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The exponential decay function that satisfies the given conditions is:

[tex]f(x) = 4 * (1/2)^x[/tex].

In this equation, the y-intercept is 4, which means that when x = 0, the function value is 4. As x increases by 1, the function decreases by a factor of one-half. This behavior is captured by raising 1/2 to the power of x in the equation.

The base of the exponent, 1/2, ensures that the function decreases exponentially. When x = 1, the exponent becomes 1, and[tex]1/2^1[/tex] equals 1/2. This means that the function value decreases to half of its previous value. Similarly, when x = 2, the exponent becomes 2, and[tex]1/2^2[/tex] equals 1/4. The function value decreases to one-fourth of its previous value, and so on.

By multiplying the exponential term by 4, we ensure that the y-intercept is 4. This scaling factor allows us to control the initial value of the function and match the given condition.

The exponential decay function[tex]f(x) = 4 * (1/2)^x[/tex] represents a decaying process where the y-values decrease exponentially as x increases, while starting at a y-intercept of 4.

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At the end of her fifth round, Charimaya would have covered a distance of 1500 meters.

To find the distance covered by Charimaya at the end of her fifth round, we need to calculate the total distance covered in one round and then multiply it by five.

Given that the track is square-shaped with a length of 75 m, we know that all four sides of the track are equal in length.

To calculate the distance covered in one round, we need to find the perimeter of the square track. Since all sides are equal, we can simply multiply the length of one side by 4.

The length of one side of the square track is 75 m. Therefore, the perimeter of the track is:

Perimeter = 4 × 75 m = 300 m

So, Charimaya covers a distance of 300 m in one round.

To find the distance covered at the end of her fifth round, we multiply the distance covered in one round by 5:

Distance covered in 5 rounds = 300 m × 5 = 1500 m

Therefore, at the end of her fifth round, Charimaya would have covered a distance of 1500 meters.

It's worth noting that since the track is square-shaped, each round consists of running along all four sides of the track.

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Answer:

[tex](x+5)^2+(y+5)^2=25[/tex]

Step-by-step explanation:

Recall that the equation of a circle with center (h,k) and radius "r" is [tex](x-h)^2+(y-k)^2=r^2[/tex]

Since the center of the circle is (h,k)=(-5,-5) and the radius is r=5, then our equation will be [tex](x-(-5))^2+(y-(-5))^2=5^2[/tex] which can be simplified into [tex](x+5)^2+(y+5)^2=25[/tex]

The given sequence can be generated by multiplying each term by 2, starting from the initial term of 6.

The pattern that fits the given sequence 6, 12, 24, 48, 96, ... is that each term is 2 times the previous term.

In the sequence 6, 12, 24, 48, 96, ... there are multiple possible patterns, each resulting from a different rule applied to generate the next term. Let's examine each of the proposed patterns:

Each term is 6 more than the previous term:

Starting with 6, if we add 6 to each term, we get:

6 + 6 = 12

12 + 6 = 18

18 + 6 = 24

24 + 6 = 30

30 + 6 = 36

...

This pattern does not match the given sequence since it does not produce the subsequent terms.

Each term is 12 more than the previous term:

Starting with 6, if we add 12 to each term, we get:

6 + 12 = 18

18 + 12 = 30

30 + 12 = 42

42 + 12 = 54

54 + 12 = 66

...

This pattern also does not match the given sequence.

Each term is 1/2 the previous term:

Starting with 6, if we multiply each term by 1/2, we get:

6 [tex]\times[/tex] 1/2 = 3

3 [tex]\times[/tex] 1/2 = 1.5

1.5 [tex]\times[/tex] 1/2 = 0.75

0.75 [tex]\times[/tex] 1/2 = 0.375

0.375 [tex]\times[/tex] 1/2 = 0.1875

...

This pattern does not match the given sequence.

Each term is 2 times the previous term:

Starting with 6, if we multiply each term by 2, we get:

6 [tex]\times[/tex] 2 = 12

12 [tex]\times[/tex] 2 = 24

24 [tex]\times[/tex]2 = 48

48 [tex]\times[/tex]2 = 96

96 [tex]\times[/tex]2 = 192

This pattern perfectly matches the given sequence. Each term is indeed 2 times the previous term, resulting in the next term.

