find the volume of a pyramid with height 28 and rectangular base with dimensions 2 and 7 using integration

The volume of the pyramid  of height h with an equilateral triangle base with side a, by using the method of volumes by SLICES  is V = a²h/3.

We can use the method of volumes by slices to find the volume of the pyramid.

Consider a slice of the pyramid that is perpendicular to the base and at a distance x from the apex. This slice has a cross-sectional area of a square with side length (base of pyramid) s, We can use the Pythagorean theorem to find that the

Height of the slice is [tex]\sqrt{(a^2 - (a/2)^2 - x^2)} = \sqrt{(3a^2/4 - x^2).[/tex]

Therefore, the volume of the slice is given by the product of its cross-sectional area and height:

dV = s^2 dh = (a^2 - 4x^2)/4 dh

To find the total volume of the pyramid,

we integrate dV from x=0 to x=h:

V = ∫₀ʰ (a²-4x²)/4 dx

Simplifying the integrand, we get:

V = (1/4) ∫₀ʰ (a² - 4x²) dx

V = (1/4) [a²x - (4x³)/3] from x=0 to x=h

V = (1/4) [a²h - (4h³)/3]

V = a²h/3

Therefore, the volume of the pyramid is V = a²h/3.

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The equation of the plane that passes through the line of intersection of the two planes and passes through the point (3,-1,2) is x - 2y + 7z - 17 = 0  .

The Plane we want to find passes through the line of intersection of the planes, so it is perpendicular to the normal vectors of both planes.

The Normal Vector of the plane 2x - 3y - z + 1 = 0 is (2, -3, -1), and

the normal vector of the plane 3x + 5y - 4z + 2 = 0 is (3, 5, -4).

The cross product of these two vectors gives a vector which is perpendicular to both of them, and

hence , it lies along the line of intersection of the two planes:

the cross product of (2, -3, -1) x (3, 5, -4) is  = (-7, -2, 1)  ;

The vector (-7, -2, 1) is the direction vector of the line of intersection.

Now to find the equation of the plane that passes through the point (3, -1, 2) and is perpendicular to this direction vector.

we let , equation of the plane be ax + by + cz + d = 0. We know that the point (3, -1, 2) lies on the plane, so we have:

⇒ a(3) + b(-1) + c(2) + d = 0  ;

⇒ 3a - b + 2c + d = 0  ;

We also know that the plane is perpendicular to the direction vector (-7, -2, 1), so we have:

⇒ a(-7) + b(-2) + c(1) = 0  ;

⇒ -7a - 2b + c = 0  ;

Let  say a = 1. Then we have :

⇒ a = 1

⇒ -7a - 2b + c = 0

⇒ 3a - b + 2c + d = 0

Substituting a = 1 into the first equation gives:

⇒ -7 - 2b + c = 0  ;

Solving for c in terms of b gives:

⇒ c = 2b + 7

Simplifying ,

we get ;

⇒ 3 - b + 2(2b + 7) + d = 0  ;

Solving for d in terms of b gives:   ⇒ d = -2b - 17  ;

Therefore, the equation of the plane is x - 2y + 7z - 17 = 0 .

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The equation solved for the variable h is h = [tex]\sqrt[]{\frac{2A -b^2}{b} }[/tex]

To solve for a variable in a given equation is to make it the subject of formula.

The variable is made to stand alone on one end of the equality sign.

We have the equation given as;

A=1/2bh(h+b)

First, cross multiply

2A = bh(h+b)

2A = bh² + b²

collect like terms

2A - b² = bh²

Now divide by the coefficient of h²

h² = 2A - b²/b

Find the square root of both sides

h = [tex]\sqrt[]{\frac{2A -b^2}{b} }[/tex]

Hence, the equation is h = [tex]\sqrt[]{\frac{2A -b^2}{b} }[/tex]

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There is a probability of 98.1% or around 0.98 of making at least one shot.

We can solve this problem using the binomial distribution. Let X be the number of shots the basketball player makes out of 6 attempts. Then X follows a binomial distribution with parameters n = 6 (number of trials) and p = 0.553 (probability of success).

