angles theorem is also referred to as vertically opposite angles theorem and it states that two (2) opposite vertical angles that are formed whenever two (2) lines intersect each other are always congruent, which simply means being equal to each other.

By applying the vertical angles theorem to the geometric figure, we have the following:

37 = 3x + 1

3x = 37 - 1

3x = 36

x = 36/3

x = 12.

For the measure of ∠BCD, in degrees, we have:

∠BCD + (3x + 1) = 180

∠BCD + (3(12) + 1) = 180

∠BCD + 37 = 180

∠BCD = 143°.

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Perimeter of the given polygon figure will be 38 units.

What exactly is a polygon in mathematics?

A closed, two-dimensional, flat, closed polygon is a shape that is constrained by geometry and has straight sides. Its sides don't curve inward at all. Another term for a polygon's sides is its polygonal edges. The points where two sides converge are known as a polygon's vertices (or corners).

Perimeter of a polygon = Measure of the circumference of the polygon

From the figure attached,

Perimeter of the figure = measure of the three straight lines + measure of the circumference of the quarter circle

Measure of linear sides = 5 + 10 + 10 + 5 = 30 units

Circumference of the quarter circle = 1/4(2πr)

                                                          =  1/2(π)(5)

                                                          = 7.85 units

Perimeter of the figure = 30 + 7.85

                                     = 37.85

                                     ≈ 38 units

  Therefore, perimeter of the given figure will be 38 units.

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Answer:

-106f+56

Step-by-step explanation:

7(-8f-2)= -56f-14

10(-5f+7)= -50f+70

-56f-14. +. 50f+70

=-106f+56

Answer:

-2(53f-28)

Step-by-step explanation:

7(−8f−2)+10(−5f+7)

7x-8f=-56f  7x-2=-14

10x-5f=-50f   10x7=70

-56f-14+-50f+70       Adding a negative to something means to subtract. Like right here at -14 + -50 is the same as -14 - 50

   2. Add the numbers

-56f-14+-50f+70   -14+70=56

-56f+56-50f

   3. Combine like terms

-56f+56-50f   -56+ -50= -106

-106f +56

   4. Common Factor

-106f + 56

-2(53f-28)

Answer:

Step-by-step explanation:

Logically, if a polynomial of real coefficients has a complex root, it’s conjugate is a root as well. That means if -20-7i is a root, -20+7i is also a root. Also, since we’re dealing with minimal degree, we wish to construct a quadratic expression with such roots. To do this, we can use Vieta’s formulas, which relates the trinomial coefficients with its roots. The two roots added together is -b/a in ax^2 + bx + c, and the two roots multiplied together is c/a. The two roots added together is -40. The two roots multiplied together is 449. So, our quadratic expression is x^2+40x+449

Check the picture below.

Answer:

Step-by-step explanation:

To find P(x<4), we need to use the cumulative distribution function (CDF) of the binomial distribution.

For N = 6 and p = 0.3, the probability mass function (PMF) is:

P(X = k) = (6 choose k) * 0.3^k * 0.7^(6-k)

where (6 choose k) is the binomial coefficient.

Using this formula, we can find the probabilities for each value of X less than 4:

P(X = 0) = (6 choose 0) * 0.3^0 * 0.7^6 ≈ 0.1176

P(X = 1) = (6 choose 1) * 0.3^1 * 0.7^5 ≈ 0.3025

P(X = 2) = (6 choose 2) * 0.3^2 * 0.7^4 ≈ 0.3241

P(X = 3) = (6 choose 3) * 0.3^3 * 0.7^3 ≈ 0.1852

Therefore, P(x<4) is the sum of these probabilities:

P(x<4) = P(X=0) + P(X=1) + P(X=2) + P(X=3) ≈ 0.9294

So the probability of getting less than 4 successes in 6 trials with a success probability of 0.3 is approximately 0.9294.

Answer:

it cant be if the term 9p and 9c is together but if it 9p and 9c both are in product form then your answer is -p+c

Answer:

(a) To find the probability that a randomly selected individual is at least 55 years of age, we need to add up the values in the last row of the contingency table corresponding to the age group 55+ and divide it by the total number of respondents:

P(age 55+) = (515 + 617 + 567 + 560) / 2259 ≈ 0.2934

So the probability is approximately 0.2934.

(b) To find the probability that a randomly selected individual is more likely to buy a product emphasized as "Made in our country," given the individual is at least 55 years of age, we need to add up the values in the second row of the contingency table (corresponding to "Yes, more likely") for the age group 55+ and divide it by the total number of respondents in that age group:

P(more likely|age 55+) = 515 / (515 + 617 + 567 + 560) ≈ 0.2281

So the probability is approximately 0.2281.

(c) To determine whether 18-34-year-olds are more likely to buy a product emphasized as "Made in our country" than individuals in general, we need to compare the proportion of respondents who answered "Yes, more likely" in the 18-34 age group to the proportion in the entire sample. We can calculate these proportions by adding up the values in the second row of the contingency table for the 18-34 age group and for the entire sample, respectively, and dividing each by the total number of respondents in each group:

P(more likely|age 18-34) = 206 / (206 + 388 + 393 + 410) ≈ 0.1913

P(more likely|entire sample) = 1397 / (1397 + 69 + 793) ≈ 0.6251

The proportion of 18-34-year-olds who are more likely to buy a product emphasized as "Made in our country" is approximately 0.1913, while the proportion in the entire sample is approximately 0.6251. Therefore, it appears that individuals in general are more likely to buy a product emphasized as "Made in our country" than 18-34-year-olds.

Step-by-step explanation:

It will run dry after 141.7 days.

The rainwater tank has a diameter of 160 cm and a height of 190 cm, then its volume is:

V = 3.14*(180cm)*(160cm/2)² = 3,617,280 cm³

We know that 1 L = 1000 cm³

then:

3,617,280 cm³ = 3,617.280 L

The water is used at a rate of 25,5 litres per day, so after x days there is:

W =  3,617.280  - 25.5*x liters

It is zero when:

0 = 3,617.280  - 25.5*x

x =  3,617.280/25.5

x = 141.7 days.

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Since two-way table shows the distribution of high school students The correct option is C: No, events A and B are not independent because P(A) ≠ P(A|B).

To determine whether events A and B are independent, we need to compare the probability of event A occurring without any knowledge of event B (i.e., P(A)) to the probability of event A occurring given that event B has occurred (i.e., P(A|B)).

From the table, we see that P(A) = 8/32 = 1/4, since there are 8 juniors out of a total of 32 students. Also, P(B) = 8/32 = 1/4, since there are 8 students who prefer earbuds out of a total of 32 students.

From the table, we see that the probability of both events A and B occurring is P(A and B) = 4/32 = 1/8, since there are 4 juniors who prefer earbuds out of a total of 32 students.

Now, to calculate P(A|B), we use the formula:

P(A|B) = P(A and B) / P(B)

Substituting the values we have, we get:

P(A|B) = (1/8) / (1/4) = 1/2

Since P(A) = 1/4 and P(A|B) = 1/2, we can see that P(A) ≠ P(A|B). Therefore, events A and B are not independent.

Therefore, the answer is No, P(A) ≠ P(A|B).

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