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Answer:

(f + g)(1) = 5

Step-by-step explanation:

(f + g) means we are going to add f(x) and g(x). But also, the (1) part means we are going to let x be equal to 1. We're going to fill in 1 in place of x. You can do this in either order.

Generally speaking its "easier" to fill in the 1 for x first and then do the adding part.

f(x) = x + 6

f(1) = 1 + 6 = 7

and,

g(x) = x - 3

g(1) = 1 - 3 = -2

add the 7 and -2 together:

7 + - 2

= 5

It works out the same if you add first:

f(x) + g(x)

= x + 6 + x - 3

= 2x + 3

then put the 1 in:

= 2×1 + 3

= 2 + 3

= 5

Hope this helps!

Answer:

Step-by-step explanation:

If {0, 3, 6, 9} are are your x's or domain  or input and there are no repeats, then yes TRUE it is a function.

We have created three different linear systems using the equation 20x + 12y = 24.

System 1 has infinitely many solutions, System 2 has no solution, and System 3 has a unique solution.

Let's create three different linear systems using the equation 20x + 12y = 24 and ensure that each system has a different number of solutions.

System 1:

Equation 1: 20x + 12y = 24 (given)

Equation 2: 40x + 24y = 48

Explanation: In this system, we multiplied both sides of the given equation by 2 to create Equation 2.

By doing so, we have essentially created two equations that are multiples of each other.

Since the equations are equivalent, they represent the same line, and the system has infinitely many solutions.

Any values of x and y that satisfy the first equation will automatically satisfy the second equation as well.

System 2:

Equation 1: 20x + 12y = 24 (given)

Equation 2: 20x + 12y = 48

Explanation: In this system, we changed the constant term in Equation 2 to 48.

By doing so, we have created two parallel lines with the same slope. Since the lines are parallel, they will never intersect, and the system has no solution.

There are no values of x and y that satisfy both equations simultaneously.

System 3:

Equation 1: 20x + 12y = 24 (given)

Equation 2: 40x + 24y = 48

Explanation: In this system, we multiplied both sides of Equation 2 by 2 to create Equation 2.

By doing so, we have created two equations that have the same slope but different y-intercepts.

Since the lines are not parallel and have different y-intercepts, they will intersect at a single point, and the system has a unique solution.

There will be one specific pair of values for x and y that satisfy both equations simultaneously.

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Answer:

volume=60cm3, surface area=96cm2

Step-by-step explanation:

volume=1/3×(6×6)×5

=60cm3

surface area= 4(1/2×6×5)+(6×6)

=96cm2

Answer:

1 - 3√(5/35) = 1 - 3√(1/7) = 1 - 3*(1/sqrt(7)) ≈ 0.0755
0.0755 * 100 = 7.55%

Step-by-step explanation:

To find the percentage of 1 - 3√(5/35), we need to first evaluate the expression.

1 - 3√(5/35) = 1 - 3√(1/7) = 1 - 3*(1/sqrt(7)) ≈ 0.0755

To convert this decimal to a percentage, we simply multiply by 100:

0.0755 * 100 = 7.55%

Answer:

44

Step-by-step explanation:

The sum of the series [tex]\sum_{k=1}^4[/tex] (2k²−4) is 44.

The series is: [tex]\sum_{k=1}^4[/tex] (2k²−4)

Let's find the value of each term for k=1, k=2, k=3, and k=4, and then add them up:

For k=1:

2(1)² - 4 = 2(1) - 4 = 2 - 4 = -2

For k=2:

2(2)² - 4 = 2(4) - 4 = 8 - 4 = 4

For k=3:

2(3)² - 4 = 2(9) - 4 = 18 - 4 = 14

For k=4:

2(4)² - 4 = 2(16) - 4 = 32 - 4 = 28

Now, let's add all the terms:

-2 + 4 + 14 + 28 = 44

So, the sum of the series [tex]\sum_{k=1}^4[/tex] (2k²−4) is 44.

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Answer:

b. 160°

d. 55°

Step-by-step explanation:

The Inscribed Angle Theorem states that an inscribed angle is half of the central angle that subtends the same arc.

In other words, if an angle is inscribed in a circle and it intercepts an arc, then the measure of the inscribed angle is equal to half the measure of the central angle that also intersects that arc.

