use the equation that you wrote in excerise 21 to find the number of vertices of a cube, which has 12 edges and 6 faces

Answer:

The given function is an exponential growth function, not an exponential decay function because as the exponent x increases, the value of y also increases instead of decreasing.

To describe its end behavior using limits, we need to find the limit of the function as x approaches infinity and as x approaches negative infinity.

As x approaches infinity, the exponent -x approaches negative infinity, and the base (1/6) is raised to increasingly larger negative powers, causing the function to approach zero. So, the limit as x approaches infinity is 0.

As x approaches negative infinity, the exponent -x approaches infinity, and the base (1/6) is raised to increasingly larger positive powers, causing the function to approach infinity. So, the limit as x approaches negative infinity is infinity.

Therefore, the end behavior of the function is that it approaches zero as x approaches infinity and approaches infinity as x approaches negative infinity.

The open circle at 3 indicates that 3 is not included in the solution set. This inequality can be read as "X is less than 5" or "X is strictly less than 5."

In mathematics, an expression is a combination of numbers, symbols, and operators that represents a value. Expressions can be as simple as a single number or variable, or they can be complex combinations of mathematical operations. For example, 2 + 3 is a simple expression that represents the value 5, while (2 + 3) x 4 - 1 is a more complex expression that represents the value 19. Expressions can be evaluated or simplified using the rules of arithmetic and algebra.

Here,

1. Simplify:

3(4x-2)+ 7X (2-1) + 4 (6+4)+(-8)

Multiplying inside the parentheses first:

12x - 6 + 7x + 4 + 40 - 8

Combining like terms:

19x + 30

Final answer: 19x + 30

2. Graph:

3 > X

This is a simple inequality in one variable (X). To graph it on a number line, we first draw a dot at 3 (since the inequality is strict), and then shade all values less than 3:

<=========o---

The open circle at 3 indicates that 3 is not included in the solution set.

3. Write the inequality:

X < 5

This is a simple inequality in one variable (X). The inequality sign is "less than," and the number on the right-hand side is 5. This inequality can be read as "X is less than 5" or "X is strictly less than 5."

4. Solve for x:

3x - 7 = 42

Adding 7 to both sides to isolate the variable:

3x = 49

Dividing both sides by 3 to solve for x:

x = 16.33 (rounded to two decimal places)

Final answer: x = 16.3

5. Find 32% of $542.50:

To find 32% of $542.50, we can use the formula:

percent * amount = part

where "percent" is the percentage expressed as a decimal, "amount" is the whole amount, and "part" is the result we're looking for.In this case, we have:

0.32 * $542.50 = part

Multiplying:

$173.60 = part

Final answer: $173.60

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For the standard normal curve, the area to the left of 0.1 is the same as the area to the right of 0.9 is true . So, the correct answer is D.

The standard normal curve is a normal distribution with a mean of 0 and a standard deviation of 1. This curve is often used in statistics to model natural phenomena, and it has many important properties.

Option A is incorrect because the area under the standard normal curve between 0 and 2 is not twice the area between 0 and 1. The area under the curve increases as we move away from the mean, so the area between 0 and 2 will be greater than the area between 0 and 1.

Option B is also incorrect because the area under the standard normal curve between 0 and 2 is not half the area between -2 and 2. The area between -2 and 2 covers more of the curve than the area between 0 and 2, so the area between 0 and 2 will be smaller than half the area between -2 and 2.

Option C is incorrect because the standard normal curve does not have a fixed IQR (interquartile range). The IQR depends on the quartiles of the distribution, which can vary depending on the sample size and the distribution's shape.

Option D is the correct answer because the standard normal curve is symmetric around the mean of 0. This means that the area to the left of any point on the curve is the same as the area to the right of its negative counterpart. Therefore, the area to the left of 0.1 is equal to the area to the right of 0.9.

Therefore, Correct option is D.

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The relative maximum of the function is (0, 4), and the relative minimum is (6, -152).

To find the relative extrema of the function f(x) = x^4 - 8x^3 + 4, we first take the derivative of the function:

f'(x) = 4x^3 - 24x^2

Then we set f'(x) = 0 to find the critical points:

4x^3 - 24x^2 = 0

4x^2(x - 6) = 0

This gives us two critical points: x = 0 and x = 6.

Next, we find the second derivative of f(x):

f''(x) = 12x^2 - 48x

We can use the Second-Derivative Test to determine the nature of the critical points.