The probability of making at least one shot can be calculated as the complement of the probability of missing all six shots. That is:

P(X ≥ 1) = 1 - P(X = 0)

Using the binomial probability formula, we can calculate P(X = 0) as:

P(X = 0) = (6 choose 0) * (0.553)^0 * (1 - 0.553)^(6-0) = 0.019

where (6 choose 0) = 1 is the number of ways to choose 0 shots out of 6.

Therefore,

P(X ≥ 1) = 1 - P(X = 0) = 1 - 0.019 = 0.981

So the probability of making at least one shot is approximately 0.981 or 98.1%.

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Answer: 12

Step-by-step explanation:

The probability of waiting less than 12 minutes between successive speeders is 0.02.

The time elapsed between successive speeders detected by a radar unit is a  random variable with a cumulative distribution function:

F(x) = 0 when x = 0.

x/10, for 0 x ten 1, for x ten

a) Using the cumulative distribution function, to calculate the probability of waiting less than 12 minutes (0.2 hours) between successive speeders: F(0.2):

F(0.2) = 0.2/10 = 0.02

So, the probability of waiting less than 12 minutes between speeders is 0.02.

b) Using the probability density function, we can differentiate the cumulative distribution function with respect to x to find the probability:

For x = 0, f(x) = dF(x)/dx = 0.

1/10 for 0 x 10 0, 0 for x 10

The area under the probability density function up to x=0.2 represents the probability of waiting less than 12 minutes:

P(0 < w < 0.2) = ∫0.2 0 f(x) (x) dx = ∫0.2 0 (1/10) dx = (1/10) * [x] 0.2 = 0.02

Using either the cumulative distribution function or the probability density function, the probability of waiting less than 12 minutes between successive speeders is 0.02.

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Based on the information, it can be inferred that the volume of a cube with a side length of 30 cm is 27,000cm³ and the area would be 90cm².

To calculate the volume and area of the cube we must perform the following mathematical procedures:

Volume:

We must multiply side * base * depth = volume.

Area:

We must multiply the base by the side

According to the above, the area and volume of this cube would be 90cm² and 27,000cm³ respectively.

Note: This question is incomplete. Here is the complete information:

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Answer:

= 5n³ + 3n² - 4

Step-by-step explanation:

(-3n³-3n²) + (8n³ + 6n² - 4)

= -3n³-3n²+8n³+6n²-4

= -3n³+8n³-3n²+6n²-4

= 5n³+3n²-4.

Answer:

[tex]\huge\boxed{\sf x\³ - 10x\² + 19x + 6}[/tex]

Step-by-step explanation:

= (x - 3)(x² - 7x - 2)

= x(x² - 7x - 2) - 3(x² - 7x - 2)

= x³ - 7x² - 2x - 3x² + 21x + 6

= x³ - 7x² - 3x² - 2x + 21x + 6

[tex]\rule[225]{225}{2}[/tex]

Based on the data, we can infer that the following numbers fill in the blanks: 100 is greater than 10.

To fill in the blanks we must carefully read the information in the fragment. Once we understand this information we can select the two numbers that meet the rule mentioned in the statement.

According to the above, a correct statement would be:

Additionally, we could put other values that comply with this rule, for example:

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The areas, diameters, and radii can be grouped as follows:

1. Area: 221.5584 square units

Diameter: 16. 8 units

Radius: 8.4 units

2. Area: 78.5 square units

Diameter: 10 units

Radius: 5 units

3. Area: 452.16 square units

Diameter: 24 units

Radius: 12 units

4. Area: 36.2984 square units

Diameter: 6.8 units

Radius: 3.4 units

5. Area: 314 square units

Diameter: 20 units

Radius: 10 units

6. Area: 886.2336 square units

Diameter: 33.6 units

Radius: 16.8 units

First, we are given the area of the circles. Next, we need to note the formula of the area of a circle and this is: πr²

To get the radius, we can use the formula: √(A/π)

For the first case, we have:

Area = 221.5584 square units

Thus, radius = √(221.5584/3.14)

= 8.4

Since diameter is = 2r, then diameter in this case

= 16.8 units.