For question:

b.

By using above theorem:

m arc XW=2* m arc XYW

m arc XW= 2*80=160°

d.

m arc WV=125°

The Inscribed Angle Diameter Right Angle Theorem states that any angle inscribed in a circle that intercepts a diameter is a right angle.

By using this theorem:

m arc WV+m arc XV =180°

Now

m arc XV =180°-m arc WV

m arc XV=180°-125°

n arc XV=55°

Answer:

[tex]\text{b.} \quad m\overset{\frown}{XW}=160^{\circ}[/tex]

[tex]\text{d.} \quad m\overset{\frown}{XV}=55^{\circ}[/tex]

Step-by-step explanation:

An inscribed angle is the angle formed (vertex) when two chords meet at one point on a circle.

An intercepted arc is the arc that is between the endpoints of the chords that form the inscribed angle.

[tex]\hrulefill[/tex]

From inspection of the given circle:

According to the Inscribed Angle Theorem, the measure of an inscribed angle is half the measure of the intercepted arc. Therefore:

[tex]m \angle WRX = \dfrac{1}{2}\overset{\frown}{XW}[/tex]

         [tex]80^{\circ}= \dfrac{1}{2}\overset{\frown}{XW}[/tex]

   [tex]\boxed{m\overset{\frown}{XW}=160^{\circ}}[/tex]

[tex]\hrulefill[/tex]

From inspection of the given circle:

According to the Inscribed Angle Theorem, the measure of an inscribed angle is half the measure of the intercepted arc. Therefore:

[tex]m \angle WVX= \dfrac{1}{2}\overset{\frown}{WX}[/tex]

         [tex]90^{\circ}= \dfrac{1}{2}\overset{\frown}{WX}[/tex]

    [tex]m\overset{\frown}{WX}=180^{\circ}[/tex]

The sum of the measures of the arcs in a circle is 360°.

[tex]m\overset{\frown}{VW}+m\overset{\frown}{WX}+m\overset{\frown}{XV}=360^{\circ}[/tex]

Therefore, so find the measure of arc XV, substitute the found measures of arcs VW and WX, and solve for arc XV:

[tex]125^{\circ}+180^{\circ}+m\overset{\frown}{XV}=360^{\circ}[/tex]

          [tex]305^{\circ}+m\overset{\frown}{XV}=360^{\circ}[/tex]

                    [tex]\boxed{m\overset{\frown}{XV}=55^{\circ}}[/tex]

Answer:

Step-by-step explanation:To calculate the value of x, we can use the formula for the area of a trapezoid:

Area = (1/2) * (sum of the parallel sides) * height

Given that the area of the trapezoid ABCD is 70 square units, we can set up the equation as follows:

70 = (1/2) * (AB + DC) * Height

Substituting the given values:

70 = (1/2) * ((2x + 4) + (x - 2)) * 4

Simplifying the equation:

70 = (1/2) * (3x + 2) * 4

Multiplying both sides by 2 to remove the fraction:

140 = (3x + 2) * 4

Dividing both sides by 4:

35 = 3x + 2

Subtracting 2 from both sides:

33 = 3x

Dividing both sides by 3:

x = 11

Therefore, the value of x is 11.

The angle the slide forms with the tower is approximately 41 degrees (rounded to the nearest degree).

To find the angle the slide forms with the tower, we can use trigonometric ratios. Let's consider the right triangle formed by the height of the tower (18 m), the length of the slide (20 m), and the angle we want to find.

Using the tangent function, we have:

tan(angle) = opposite/adjacent

In this case, the opposite side is the height of the tower (18 m) and the adjacent side is the length of the slide (20 m). Therefore:

tan(angle) = 18/20

To find the angle, we can take the inverse tangent (arctan) of both sides:

angle = arctan(18/20)

Using a calculator, we find that arctan(18/20) is approximately 40.56 degrees.

Therefore, the angle the slide forms with the tower is approximately 41 degrees (rounded to the nearest degree).

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By using the change of base formula: The solution to the equation log(base y) (X-2) = 5 is [tex]X = y^5 + 2.[/tex]

To solve the equation log(base y) (X-2) = 5 using the change of base formula, we can rewrite the equation as log(base b) (X-2) / log(base b) y = 5.