For x = 0, we have:

f''(0) = 0 - 0 = 0

This tells us that the Second-Derivative Test is inconclusive at x = 0.

For x = 6, we have:

f''(6) = 12(6)^2 - 48(6) = 0

Since the second derivative is zero at x = 6, we cannot use the Second-Derivative Test to determine the nature of the critical point at x = 6.

To determine whether the critical points are relative maxima or minima, we can use the first derivative test or examine the behavior of the function around the critical points.

For x < 0, f'(x) < 0, so the function is decreasing.

For 0 < x < 6, f'(x) > 0, so the function is increasing.

For x > 6, f'(x) < 0, so the function is decreasing.

Therefore, we can conclude that the critical point at x = 0 is a relative maximum and the critical point at x = 6 is a relative minimum.

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Answer: $2,162.06

Step-by-step explanation:

To calculate the total monthly payment to the bank, we need to add up the monthly mortgage payment, the monthly portion of the semi-annual property tax bill, and the monthly portion of the annual homeowner's insurance bill.

First, we need to find the monthly portion of the semi-annual property tax bill. To do this, we divide the semi-annual property tax bill by 6 (since there are 6 months in half a year):

Monthly property tax payment = Semi-annual property tax bill / 6

Monthly property tax payment = $3,239 / 6

Monthly property tax payment = $539.83

Next, we need to find the monthly portion of the annual homeowner's insurance bill. To do this, we divide the annual homeowner's insurance bill by 12 (since there are 12 months in a year):

Monthly homeowner's insurance payment = Annual homeowner's insurance bill / 12

Monthly homeowner's insurance payment = $980 / 12

Monthly homeowner's insurance payment = $81.67

Now we can add up the monthly mortgage payment, the monthly property tax payment, and the monthly homeowner's insurance payment to find the total monthly payment to the bank:

Total monthly payment = Monthly mortgage payment + Monthly property tax payment + Monthly homeowner's insurance payment

Total monthly payment = $1,711.22 + $539.83 + $81.67

Total monthly payment = $2,332.72

Therefore, the Dotkoms pay the bank $2,332.72 each month.

Answer:

Step-by-step explanation:

Using proportions, it is found that they pay the bank $2332.72 each month.

What is a proportion?

A proportion is a fraction of a total amount.

Using proportions, it is found that they pay the bank $2332.72 each month.

What is a proportion?

A proportion is a fraction of a total amount.

Their payments are given by:

Monthly mortgage of $1,711.22.

Semi-annual property tax bill is $3,239, that is, it is paid every 6 months, hence 3239/6 = $539.83.

Answer:

Step-by-step explanation:

To find the current price index number using the 1975 price of 56.7 cents as the reference value, we can use the formula:

Price Index = (Current Price / Base Price) x 100

Where "Current Price" is the current cost of gasoline, and "Base Price" is the 1975 price of 56.7 cents.

Substituting the values given in the problem, we get:

Price Index = ($2.93 / $0.567) x 100

Price Index = 516.899

Therefore, the current price index number, using the 1975 price of 56.7 cents as the reference value, is 516.899.

Answer:

Step-by-step explanation:

[tex]\{A\cup B \}=\{1,2,3,7,10,11,12,14,16,17,18,19 \}\\\\\{A \cap B \}=\{ 1,7,10,14\}[/tex]

A∪B: Any element which is in either or both sets.

A∩B: Only elements that are in both A and B.

Answer:

a)220 4 leaf clovers, 780 3 leaf clovers b)440 four leaf clovers, 1560 3 leaf clovers

Step-by-step explanation:

100:1000=1:10 22:x=1:10 x=220

220x2=440

1000-220=780

780x2=1560

V is 5.92 centimetres above the pyramid's base at its highest point.

A pyramid is a 3D pοlyhedrοn with the base οf a pοlygοn alοng with three οr mοre triangle-shaped faces that meet at a pοint abοve the base. The triangular sides are called faces and the pοint abοve the base is called the apex. A pyramid is made by cοnnecting the base tο the apex. Sοmetimes, the triangular sides are alsο called lateral faces tο distinguish them frοm the base. In a pyramid, each edge οf the base is cοnnected tο the apex that fοrms the triangular face.