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Three equations that you think would be the least difficult to solve are,

∛(t + 4) = 3

∛ (r³ - 19) = 2

4 + ∛- m + 4 = 6

And, 3 equations that you think would be the most difficult to solve are,

∛z + 9 = 0

6 - ∛b = 0

∛2n + 3 = - 5

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

Now, By all the expressions;

Some expression gives easily real solution after solving.

But here some expression have complex solution after solving and tough to solve.

Thus, By all the expressions,

Three equations that you think would be the least difficult to solve are,

∛(t + 4) = 3

∛ (r³ - 19) = 2

4 + ∛- m + 4 = 6

And, 3 equations that you think would be the most difficult to solve are,

∛z + 9 = 0

6 - ∛b = 0

∛2n + 3 = - 5

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The decimal representation of the given binary fractional number 0.000112 is 0.109375.

Binary numbers are composed of only 0s and 1s, while decimal numbers are composed of 0-9. Additionally, binary numbers follow a different place value system, where each digit represents a power of 2 rather than a power of 10.

Now, let's focus on the given problem - converting a binary fractional number 0.000112 to a decimal representation. We can use the following formula to convert any binary fraction to decimal:

(decimal equivalent) = (binary digit 1/2¹) + (binary digit 2/2²) + (binary digit 3/2³) + ...

Using this formula, we can start by breaking down the given binary fraction 0.000112 into its individual binary digits:

0.000112 = 0/2¹ + 0/2² + 0/2³ + 1/2⁴ + 1/2⁵ + 1/2⁶

Now, we can simply evaluate each term in the equation and add them up to get the decimal equivalent:

0/2¹ = 0

0/2² = 0

0/2³ = 0

1/2⁴ = 0.0625

1/2⁵ = 0.03125

1/2⁶ = 0.015625

(decimal equivalent) = 0 + 0 + 0 + 0.0625 + 0.03125 + 0.015625 = 0.109375

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If the scores of a certain population are Normally Distributed , and the mean is 104.32 , then the 95% confidence interval is (b) 104.32 ± 3.29   .

The Scores of the population of WISC are Normally Distributed ;  

and the Mean(μ) of the 25 scores is = 104.32  ;

the standard deviation(σ) for the population is = 10  ;

the sample size(n) = 25  ;

For  the 95% confidence, critical value is = 1.96 ;

So,  Confidence interval is written as =  μ ± 1.96(σ/√n)

Substituting the required values  ,

we get ;

⇒ 104.32 ± 1.96(10/√25)

⇒ 104.32 ± 1.96×2

⇒ 104.32 ± 3.92

Therefore , the required 95% confidence interval is (104.32 ± 3.92) .

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The given question is incomplete , the complete question is

The scores of a certain population on the Wechsler Intelligence Scale for Children (WISC) are thought to be Normally distributed with mean μ and standard deviation σ = 10. A simple random sample of 25 children from this population is taken and each is given the WISC. The mean of the 25 scores is = 104.32.

Based on these data, a 95% confidence interval for μ is

(a) 104.32 ± 0.78.

(b) 104.32 ± 3.29.

(c) 104.32 ± 3.92.

(d) 104.32 ± 19.60

The probability that the couple will have 3 normal girls is approximately 0.422

Since both parents are normal, they both have to be heterozygous carriers for the recessive allele that causes albinism. We can represent the normal allele with the letter N and the albino allele with the letter n. Then, we can write the genotypes of the parents as Nn x Nn.

The daughter is albino, so her genotype must be nn. The son is normal, so his genotype must be Nn.

We want to find the probability that the couple will have 3 normal girls. We can use the multiplication rule of probability to find this probability:

P(3 normal girls) = P(normal girl) x P(normal girl) x P(normal girl)

The probability of having a normal girl is 3/4 since there are three possible genotypes that result in a normal phenotype (NN, Nn, Nn) out of a total of four possible genotypes. The probability of having a normal boy is also 3/4, for the same reason.