Using the change of base formula, we can choose any base for b.

Let's choose base 10 for simplicity.

So the equation becomes log(base 10) (X-2) / log(base 10) y = 5.

We know that log(base 10) (X-2) represents the logarithm of (X-2) to the base 10, and log(base 10) y represents the logarithm of y to the base 10.

Now, to solve for X, we can isolate it by multiplying both sides of the equation by log(base 10) y:

log(base 10) (X-2) = 5 [tex]\times[/tex] log(base 10) y.

This simplifies to:

log(base 10) (X-2) [tex]= log(base 10) y^5.[/tex]

Since the logarithms on both sides have the same base, we can remove the logarithm and equate the arguments:

[tex]X - 2 = y^5.[/tex]

Now we can solve for X by adding 2 to both sides:

[tex]X = y^5 + 2.[/tex]

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Statement that is always true about obtuse triangles:

An obtuse triangle always has one angle that measures more than 90 degrees.

In the given tree diagram, the "Obtuse" category represents triangles with at least one obtuse angle.

An obtuse angle is an angle that measures more than 90 degrees. Since an obtuse triangle is defined as having one obtuse angle, it will always have an angle that measures more than 90 degrees.

Therefore, the statement that an obtuse triangle always has one angle that measures more than 90 degrees is always true.

Statement that is sometimes true about obtuse triangles:

An obtuse triangle can have different side lengths.

In the given tree diagram, the "Obtuse" category represents triangles with at least one obtuse angle.

The "Scalene" category represents triangles with different side lengths. Therefore, it is possible for an obtuse triangle to have different side lengths, making the statement "An obtuse triangle can have different side lengths" sometimes true.

However, it is also possible for an obtuse triangle to have two or more sides with the same length, which would make it an isosceles or equilateral triangle.

Hence, the statement is only sometimes true and not always true.

In summary, an always true statement about obtuse triangles is that they always have one angle that measures more than 90 degrees.

A sometimes true statement about obtuse triangles is that they can have different side lengths.

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The regression equation is y = 17.1643X - 2.47977

To solve this problem, we have to calculate the equation of regression.

Sum of X = 2.97

Sum of Y = 28.66

Mean X = 0.33

Mean Y = 3.1844

Sum of squares (SSX) = 0.3552

Sum of products (SP) = 6.0959

Regression Equation = y = bX + a

b = SP/SSX = 6.1/0.36 = 17.1643

a = MY - bMX = 3.18 - (17.16*0.33) = -2.47977

y = 17.1643X - 2.47977

The line of best fit is y = 17.1643X - 2.47977

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The sixth term of the geometric sequence is 2048.

The given geometric sequence is 2, 8, 32, 128. We can observe that each term is obtained by multiplying the previous term by 4. Therefore, the common ratio (r) of the sequence is 4.

The formula for the nth term (an) of a geometric sequence is given by:

an = a1 * r^(n-1)

where a1 is the first term and r is the common ratio.

For this sequence, a1 = 2 and r = 4. Plugging in these values into the formula, we get:

an = 2 * 4^(n-1)

To find a6, we substitute n = 6 into the formula:

a6 = 2 * 4^(6-1)

  = 2 * 4^5

  = 2 * 1024

  = 2048

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The Probable question may be:
Write an equation for the nth term of the geometric sequence 2, 8, 32, 128,

Then find a6. Round to the nearest tenth if necessary.

a = 5×4 X

a1 = n-1 X

Answer:

7.80 ounces can be bought for $1.95

Step-by-step explanation:

Step 1:  Determine how many ounces is in a pound:

Thus, 16 ounces cost $4.

Step 2:  Create a proportion to determine how many ounces can be bought for $1.95.

Since you can get 16 ounces for $4, we can create a proportion to determine how many ounces can be bought for $1.95:

16 ounces / $4 = x ounces / $1.95

Step 3:  Simplify on the left-hand side of the equation:

16/4 = x/1.95

4 = x/1.95

Step 4: multiply both sides by 1.95 to determine how many ounces can be bought for $1.95:

(4 = x/1.95) * 1.95

7.80 = x

Thus, 7.80 ounces can be bought for $1.95.

Incremental cash flows associated with the donation of a valuable painting from a corporation's private collection may include It's important to note that any direct costs associated with the donation.