Give the altitude the letter h. Next, we have:

tan(60) = h/2

Simplifying, we get:

h = 2 tan(60) = 2 √(3)

The Pythagorean theorem yields the following:

[tex]$\begin{align}{{V O^{2}+O F^{2}=V F^{2}}}\\ {{V O^{2}+1^{2}=6^{2}}}\\ {{V O^{2}=35}}\end{align}$[/tex]

Taking the square root of both sides, we get:

VO ≈ 5.92 cm

Rounding to three significant figures, we get:

VO ≈ 5.92 cm

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The correct numerical expression for the given statement "multiply the sum of 7 and 6 by the sum of 4 and 5" is (7 + 6) x (4 + 5).

Mathematical statements are called expressions if they have at least two words that are related by an operator and contain either numbers, variables, or both. There are two sorts of expressions in mathematics: numerical expressions, which only contain numbers, and algebraic expressions, which also include variables. A mathematical expression is a phrase that has a minimum of two numbers or variables and at least one mathematical operation.

The given statement, "multiply the sum of 7 and 6 by the sum of 4 and 5", can be represented as:

sum of 7 and 6 = (7 + 6)

sum of 4 and 5 = (4 + 5)

multiply: (7 + 6) x (4 + 5)

Hence, the correct numerical expression for the given statement is (7 + 6) x (4 + 5).

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Answer:

(7 + 6) x (4 + 5).

Step-by-step explanation:

hope this helps!

Answer:

Step-by-step explanation:

To find the zeros of the function, we set y to zero and solve for x:

y = (x+1)(x-2)(x-6) = 0

Setting each factor equal to zero and solving for x gives us the zeros:

x+1 = 0 or x-2 = 0 or x-6 = 0

x = -1, x = 2, x = 6

So the zeros of the function are -1, 2, and 6.

To graph the function, we can use the zeros and the leading coefficient to sketch a rough graph. The leading coefficient is positive, so the graph will open upward. The zeros are -1, 2, and 6, so the graph will intersect the x-axis at those points. We can also find the y-intercept by plugging in x = 0:

y = (0+1)(0-2)(0-6) = 12

So the y-intercept is (0, 12).

Using this information, we can sketch the graph:

Using the two points you have plotted, draw the parabola. It should look like a downward-facing curve opening at the vertex (0, 7).

What is parabola?

A parabola is a symmetrical, U-shaped curve that is formed by the graph of a quadratic function.

Assuming you meant [tex]f(x) = -x^2 + 7[/tex], here's how you can graph the parabola using the parabola tool:

Find the vertex

The vertex of the parabola is located at the point (-b/2a, f(-b/2a)), where a is the coefficient of the [tex]x^2[/tex] term and b is the coefficient of the x term. In this case, a = -1 and b = 0, so the vertex is located at the point (0, 7).

Plot the vertex

Using the parabola tool, plot the vertex at the point (0, 7).

Plot a second point

To plot a second point, you can choose any x value and find the corresponding y value using the quadratic function. For example, if you choose x = 2, then [tex]f(2) = -2^2 + 7 = 3[/tex]. So the second point is located at (2, 3).

Therefore, Using the two points you have plotted, draw the parabola. It should look like a downward-facing curve opening at the vertex (0, 7).

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Complete Question:

Use the parabola tool to graph the quadratic function.

f(x) = -√² +7

Graph the parabola by first plotting its vertex and then plotting a second point on the parabola.

The steps for finding out the third derivative of y (given as y''') are explained and given below.

To find the third derivative of y (y'''), you would need to differentiate the function y three times with respect to the independent variable. Here are the steps:

Start by differentiating y once to get the first derivative, y'.

Differentiate y' again to get the second derivative, y''.

Finally, differentiate y'' to get the third derivative, y'''.

You can use the chain rule, product rule, quotient rule, and other differentiation rules as needed to find each derivative.

Here's an example of finding the third derivative of y for the function y = x^4 + 2x^3 - 5x:

y' = 4x^3 + 6x^2 - 5

y'' = 12x^2 + 12x

y''' = 24x + 12

So the third derivative of y is y''' = 24x + 12.

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Using binomial theorem, the generating function is G(x) = x^4 + 16x^3 + 96x^2 + 256x + 256 while the associated sequence of (x+4)^4 is {1, 16, 96, 256, 256}.