Therefore, the probability of having 3 normal children (in this case, girls) is:

P(3 normal girls) = (3/4) x (3/4) x (3/4) = 27/64 ≈ 0.422

So, the probability that the couple will have 3 normal girls is approximately 0.422, or about 42.2%.

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The time to travel 279 miles is 4.5 hours

We can use the formula:

time = distance ÷ speed

To find the time it will take to travel 279 miles, given that it takes 6 hours to travel 372 miles.

The speed is constant, so we can find it by dividing the distance by the time for the first trip:

speed = distance ÷ time = 372 miles ÷ 6 hours = 62 miles/hour

Now we can use the speed to find the time it will take to travel 279 miles:

time = distance ÷ speed = 279 miles ÷ 62 miles/hour ≈ 4.5 hours

Hence, it will take approximately 4.5 hours to travel 279 miles at this speed.

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The effect on the graph for the given function if the slope of the function is changed to -10 include the following: D. the line is less steep.

The effect on the graph for the given function if the y-intercept of the function is changed to a smaller positive number is: C. the line shifts down.

In Mathematics, a slope is sometimes referred to as rate of change and it is typically used to describe both the direction, ratio, and steepness of the function of a straight line based on its coordinates or points.

Generally speaking, a slope simply refers to a measure of the steepness of a straight line. This ultimately implies that, the higher the slope of a line, the more steep it is and vice-versa.

Based on the information provided about this function y = 8 - 2x, we can logically deduce that the line would be shifted downward when its y-intercept is changed to a smaller positive number.

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The expression value of 3x^2 - 2x+3 is 6x^2 - 13x + 6

Value of an Expression The value of the expression is the result of the calculation described by this expression when the variables and constants in it are assigned values. Letters can be used to represent numbers in an algebraic expression.

Apply the distributive property:

→ 3x (2 - 3) - 2 (2x - 3)

Apply the distributive property:

→ 3x (2x) + 3x ⋅ -3 - 2 (2x - 3)

Apply the distributive property:

→ 3x (2x) + 3x ⋅ -3 - 2 (2x) - 2 ⋅ -3

Simplify each term:

→ 6x^2 - 9x - 4x  + 6x

Subtract 4x from -9x

→ 6x^ - 13x + 6

Therefore, the expression value of 3x^2 - 2x+3 is 6x^2 - 13x + 6

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The value of the expression given is 136.

Expressions are mathematical statements which consist of two or more terms and terms are connected to each other using mathematical operators like addition, multiplication, subtraction and so on.

Given expression is,

3x² - 2x + 3

We have to find the value of the expression when x = 7.

Substitute x = 7.

(3 × 7²) - (2 × 7) + 3 = (3 × 49) - 14 + 3

                               = 147 - 14 + 3

                               = 136

Hence the value of the expression is 136.

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Depends how the answer "should" be formatted but 114 2/3 feet should do the trick

Answer:

Charlotte's mix has a bluer shade of purple

Step-by-step explanation:

Since we are asked whose paint is a bluer shade it would be easier to express the ratios of the paint mix as blue:red

Charlotte: 1 cup red for every 2 cups blue ==> 2 cups blue for every 1 cup red

This can be expressed as the ratio
[tex]\dfrac{ 2 \;blue}{1\;red} = \dfrac{2}{1} = 2[/tex]

[tex]-------------------[/tex]

Pablo : 2 cups red for every 3 cups blue ==> 3 cups blue for every 2 cups red

The ratio of blue to red
= [tex]\dfrac{3 \;blue}{2\;red} = \dfrac{3}{2} = 1\dfrac{1}{2} \;\; (or\; 1.5)[/tex]

Since
[tex]2 > 1\dfrac{1}{2} \;\;\;(or\; 2 > 1.5)[/tex]  

Charlotte's mix has a bluer shade of purple

The velocity of the particle at times t = a, t = 1, t = 2, and t = 3 are:

a) 4/a^2 m/s, 4 m/s, 1 m/s, 4/9 m/s.

b) The instantaneous velocity of the particle when t=1 is 4 m/s.