Tax benefits: The corporation may be eligible for tax deductions or credits for charitable donations, which could result in a reduction in its tax liability and generate cash flow savings.

Opportunity cost: If the corporation could have sold the painting instead of donating it, the incremental cash flow would be the potential proceeds from the sale.

Storage and maintenance cost savings: By donating the painting to the art museum, the corporation no longer has to incur expenses for storing, insuring, and maintaining the artwork, resulting in cost savings.

Public relations and marketing benefits: Donating the painting can enhance the corporation's reputation and generate positive publicity, potentially leading to increased customer goodwill and brand value, which can translate into future cash flows.

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a. Mean = 3

Standard deviation = 2

b. The z-scores for each score in the sample are: 1, -1.5, 0.5, 1, -1, -0.5, 0.5.

a. To compute the mean and standard deviation for the sample, we follow these steps:

Calculate the mean (average)

Mean = (sum of all scores) / (number of scores)

Mean = (5 + 0 + 4 + 5 + 1 + 2 + 4) / 7

Mean = 21 / 7

Mean = 3

The mean of the sample is 3.

Calculate the standard deviation

The formula for standard deviation for a sample is given by:

Standard deviation = sqrt((sum of squared differences from the mean) / (number of scores - 1))

First, calculate the squared differences from the mean for each score:

(5 - 3)^2 = 4

(0 - 3)^2 = 9

(4 - 3)^2 = 1

(5 - 3)^2 = 4

(1 - 3)^2 = 4

(2 - 3)^2 = 1

(4 - 3)^2 = 1

Next, sum up these squared differences:

4 + 9 + 1 + 4 + 4 + 1 + 1 = 24

Now, divide this sum by (number of scores - 1):

24 / (7 - 1) = 24 / 6 = 4

Finally, take the square root of this result:

Standard deviation = sqrt(4) = 2

The standard deviation of the sample is 2.

b. To find the z-score for each score in the sample, we use the formula:

z = (X - Mean) / Standard deviation

For each score, we substitute the values into the formula:

X = 5, z = (5 - 3) / 2 = 2 / 2 = 1

X = 0, z = (0 - 3) / 2 = -3 / 2 = -1.5

X = 4, z = (4 - 3) / 2 = 1 / 2 = 0.5

X = 5, z = (5 - 3) / 2 = 2 / 2 = 1

X = 1, z = (1 - 3) / 2 = -2 / 2 = -1

X = 2, z = (2 - 3) / 2 = -1 / 2 = -0.5

X = 4, z = (4 - 3) / 2 = 1 / 2 = 0.5

The z-scores for each score in the sample are:

z = 1, z = -1.5, z = 0.5, z = 1, z = -1, z = -0.5, z = 0.5

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The amount of snow accumulated between 2 am and 5 am is: 1.25 inches

The general formula for the equation of a line in slope intercept form is:

y = mx + c

where:

m is slope

c is y-intercept

From the given graph attached, we see that the y-axis gives the amount of snow at different specific times.

Meanwhile the x-axis gives the time in hours after midnight

At 2am, the y-axis value is 1.25 inches, and as such at 2am snow accumulation was 1.25 inches.

At 5 am, the y-axis value reads 2.5 inches, and as such at 5am snow accumulation was 2.5 inches.

The difference in both snow accumulations is:  2.5 - 1.25 = 1.25

Hence, 1.25 inches snow accumulated between 2 am and 5 am.

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The name of the Platonic solid that resembles a cuboid is the hexahedron, or more commonly known as a cube.

The correct answer is option C.

The name of the Platonic solid that resembles a cuboid is the hexahedron, also known as a cube. The hexahedron is one of the five Platonic solids, which are regular, convex polyhedra with identical faces, angles, and edge lengths. The hexahedron is characterized by its six square faces, twelve edges, and eight vertices.

The term "cuboid" is often used in general geometry to describe a rectangular prism with six rectangular faces. However, in the context of Platonic solids, the specific name for the solid resembling a cuboid is the hexahedron.

The hexahedron is a highly symmetrical three-dimensional shape. All of its faces are congruent squares, and each vertex is formed by three edges meeting at right angles. The hexahedron exhibits symmetry under several transformations, including rotations and reflections.

