Chapter 6
Lesson 2 skills practice numerical expressions

Each person buys a number of cakes

The term "operation" in mathematics refers to the process of computing a value utilizing operands and a math operator. For the specified operands or integers, the math operator's symbol has predetermined rules that must be followed. In mathematics, there are five basic operations: addition, subtraction, multiplication, division, and modular forms.

Given, Andy buys x cakes, Betty buys 4 times as many cakes as andy, and colin buys 3 more cakes than andy

So,

Andy has = x Cakes

Betty have = 4x Cakes

Clin has = x +3 Cakes

Since each cake costs 65p and the total cost of the cakes is £52.65

thus,

0.65( x + 4x + x +3) = 52.62

x + 4x + x +3 = 81

6x = 78

x = 13

Therefore,

Andy has = 13 cakes

Betty has = 4x = 4*13 = 52 cakes

Colin has = x + 3 = 16 cakes

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The region enclosed by the curve y=3x-x2 and the line y=2x can be rotated about the y-axis to determine the volume of the solid of revolution. The cylindrical shell method can be used to determine this volume.

The cylindrical shell method can be used to determine the volume of the solid of revolution created by rotating the area enclosed by the curve y=3x-x2 and the line y=2x about the y-axis.

We must first determine the region's boundaries.

At x=2 and y=4, y=2x.

When y=3x-x2, x=0, and y=0.

As a result, the region's boundaries are x=0 to x=2 and y=0 to y=4.

The following formula determines the solid's volume:

Volume equals b a 2 y r2 d y

where x is a constant and r is the cylindrical shell's radius.

As a result, the solid of revolution's volume is given by:

Volume = 4.002y x 2.2dy

= 2π ∫4 0 (3x2-x3)dx

= 2π [x3-x4/4]

4 0

= 2π (16/4)

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Answer:

  all are True

Step-by-step explanation:

You want to know if the end behavior is properly described for the polynomial functions shown in the table.

A function of odd degree with a positive leading coefficient has a graph that has a generally positive slope (/).

When the leading coefficient is negative, the slope is generally negative (\). In this case, the sign of the end behavior is opposite the sign of x.

  x → -∞, -2x³ → ∞ . . . . . True

A function of even degree with a positive leading coefficient has a graph that generally opens upward (U). The sign of the end behavior will match the sign of the leading coefficient.

  x → ∞, 2x⁴ → ∞ . . . . . True

This function is of odd degree with a negative leading coefficient. The sign of the end behavior is opposite the sign of x.

  x → ∞, -9x⁵ → -∞ . . . . . True

All the Limit statement are True

If the polynomial functions indicated in the table's final behavior are appropriately explained, you wish to know.

-2x³ at -∞

The graph of an odd-degree function with a positive leading coefficient has a generally upward slope (/).

The slope is often negative when the leading coefficient is negative. The sign of the end behavior is thus the opposite of the sign of x.

x -, -2x3 -∞ are True

2x⁴ at ∞

The graph of an even degree function with a positive leading coefficient typically opens upward (U). The final behavior's sign will coincide with the leading coefficient's.

x, 2x4 ∞ are True

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Their mom squeezed 8 lemons and Elvis and Costello bought 36 lemons.

The equation is defined as mathematical statements that have a minimum of two terms containing variables or numbers that are equal.

Let's represent the number of lemons Elvis squeezed "E", the number of lemons Costello squeezed "C", and the number of lemons their mom squeezed "M".

We know that:

E + C + M = 36 (because they bought 36 lemons in total)

E = (1/3) x 36 = 12 (because Elvis squeezed 1/3 of the lemons)

C = (4/9) x 36 = 16 (because Costello squeezed 4/9 of the lemons)

We can use the first equation to find the value of M:

M = 36 - E - C

Substituting in the values we know:

M = 36 - 12 - 16

M = 8

So their mom squeezed 8 lemons.

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The variable a log of all the ages in years  is best described as quantitative and continuous.

As we know the variables are classified as qualitative and quantitave.