To find the generating function of (x+4)^4, we expand it using the binomial theorem:

[tex](x+4)^4 = C(4,0)x^4 + C(4,1)x^3(4) + C(4,2)x^2(4^2) + C(4,3)x(4^3) + C(4,4)(4^4)[/tex]

where C(n,k) denotes the binomial coefficient "n choose k".

Simplifying the terms, we get:

[tex](x+4)^4 = x^4 + 16x^3 + 96x^2 + 256x + 256[/tex]

Therefore, the generating function of (x+4)^4 is:

[tex]G(x) = x^4 + 16x^3 + 96x^2 + 256x + 256[/tex]

The associated sequence can be read off by finding the coefficients of each power of x:

To find the generating function of (x+4)^4, we expand it using the binomial theorem:

(x+4)^4 = C(4,0)x^4 + C(4,1)x^3(4) + C(4,2)x^2(4^2) + C(4,3)x(4^3) + C(4,4)(4^4)

where C(n,k) denotes the binomial coefficient "n choose k".

Simplifying the terms, we get:

(x+4)^4 = x^4 + 16x^3 + 96x^2 + 256x + 256

Therefore, the generating function of (x+4)^4 is:

G(x) = x^4 + 16x^3 + 96x^2 + 256x + 256

The associated sequence can be read off by finding the coefficients of each power of x:

The coefficient of x^k is the k-th term of the sequence.

In this case, the sequence is given by the coefficients of G(x):

a₀ = 256

a₁ = 256

a₂ = 96

a₃ = 16

a₄ = 1

Therefore, the associated sequence of (x+4)^4 is {1, 16, 96, 256, 256}.

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On solving the question we can say that so the other side of triangle is [tex]B = \sqrt324[/tex], therefore the angle will be [tex]cos^{-1} (0.38)[/tex].

A triangle is a closed two-dimensional geometric object consisting of three line segments, called edges, that intersect at three places called vertices. Triangles are distinguished by their sides and angles. A triangle can be equilateral (all sides equal), isosceles, or odd, depending on the sides. Triangles are classified as acute (any angle less than 90 degrees), right (angles equal to 90 degrees), or obtuse (any angle greater than 90 degrees). The area of ​​a triangle can be calculated using the formula A =​​ (1/2)bh. where A is the area, b is the base of the triangle, and h is the height of the triangle.  

here two sides of the triangle are given that are 19.5 and 7.5

so by

[tex]A^2 = B^2 + C^2\\B^2 = 19.5^2 - 7.5^2\\B^2 = 380.25 - 56.25\\B^2 = 324\\B = \sqrt324[/tex]

so the other side is [tex]B = \sqrt324[/tex], therefore the angle will be [tex]cos^{-1} (0.38)[/tex].

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The difference in length between the two pieces of scrap wood is 7/8 inches.

To get the difference we just need to take the difference between the two lenghs.

Remember that we only have pieces of scraph wood if we have an "x" over the correspondent value in the line diagram.

By looking at it we can see that the longest pice measures 5 inches, while the shortest one (there are two of these) measure (4 + 1/8) inches.

The difference is:

5 - (4 + 1/8) = 7/8

The longest piece is 7/8 inches longer.

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Answer:To simplify the expression, you need to combine the like terms, which are the terms that have the same variable and power. In this case, the like terms are -3a and 5a:

(-3a + 56) + (5a + 40)

= (-3a + 5a) + (56 + 40)

= 2a + 96

Therefore, the simplified expression is 2a + 96.

Enjoy (:

Step-by-step explanation:

When g be continuous on [1,6], where g(1) = 18 and g(6): = 11. Does a value 1 < c < 6 exist such that g(c) = 12

The intermediate value theorem is a fundamental theorem in calculus that states that if a continuous function f(x) is defined on a closed interval [a, b], and if there exists a number y between f(a) and f(b), then there exists at least one point c in the interval [a, b] such that f(c) = y.

According to the intermediate value theorem,

since g(x) is a continuous function on the closed interval [1, 6]

since g(1) = 18 is greater than 12, and

g(6) = 11 is less than 12,

there must be at least one value c between 1 and 6 where g(c) = 12.

Therefore, we can conclude that a value of c does exist such that g(c) = 12.

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Descriptive statistics are used to summarize and describe data, making them useful for providing a clear understanding of a dataset's important features.