Its regularity and symmetry make the hexahedron an important geometric shape in mathematics and design. It has numerous applications in architecture, engineering, and computer graphics. The cube, as a special case of the hexahedron, is particularly well-known and widely used in everyday life, from dice and building blocks to cubic containers and architectural structures.

Therefore, the option which is the correct is C.

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The question probable may be:

What is the name of the Platonic solid which resembles a cuboid?

A. Dodecaheron  

B. Tetrahedron

C. Hexahedron

D. Octahedron  

17x = 425

x = 25

8x = 200 boys

9x = 225 girls

If there are 200 high school students in the district, the number of high school students expected to be in Chemistry is 60 because the percentage who offer Chemistry in the district is 30%.

The number of high school students who offer Chemistry in the district can be determined by multiplying the total number of high school students and the percentage of students who offer Chemistry.

The result of a multiplication operation (multiplicand and multiplier), which is one of the basic mathematical operations, is known as the product.

The total number of high school students in the district = 200

The percentage of students who offer Chemistry in the district = 30%

The number of students likely to be offering Chemistry in the district = 60 (200 x 30%).

Thus, we can conclude that 60 high school students are in Chemistry based on the Chemistry percentage.

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The percentage of high school students in the district who offer Chemistry is 30%.  If there are 200 high school students in the district, how many would you expect to be in Chemistry?

Answer:

(2, 7) and (-6, -1)

Step-by-step explanation:

y = x + 5

y = x² + 5x − 7

Equatig the above,

x² + 5x − 7 = x + 5

⇒ x² + 4x −12 = 0

⇒ x² + 6x - 2x - 12 = 0

⇒ x(x + 6) - 2(x + 6) = 0

⇒  (x - 2)(x + 6) = 0

⇒ x = 2 or x = -6

Eq(1) : y = x + 5 (given)

When x = 2

y = 2 + 5 = 7

Point : (2, 7)

When x = -6

y = -6 + 5 = -1

Point: (-6, -1)

Hola!

-4 - 6

= -10

the answer is -10

Answer:

x = 8

Step-by-step explanation:

The given diagram shows a triangle with the length of two sides and its included angle.

To find the value of the missing variable x, we can use the Law of Cosines.

[tex]\boxed{\begin{minipage}{6 cm}\underline{Law of Cosines} \\\\$c^2=a^2+b^2-2ab \cos C$\\\\where:\\ \phantom{ww}$\bullet$ $a, b$ and $c$ are the sides.\\ \phantom{ww}$\bullet$ $C$ is the angle opposite side $c$. \\\end{minipage}}[/tex]

From inspection of the given triangle:

Substitute the values into the formula and solve for x:

[tex]\begin{aligned}x^2&=18^2+21^2-2(18)(21)\cos 22^{\circ}\\x^2&=324+441-756\cos 22^{\circ}\\x^2&=765-756\cos 22^{\circ}\\x&=\sqrt{765-756\cos 22^{\circ}}\\x&=8.00306228...\\x&=8\end{aligned}[/tex]

Therefore, the value of the missing variable x is x = 8, rounded to the nearest hundredth.

The final answer after evaluating the expression 3.14([tex]a^{2}[/tex] + ab) (by putting the value a = 3 and b = 4) is 65.94.

When a = 3 and b = 4, we substitute the supplied values into the expression to assess 3.14([tex]a^{2}[/tex] + ab):

3.14([tex]3^{2}[/tex] + 3 * 4)

We begin by solving the exponent:

[tex]3^{2}[/tex] = 3 * 3 = 9

The values are then entered into the expression:

3.14(9 + 3 * 4)

Inside the brackets, multiply the result:

3.14(9 + 12)

The numbers in the brackets are added:

3.14(21)

The decimal number is now multiplied by 21:

3.14 * 21 = 65.94

The evaluated expression is 65.94 as a result.

Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation.

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The answer is:

Work/explanation:

We're asked to evaluate the expression [tex]\sf{3.14(a^2+ab)}[/tex] for a = 3 and b = 4.

Plug in the data:

[tex]\sf{3.14(3^2+3*4)}[/tex]

[tex]\sf{3.14(9+12)}[/tex]

[tex]\sf{3.14(21)}[/tex]

[tex]\bf{65.94}[/tex]