The quantitative variables are further classified as: discrete and continuous.

A discrete variable is a variable that can only take specific numeric values but values have a clear quantitative interpretation.

This variable takes the values that are countable and have a finite number of values.

In case of  continuous quantitative variable, it can take any value in an interval.

The values of  continuous quantitative variable are not countable and they have an infinite number of possibilities.

As we know the age is a discrete variable as it is commonly expressed as an integer.

Here, a demographer they keep a log of all the ages in years.

So, variable is continuous variable.

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The Maclaurin series is xln(2x + 1) - x^2 + (2x^3)/3 - (2x^4)/4 + ... and the arc is (-1/2, 1/2).

From the question, we have the following parameters that can be used in our computation:

f(x) = x ln(2x+1)

The Maclaurin series of f(x) = x ln(2x + 1) can be found by using the power series expansion of the logarithm function and the definition of a derivative.

Starting with the power series expansion of ln(1 + x):

ln(1 + x) = x - (x^2)/2 + (x^3)/3 - (x^4)/4 + ...

We can then substitute x = 2x + 1 - 1 to obtain the power series expansion of ln(2x + 1):

ln(2x + 1) = (2x + 1 - 1) - (2x + 1 - 1)^2/2 + (2x + 1 - 1)^3/3 - (2x + 1 - 1)^4/4 + ...

= 2x - x^2 + (2x^3)/3 - (2x^4)/4 + ...

Next, we use the definition of a derivative to find the Maclaurin series of f(x) = x ln(2x + 1):

f'(x) = d/dx (x ln(2x + 1)) = ln(2x + 1) + x * d/dx ln(2x + 1)

= ln(2x + 1) + x * (2 - 2x + 2x^2 - 2x^3 + ...)

= ln(2x + 1) + 2x - 2x^2 + 2x^3 - 2x^4 + ...

Therefore, the Maclaurin series of f(x) = x ln(2x + 1) is:

f(x) = ∫ f'(x) dx = ∫ (ln(2x + 1) + 2x - 2x^2 + 2x^3 - 2x^4 + ...) dx

= xln(2x + 1) - x^2 + (2x^3)/3 - (2x^4)/4 + ...

The arc of a Maclaurin series refers to the range of values for x for which the series converges to the actual value of the function.

For the Maclaurin series above, the range of convergence depends on the function ln(2x + 1) which has a singularity at x = -1/2.

This implies that the Maclaurin series converges for all x such that |x| < 1/2.

Rewrite as

(-1/2, 1/2)

So, the arc of the series is (-1/2, 1/2).

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Answer:

Step-by-step explanation:

I DONT UNDERSTAND

Answer:

Below

Step-by-step explanation:

Squares have equal side lengths

area = side X side

      = 2.4 m  X  2.4 M = 5.76 m^2

The volume of the pyramid is, 72 cm³.

To multiply means to add a number to itself a particular number of times. Multiplication can be viewed as a process of repeated addition.

We have to given that;

Height of pyramid = 12 cm

Base of pyramid = 12 cm

Weight of pyramid = 14 cm

Now, We know that;

⇒ Volume of pyramid = 1/2 × Base × Height

Substitute all the values, we get;

⇒ Volume of pyramid = 1/2 × 12 × 12

⇒ Volume of pyramid = 6 × 12

⇒ Volume of pyramid = 72 cm³

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The type of sampling used in this example is a. random sampling

One of the most straightforward methods for gathering information from the entire population is random sampling. Every member of the subset has an equal chance of being chosen as part of the sampling process when using random sampling. For instance, if an organization employs 300 people overall, a sample group of 30 workers will be chosen to participate in the survey. In this instance, the population is the entire workforce of the organization, and the sample is 30 employees. Since all of the employees who were picked to participate in the survey were chosen at random, every worker has an equal chance of getting chosen.

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The value of k in the cuboid is 7 cm when the volume is 196cm³.

The space taken up by any three-dimensional solid constitutes a volume, to put it simply. They can be a cube, cuboid, cone, cylinder, or sphere.