To find the velocity, we need to take the derivative of the displacement function with respect to time. The derivative of s = 4/t^2 is ds/dt = -8/t^3. So, the velocity of the particle at time t is given by v = ds/dt = -8/t^3.

For t = a, the velocity is v = -8/a^3 m/s. For t = 1, the velocity is v = -8 m/s. For t = 2, the velocity is v = -2 m/s. For t = 3, the velocity is v = -8/27 m/s.

To find the average velocity during the time period [1, 2], we need to find the displacement at t = 2 and t = 1, then calculate the change in displacement divided by the time interval: (4/4 - 4/1)/1 = 0 cm/s. To find the average velocity during the time period [1, 1.1], we need to find the displacement at t = 1.1 and t = 1, then calculate the change in displacement divided by the time interval: (4/1.1^2 - 4/1)/0.1 = -19.60 cm/s.

To estimate the instantaneous velocity of the particle at t = 1, we can plug in t = 1 to the derivative we found earlier: v = -8/1^3 = -8 m/s. Therefore, the instantaneous velocity of the particle when t = 1 is 4 m/s.

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The solution to the system of equation y = -2x + 3 and y = 5x - 4 is (1,1).

Given the system of equation in the question;

y = -2x + 3

y = 5x - 4

To solve the system of equation, plug equation one to into equation two and solve for x.

y = 5x - 4

Plug in y = -2x + 3

-2x + 3 = 5x - 4

Solve for x

-2x - 5x = - 4 - 3

-7x = -7

x = -7 / -7

x = 1

Now, plug x = 1 into equation one and solve for x.

y = -2x + 3

Plug in x = 1

y = -2( 1 ) + 3

y = -2 + 3

y = 1

Therefore, the value of x is 1 and y is also 1.

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The probability that exactly 90 defective washers are found is (400 choose 90) * (1100 choose 110) / (1500 choose 200). and the probability that at least 2 defective items are found is (400 choose 1) * (1100 choose 199) / (1500 choose 200).

Let X be the number of defective washers in a sample of 200 washers.

We can model X as a hypergeometric distribution with parameters N = 1500 (total number of washers), K = 400 (total number of defective washers), and n = 200 (sample size).

a) The probability of finding exactly 90 defective washers is:

P(X = 90) = (400 choose 90) * (1100 choose 110) / (1500 choose 200)

This is because we need to choose 90 defective washers from the 400 defective washers, and 110 non-defective washers from the 1100 non-defective washers, out of the total of 200 washers chosen.

b) The probability of finding at least 2 defective items can be calculated as the complement of the probability of finding 0 or 1 defective item in the sample:

P(X >= 2) = 1 - P(X = 0) - P(X = 1)

To compute P(X = 0), we need to choose 0 defective washers from the 400 defective washers, and 200 non-defective washers from the 1100 non-defective washers, out of the total of 1500 washers:

P(X = 0) = (400 choose 0) * (1100 choose 200) / (1500 choose 200)

To compute P(X = 1), we need to choose 1 defective washer from the 400 defective washers, and 199 non-defective washers from the 1100 non-defective washers, out of the total of 1500 washers:

P(X = 1) = (400 choose 1) * (1100 choose 199) / (1500 choose 200)

Once we have computed P(X = 0) and P(X = 1), we can substitute these values into the expression for P(X >= 2) to obtain the desired probability.

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The simplified form of the expression (4 - a²b³)/5 = (9 - 3b³)/7 is

b³(15 - 7a²) = 17.

An equation is written in the form of variables and constants separated by the operation of multiplication and division,

An equation states that terms in different forms on both sides of the equality sign are equal.

Multiplication and division do not separate the terms of an equation.

Given, An equation (4 - a²b³)/5 = (9 - 3b³)/7,  in variables 'a' and 'b'.

7(4 - a²b³) = 5(9 - 3b³).

28 - 7a²b³ = 45 - 15b³.