When a researcher wants to report the average cost of college tuition from the 1950s until present time, descriptive statistics are the appropriate method to use. Descriptive statistics are used to summarize and describe data, making them useful for providing a clear understanding of a dataset's important features. By using descriptive statistics, the researcher can calculate measures of central tendency, such as the mean, median, and mode, to determine the typical or average cost of college tuition over time. Additionally, measures of variability, such as the range and standard deviation, can be calculated to understand the spread of the data. Descriptive statistics are commonly used in many fields, including business, economics, psychology, and education, and can provide valuable insights into trends, patterns, and distributions within a dataset.

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Answer:

Step-by-step explanation:

If the zeros and their multiplicities are given, we can write the polynomial as the product of linear factors corresponding to each zero.

For this problem, the polynomial has zeros of -9 (multiplicity 1) and -1 (multiplicity 2), so the linear factors are:

(x + 9) and (x + 1)^2

To find the third factor, we use the fact that the degree of the polynomial is 3. We can multiply the linear factors together and then simplify:

(x + 9)(x + 1)^2 = (x^2 + 10x + 9)(x + 1)

= x^3 + 11x^2 + 19x + 9

Therefore, the polynomial with zeros of -9 (multiplicity 1), -1 (multiplicity 2), and degree 3 is:

f(x) = x^3 + 11x^2 + 19x + 9

Continuous numerical values make up the data type for the variable "Age of a tennis player selected at random in the Wimbledon tennis tournament."

Discrete numerical, continuous numerical, and categorical data are the three basic types that can be identified.

- Non-numerical categorical variables, such as gender or eye colour, represent categories or groups.

- Discrete numerical data, such as the number of siblings or pets, are numerical data that can only take on specified values.

Continuous numerical data, like age or weight, are numerical data that can have any value within a range.

Because age can have any value within a range, the data for the variable "Age of a randomly chosen tennis player in the Wimbledon tennis competition" is continuous numerical (for example, a player could be 18.5 years old or 25.2 years old). Hence, continuous numerical data is the right response.

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Answer:

A. To find the derivative of h(x), we can use the chain rule:

h(x) = a(-2x + 1)^5 - b

h'(x) = a * 5(-2x + 1)^4 * (-2) = -10a(-2x + 1)^4

To find the second derivative, we can again use the chain rule:

h''(x) = -10a * 4(-2x + 1)^3 * (-2) = 80a(-2x + 1)^3

B. To show that h is monotonic, we need to show that h'(x) is either always positive or always negative. Since h'(x) is a multiple of (-2x + 1)^4, which is always non-negative, h'(x) is always either positive or negative depending on the sign of a. If a > 0, then h'(x) is always negative, which means that h(x) is decreasing. If a < 0, then h'(x) is always positive, which means that h(x) is increasing.

C. To find the critical points, we need to find where h'(x) = 0:

h'(x) = -10a(-2x + 1)^4 = 0

-2x + 1 = 0

x = 1/2

Thus, the critical point is at x = 1/2. This value is independent of a and b, as neither a nor b appear in the calculation of the critical point.

The scale of the map in the question is 1 inch = 50 miles.

A scale in a map is a relation that tells us how many units each unit in the map represents. In this case, we know that the distance between two points A and B on the map is 5 inches, while the actual distance between these two places is 250 miles.

Then we start with the relation:

5 inches = 250 miles.

But to get the scale of the map we need to see how many miles one inch represents in the map, then we can divide both sides of the equation by 5 to geT:

5 in = 250 mi

1 in = 250mi/5

1 in = 50 mi

The scale of the map is 1 inch to 50 miles.

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The relationship between AB and BC is given as follows:

AB > BC.

Two angles are defined as supplementary angles when the sum of their measures is of 180º.

The supplementary angles for this problem are given as follows:

By the law of sines, we have that:

AB/sin(110º) = BC/sin(70º).

As sin(110º) > sin(70º), the inequality for this problem is given as follows:

AB > BC.

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Answer:

AB>BC

Step-by-step explanation:

AI-generated answer

Based on the given information, the relationship between AB and BC depends on the measure of angle ZADB. If mZADB is 110°, we can determine the relationship as follows:

Since triangle ABD and triangle CBD share side AB, the larger the angle ZADB, the longer the side AB will be compared to side BC. Therefore, if mZADB is 110°, we can conclude that AB is greater than BC.

In summary, when mZADB is 110°, the relationship between AB and BC is:

AB > BC.

The probability of a pitcher throwing exactly 5 pitches in an at-bat is 0.1 or 10%.