The volumes of various shapes vary. In three-dimensional geometry, we have examined a variety of solids and shapes that are defined in three dimensions, including cubes, cuboids, cylinders, cones, etc. We are going to learn to calculate the volume for each of these shapes.

The volume of cuboid = l × b × h

Here,

4 × k × k = 196cm³

k² =  196/4

k² = 49

k = √49

k = 7

Thus, The value of k in the cuboid is 7 cm when the volume is 196cm³.

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The estimate of the integral by parts ∫₀³f(x)g'(x) = 62.1

Integration is the process of reverse differentiation. It is the process of finding the anti-derivative of a function.

Since we have the integral  [tex]\int\limits^3_0 {f(x)g'(x)} \, dx[/tex] and f(x) = x³. To evaluate the integral within the limits, we use integration by parts.

So, for any integral by parts

 [tex]\int\limits^a_b {u(x)v'(x)} \, dx = [u(x)v(x)]^{a} _{b} - \int\limits^a_b {v(x)u'(x)} \, dx[/tex]

So, for the given integral   [tex]\int\limits^3_0 {f(x)g'(x)} \, dx[/tex], we have that

 [tex]\int\limits^3_0 {f(x)g'(x)} \, dx = [f(x)g(x)]^{3} _{0} - \int\limits^3_0 {g(x)f'(x)} \, dx[/tex]

So, we have that

 [tex]\int\limits^3_0 {f(x)g'(x)} \, dx = [f(x)g(x)]^{3} _{0} - \int\limits^3_0 {g(x)f'(x)} \, dx\\= [f(x)g(x) - f(0)g(0)] - [{g(3)f'(3) - g(0)f'(0)}][/tex]

Now, we have that f(x) = x³. So,f'(x) = 3x².

So,

Also, form the table

Substituting these into the equation, we have that  

[tex]\int\limits^3_0 {f(x)g'(x)} \, dx = [f(3)g(3) - f(0)g(0)] - [{g(3)f'(3) - g(0)f'(0)}]\\= [27 \times 4.9 - 0 \times 2.3] - [{4.9 \times 27 - 2.3\times 27}]\\= (132.3 - 0) - (132.3 - 62.1)\\= 132.3 - 132.3 + 62.1\\= 0 + 62.1\\= 62.1[/tex]

So, integral [tex]\int\limits^3_0 {f(x)g'(x)} \, dx = 62.1[/tex]

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To purchase the Apple iPad Pro, based on an equation, Sandra contributed $257 while her parents contributed $643.

An equation is a mathematical statement that states that two or more mathematical expressions are equal or equivalent.

Mathematical expressions are depicted as equations using variables, constants, numbers, and values and the equation symbol (=).

The total cost of the Apple iPad Pro = $900

The contribution of Sandra's parents per $2 = $5

The contribution of Sandra = $2

Let x = the variable contributed by each party

2x + 5x = 900

7x = 900

x = 128.57

Sandra's contribution = $257 (2 x 128.57)

Parents contribution = $643 (5 x 128.57)

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Answer:

  11

Step-by-step explanation:

You have an average of 7 on 3 tests, and you want to know the score required of the 4th test to make the average be 8.

The average is the sum, divided by the number.

If you have an average of 7 on the first 3 tests, the sum of those scores is ...

  average = sum/number

  7 = sum/3

  21 = sum

If you want an average of 8 on 4 tests, the sum of the scores must be ...

  8 = sum/4

 32 = sum

In order to increase the sum of scores from 21 to 32, you must have a score of ...

  32 -21 = 11

You need to score 11 on the 4th test to bring the average to an 8.

__

Alternate solution

The difference between the scores you have and the average you want is ... 3(7 -8) = -3.

In order to bring your average up to 8, you need to have a total of 3 more points than that average on your 4th test: 8 + 3 = 11. (This brings the total of differences from average to zero: (-1) +(-1) +(-1) +(3) = 0.)

Your 4th test must have a score of 11 to bring your average on 4 tests up to 8.