15b³ - 7a²b³ = 45 - 28.

b³(15 - 7a²) = 17.

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The measure of XZ is equivalent to 24. Option C is correct

The given triangles in question are similar in nature. For every similar triangles, the ratio of the similar sides of the triangle is equal to a constant as shown:

k+3/36 = k+5/44

Cross multiply and expand

36(k + 5) = 44(k + 3)

36k + 180 = 44k + 132

36k - 44k = 132-180|

-8k = -48

k = 6

For the measure of XZ

k/XZ = k+5/44

6/XZ = 11/44
6/XZ = 1/4
XZ = 24

Hence the measure of XZ from the given diagram is equivalent to 24

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Mack should make 4 necklaces and 8 wristbands to maximize his profit while staying within his time and quantity constraints.

His profit would be:

Profit = 3(4) + 2(8) = $20.

One or many equations having the same number of unknowns that can be solved simultaneously are called simultaneous equations. And the simultaneous equation is the system of equations.

From the information given, we know:

a) Necklace requires 40 minutes to make.

Let x be the number of necklaces made, then the total time spent on necklaces is 40x.

b) Wristband requires 25 minutes to make.

Let y be the number of wristbands made, then the total time spent on wristbands is 25y.

c) Mack has a total of 360 minutes to make necklaces and wristbands. Therefore, the equation for time is:

40x + 25y = 360

d) Mack wants to make no more than 12 items, so we have the inequality:

x + y ≤ 12

e) Mack makes a profit of $3.00 per necklace and $2.00 per wristband. Therefore, the equation for profit is:

Profit = 3x + 2y

Now we have the following system of equations:

40x + 25y = 360

x + y ≤ 12

Profit = 3x + 2y

From equation (2), we have y = 12 - x.

Substitute y = 12 - x into equations (1) and (3):

40x + 25(12 - x) = 360

Profit = 3x + 2(12 - x)

Simplify and solve for x:

40x + 300 - 25x = 360

15x = 60

x = 4

Substitute x = 4 into equation (2) to find y:

y = 12 - x = 12 - 4 = 8

Therefore, His profit would be, Profit = 3(4) + 2(8) = $20.

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The quotient is x² - 3 - 4 and remainder is 1.

So, The value of f (3) is, 1

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

The value of f(3) is told by the remainder theorem to be the remainder from division of f(x) by (x -3). That division is shown in the attachment. The remainder is 1, hence we have;

⇒ f(3) = 1

The number at upper left in the synthetic division table is the value of x for which we are evaluating f(x).

When the polynomial is written in Horner Form:

⇒ f(x) = ((x -6)x +13)x -11

Here, the coefficient of x at each point in the evaluation is the same as the number on the bottom row of the synthetic division. The result of the evaluation is the same as the remainder from the division.

Now, Add the obtained result to the next coefficient of the dividend and write down the sum as;

3 | 1  - 6  13         - 11

         3  - 9        3x4 = 12

-----------------------

   1  - 3    4    (- 11)  + 12 = 1

Thus, We have complete table and have obtained the resulting coefficient are,

1 , - 3, 4, 1

Thus, The quotient is x² - 3 - 4 and remainder is 1.

So, The value of f (3) is, 1

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The value of the function f(x) = 3x^2 - 2x + 1 at x = -1, will be 6

In mathematics, an expression is a phrase that has at least two numbers or variables and at least one arithmetic operation. Addition, subtraction, multiplication, or division are all examples of math operations. An expression's structure is as follows: (Number/variable, Math Operator, Number/variable) is an expression.

Algebraic expressions are expressions that are made up of variables and constants. Any value can be assigned to a variable. The value of an expression changes depending on the values assigned to the variables it includes. There are an unlimited number of points on a number line.

To evaluate the function f(x) = 3x^2 - 2x + 1 at x = -1,

we simply substitute -1 for x in the expression:

f(-1) = 3(-1)^2 - 2(-1) + 1

= 3(1) + 2 + 1

= 6

Therefore, f(-1) = 6.

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