What is probability and how is it calculated?

Probability is a measure of the likelihood of an event occurring. It is expressed as a number between 0 and 1, where 0 indicates that the event is impossible and 1 indicates that the event is certain. The probability of an event A is calculated as the ratio of the number of outcomes that correspond to event A to the total number of possible outcomes.

Calculating probability of a pitcher throwing exactly 5 pitches :

To calculate the probability of a pitcher throwing exactly 5 pitches in an at-bat, we need to add up the frequencies of all the at-bats that have exactly 5 pitches. From the given table, we see that there are 8 at-bats that have exactly 5 pitches.

The total number of at-bats is the sum of the frequencies of all pitch counts.

Total number of at-bats = 12+16+32+12+8 = 80

Therefore, the probability of a pitcher throwing exactly 5 pitches in an at-bat is:

P(5) = Frequency of 5-pitch at-bats / Total number of at-bats

P(5)= 8/80 = 0.1 or 10%

Hence, the probability that a pitcher will throw exactly 5 pitches in an at-bat is 10%.

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The surface area is  1305. 96 square units

It is important to note that the formula for calculating the surface area of a cylinder is expressed with the equation;

SA = 2πrh + 2πr²

Given that the parameters are;

Now, substitute the values, we have;

Surface area = 2 × 3.14 × 9 ×14 + 2 × 3.14 × 9²

Multiply the values

Surface area = 791. 28 + 508. 68

add the values

Surface area = 1305. 96 square units

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The Sunday game brought in $2232.50, while the Thursday game brought in $1832.00.

A company's income, costs, and profit are compiled in a profit and loss (P&L) statement, a financial report. It provides information to investors and other interested parties about a company's operations and financial viability.

The issue informs us that the combined revenue from the two games was $4064.50.

S + (S - 400.50) = 4064.50

Simplifying the left side, we get:

2S - 400.50 = 4064.50

Adding 400.50 to both sides, we get:

2S = 4465

Dividing both sides by 2, we get:

S = 2232.50

So the Sunday game generated $2232.50, and the Thursday game generated $2232.50 - $400.50 = $1832.00.

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If we round up, we get x = 12, which means that 12 students took history and 36 students took mathematics.

A Venn diagram is a graphical representation of sets, often used to show relationships between different sets of data. It consists of one or more circles, where each circle represents a set. The circles are usually overlapping to show the relationships between the sets. The region where the circles overlap represents the elements that belong to both sets. The non-overlapping parts of each circle represent the elements that belong to only one of the sets.

Here,

A. To illustrate the data on a Venn diagram, we need to draw two overlapping circles, one for history and one for mathematics. We can label the regions as follows:

The region where the circles overlap represents the students who took both history and mathematics.

The region outside both circles represents the students who did not take either history or mathematics.

The region within the history circle but outside the mathematics circle represents the students who took history but not mathematics.

The region within the mathematics circle but outside the history circle represents the students who took mathematics but not history.

In this diagram, let's assume that x students took history and 3x students took mathematics. Then the total number of students who took the examination is x + 3x = 4x. We can label the regions on the diagram accordingly:

The region where the circles overlap represents the students who took both history and mathematics. Since there are 3x students who took mathematics, and all of them are included in the overlap region, the number of students who took both is 3x.

The region outside both circles represents the students who did not take either history or mathematics. Since there are 46 students in total, and 4x of them took the examination, the number of students who did not take either is 46 - 4x.

The region within the history circle but outside the mathematics circle represents the students who took history but not mathematics. Since x students took history, and 3x took mathematics, the number of students who took history but not mathematics is x - 3x = -2x. However, since we cannot have a negative number of students, we can assume that this region is empty.

The region within the mathematics circle but outside the history circle represents the students who took mathematics but not history. Since there are 3x students who took mathematics, and some of them also took history, the number of students who took mathematics but not history is 3x - 3x = 0. Therefore, this region is also empty.

B. We know that the total number of students who took the examination is 46. Therefore, we have:

x + 3x = 46

4x = 46

x = 11.5

However, since x represents the number of students who took history, it must be a whole number. Therefore, we can round x up or down to the nearest whole number. If we round down, we get x = 11, which means that 11 students took history and 33 students took mathematics.

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Answer:

3x(x - 4y)

Step-by-step explanation:

3x² - 12xy ← factor out 3x from each term

= 3x(x- 4y)