Answer: 3.4c - 1

Step-by-step explanation:

multiply each part within the parentheses by the negative sign

so -3.4c x -1 = 3.4c

and 1 x -1 = -1

add those two together and we get 3.4c - 1

The true statements are the study does not involve a voluntary response, the first stage creates a stratified random sample and the whole study creates a multistage random sample.

In the given question, the study done ideally creates a multistage random sample. There are two parts to the sampling process for the study in the given question. Initially, a combined sample of thousand of individuals was assembled by combining a randomly selected sample of fifty individuals, that were from each of the 20 regions of each nation. The next step was to choose a random sample of fifty individuals from every one of the five income groups represented among those 1,000 individuals.

Additionally, because the study doesn't entirely rely on a discretionary answer, individuals were picked by a random sample approach rather than explicitly or through self-selection. A stratified random sample is also created during the initial step. This shows that a random sample was drawn from each subgroup after the dataset was separated into groups. 

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Answer:

  a. D1AF

  b. 1F

  c. 1

  d. EDB2

Step-by-step explanation:

You want these unsigned binary numbers converted to hexadecimal.

  a. 1101 0001 1010 1111

  b. 0001 1111

  c. 0001

  d. 1110 1101 1011 0010

Each group of 4 bits can take on any of 16 different values. Conveniently, each of those corresponds to a hexadecimal digit, as shown in the attachment.

To do the conversion, we replace each group of 4 bits by its hexadecimal equivalent.

  a. D1AF

  b. 1F

  c. 1

  d. EDB2

__

Additional comment

A couple of the given numbers are one or more bits short of a group of 4 bits. You may want to check your actual problem statement to see if the numbers here are the same. The attached table applies in any event.

Answer:

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

We know that the equation for the time is:

Find the time at travel in the first day:

Find the time at travel in the second day:

To find the total time add app the two expressions we got above:

There matching choice is C.

Answer: x=-13, y=-23

Step-by-step explanation:

y in terms of x

-2x+y=3

y=3+2x

Substitute in other equation:

5x-4(3+2x)=27

5x-12-8x=27

-3x=39

x=39/-3

x=-13

Solve for y, substitute for x:

-2(-13)+y=3

26+y=3

y=-23

Answer:

see below

Step-by-step explanation:

1. study hard and go to a good college and then find a good job.

2. be intelligent on spending and save money to invest in the financial markets

3. buy a few life insurance policies with living benefits.

4. find a niche on social media and create contents to attract followers, and then invest the earnings in stock market or bond market.

5. invest in real estate

6. start a small business and then grow into franchise.

The expression \root(4)(10mn) and (10mn)^((1)/(4)) are equivalent

Index forms are described as that number written in the form of an exponential expression.

It is also described as a single number raised to another number.

The exponent of an exponential expression explains how many times to multiply the base by itself to simply the expression.

Standard forms and scientific notation are other names for index forms.

It is important to note that the fourth root of a number or variable is  the biquadratic root.

From the information given, we have that;

[tex]\sqrt[4]{10mn}[/tex]

[tex](10mn)^1^/^4[/tex]

Note the following;

√2 = 2^1/2

∛2 = 2^1/3

Hence, the expressions are equivalent

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The equation of the plane passing through the point (-8,3,7) and parallel to the given plane will be 4x + 8y - 2z = -22.

The given equation of the plane is 4x + 8y - 2z = 45.

To find the equation of the plane which passes through the point (-8,3,7) and is parallel to the given plane, we need to calculate the values of x, y and z in the given equation.

We have x = -8, y = 3 and z = 7 in the given point.

Substituting these values in the given equation, we get

4(-8) + 8(3) - 2(7) = 45

-32 + 24 - 14 = 45

-22 = 45

This is not possible.

Hence, the equation of the plane passing through the point (-8,3,7) and parallel to the given plane will be 4x + 8y - 2z = -22.

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The slope of the line of the equation y - 12 = 3/4(x + 7), is: 3/4.

The point-slope form of a linear equation is given as y - y1 = m(x - x1), where m is the slope of the line, (x1, y1) is a point on the line, and y and x are the variables.

To identify the slope of a line in point-slope form, you can simply look at the coefficient of x, which is the value of m. In the equation y - y1 = m(x - x1), m is the slope of the line.

This value tells you the rate of change or steepness of the line. A positive slope value implies that the line is increasing and a negative value implies that the line is decreasing.

Given the equation of a line in point-slope form as y - 12 = 3/4(x + 7). Therefore, the slope (m) can be identified as 3/4.

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Yes, Mabel is correct. She correctly followed all the steps and arrived at the correct answer.

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In a synthetic polynomial , the real solution of 2 is found; the year is 1996, and $20,300,000,000 was sold.

a. 20,300,000,000 = -10t^3 + 140t^2 - 20t + 18,150

b. Suggestions for the problem: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

b. To solve, use synthetic division; the solution is 2; the year is 1996.

In order to discover the year in which the specified quantity of sporting equipment was sold, we must create a polynomial equation. E(t) = -10t3 + 140t2 - 20t + 18,150, where t is the number of years since 1994, is the equation given in the problem. We must rewrite the equation so that the right side equals 20,300,000,000 in order to determine the year in which $20,300,000,000 worth of sporting equipment was sold. After that, the equation is 20,300,000,000 = -10t3 + 140t2 - 20t + 18,150, which needs to be solved. The equation can have whole-number solutions that are less than 10, including 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. We may use synthetic division to identify which of these answers is the true solution. this method, the coefficients of the equation are divided by the coefficient of the equation with the highest degree, and the quotient is then multiplied by the solution. Until the last portion is identified, this process is repeated. The answer is the quantity that equals the remainder.

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The unknow angle in the isosceles triangle is as follows:

m∠C = 80 degrees.

An isosceles triangle is a triangle that has two of its sides equal to each other and also have two if its angles equal to each other.

The base angles of the isosceles triangles are equal and the legs of the triangle are equal.

Therefore, in ΔABC

AB = BC

Hence,

m∠A ≅ m∠C

The angle A and angle C are the base angles. This makes them congruent to each other.

Therefore,

m∠C = 80 degrees

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The value of the sigma notation is 55924050

From the question, we have the following parameters that can be used in our computation:

sigma1^12 10×4^n-1

Express properly

So, we have the following representation

[tex]\sum^{12}_{1} 10*4^{n-1[/tex]

The sum of the geometric series can be calculated using the formula:

S = a(1 - r^n)/(1 - r)

From the sigma notation, we have the following parameters

a = 10,

r = 4

n = 12.

Substituting these values, we get:

S = 10 * (1 - (4)^12) / (1 - 4)

Evaluate the expression

S = 55924050

Hence, the sum is 55924050

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The probability that she will make a basket on or before her last attempt within her average waiting time are 1/8 and 2/8

Probability of an event is a measurement of how likely an event can occur as an outcome of a random experiment;

Probability ranges from 0 to 1, both inclusive. Events whose probability is closer to 0 are rarer to occur than those whose probabilities are closer to 1 (relatively);

When converted to percentage, we just need to multiply its decimal representation by 100. In percentage form, the probability ranges from 0% to 100%;

Given:

Hannah makes 5 out of 8 attempted free-throws.

Now,

The average waiting time of hannah to make her first basket is 5/8

The probability of hannah making basket at her last attempt is 1/8

The probability of hannah making basket before her last attempt is 2/8

Hence, the probabilty will be 1/8 and 2/8;

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The mistake of Elena is that the powers of the terms were not the same and the correct result is 3.83 ×10^6.

We have to note that the addition of the exponents is not just like the addition of the ordinary numbers that we know. The operation of the addition of the exponents would require an extra care to ensure that the powers of the two terms to be added is exactly the same.

In the work of Elena; (2.3×10^5)+(3.6×10^6)=5.9×10^6. This is wrong because the powers of the terms on the right hand side are not the same. As such the correct approach would be;

0.23 × 10^6 + 3.6×10^6 = 3.83 ×10^6

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