Simplify (10 + 3i)(5 – 2i) ...?

Answer:

Area and volume of pentagonal prism is 660 inches² and 1080 inches³

Step-by-step explanation:

Given : A pentagonal prism having  side length of the base is 6 inches, and the apothem of the base is 4 inches. The height of the prism is 18 inches.

We have to calculate

1) area of a cross-section of the prism

2) the volume of the prism

1) Area of cross section of pentagonal prism =5ab +5bh

Where a = apothem

b - base length

h = height.

For the given pentagon

apothem = 4 inches

base = 6 inches

Height = 18 inches

Area of cross section of pentagonal prism =5 ×4 × 6 + 5× 6 × 18

Area of cross section of pentagonal prism = 120 + 540

Area of cross section of pentagonal prism = 660 inches²

2) Volume of pentagonal prism = [tex]\frac{AP}{2} \times H[/tex]

Where, A = apothem

P is perimeter of base

H= height

Perimeter of pentagon is 5b = 30 inches

Volume of pentagonal prism = [tex]\frac{4\times 30}{2} \times 18[/tex]

Volume of pentagonal prism = 1080 inches³

Thus, Area and volume of pentagonal prism is 660 inches² and 1080 inches³.

Final answer:

To find the scale factor for enlarging a children's book from 20 cm by 24 cm to 25 cm by 30 cm, divide the enlarged dimension by the original corresponding dimension, resulting in a scale factor of 1.25.

Explanation:

We are asked to find the scale factor used to enlarge a children's book from dimensions of 20 cm by 24 cm to a larger version with dimensions of 25 cm by 30 cm. To find the scale factor, we need to divide one dimension of the enlarged book by the corresponding dimension of the original book. We can take either the length or the width to find the scale factor.

Let's use the width for this example:

Original width = 20 cm
Enlarged width = 25 cm

Scale factor = Enlarged width / Original width
Scale factor = 25 cm / 20 cm

Scale factor = 1.25

Therefore, a scale factor of 1.25 should be used to enlarge the book from the original dimensions to the new dimensions.

To solve for b in the equation A = (a + b + c) / 3, multiply both sides by 3 and then subtract a and c from both sides. The correct answer is A) 3A - a - c = b.

To solve for b in the equation A = (a + b + c) / 3, you need to isolate the variable b.

First, multiply both sides of the equation by 3 to eliminate the denominator:

3A = a + b + c

Then, subtract a and c from both sides to get b by itself:

3A - a - c = b

So, the correct answer is:

A) 3A - a - c = b

To factor the polynomial 9x^3 + 18x^2 − x − 2, group terms to factor out common factors and then factor further if possible. The factored form is (x + 2)(3x + 1)(3x - 1).

To factor the polynomial 9x3 + 18x2 − x − 2, we look for common factors and use techniques such as grouping. First, we can try to group the terms in pairs and factor out the greatest common factor from each pair.

Let's group the first two terms and the last two terms separately:

From the first group, we can factor out 9x2 and from the second group, we can factor out -1:

We now have a common factor of (x + 2) that we can factor out:

The expression 9x2 - 1 is a difference of squares, which can be factored further:

So, the fully factored form of the given polynomial is (x + 2)(3x + 1)(3x - 1).

The polynomial 9x^3 + 18x^2 - x - 2 can be factored by grouping and by recognizing the difference of squares, resulting in the factors (x + 2)(3x + 1)(3x - 1).

The student has asked to factor the polynomial 9x3 + 18x2 - x - 2. Factoring polynomials is a process of expressing a polynomial as a product of its factors, which can involve numbers, variables, or both. This can make the expression simpler or more useful for further mathematical operations such as solving equations.

To begin factoring 9x3 + 18x2 - x - 2, we look for a common factor in each term. Here, there is no common factor, so we attempt to factor by grouping. We group terms that can potentially have common factors or that can be factored further.

Let's separate the polynomial into two groups:

Our two groups now look like this:

Because the (x + 2) is a common factor in both groups, we can factor it out:

(x + 2)(9x2 - 1)

The second term, 9x2 - 1, is a difference of squares which can be factored as (3x + 1)(3x - 1).

Finally, the completely factored form of the given polynomial is:

(x + 2)(3x + 1)(3x - 1)

The denominator of the fraction is -2x.

The correct answer is option (b)

"It is a mathematical statement which consists of equal symbol between two algebraic expressions."

"In a fraction, the value placed above the horizontal line."

"In a fraction, the value placed below the horizontal line."

For given question,

The axis of symmetry for a quadratic equation can be found using the formula [tex]x=-\frac{b}{2a}[/tex]

where a and b are coefficients in the quadratic equation and x represents the values along a vertical line on the coordinate plane.

[tex]\Rightarrow x=\frac{-b}{2a}\\\\ \Rightarrow \frac{a}{x} \times x=\frac{-b}{2a}\times \frac{a}{x}\\\\\Rightarrow a=\frac{b}{-2x}[/tex]

For the fraction [tex]\frac{b}{-2x}[/tex] the numerator is b and denominator is -2x.

Therefore, the denominator of the fraction is -2x.

The correct answer is option (b)

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Answer:

Option B is correct

The average rate of change of d(t) between 2 second and 4 second is; 90 ft/s

and it represents the average speed of the object between 2 seconds and 4 seconds.

Step-by-step explanation:

Average rate of change of function is defined as the ratio of the difference in the function f(x) as it changes from a to b to the difference between a and b. Then, the average rate of change is denoted as A(x).

[tex]A(x) =\frac{f(b)-f(a)}{b-a}[/tex]

As per the given statement, the distance d(t) is in feet and t is the time in second.

To find the average rate of change of d(t) between 2 seconds and 4 seconds.

From the table we have;

at t = 2 , d(2) = 60

and

at t =4 , d(4) = 240.

Then, by the definition of average rate of change ;

[tex]A(t) = \frac{d(4)-d(2)}{4-2}[/tex] = [tex]\frac{240-60}{4-2} =\frac{180}{2}[/tex]

Simplify:

[tex]A(t) = 90 ft/s[/tex]

therefore, the average rate of change of d(t) between 2 second and 4 second is; 90 ft/s and it represents the average speed of the object between 2 seconds and 4 seconds.

rx+(9/5)=h 

rx=h-(9/5) 

x=(h-(9/5))/r

Answer:

y=5x - 4.5

Step-by-step explanation:

I took the test

To determine the side length of a Rubik's cube with a volume of 384 cubic centimeters, take the cube root of 384 to find that each side is approximately 7.24 centimeters long.

To find the length of one side of a cube when given the volume, you can use the formula for the volume of a cube, which is V = L³, where V is the volume and L is the length of a side of the cube.

In this case, the volume is 384 cubic centimeters, so we need to find the cube root of 384 to find the length of one side.

The calculation is as follows:

Therefore, each side of the Rubik's cube is approximately 7.24 centimeters long.

A

first option is correct

The answer is $112.86

Answer:

-3 ----- C

Step-by-step explanation:

Answer:

Option (c) is correct.

The system is Inconsistent.

Step-by-step explanation:

Given : A pair of lines.

We have to identify the system by type.

Consider the given system of pairs of lines.

Since, the graph shows two parallel lines.

1) Consistent :  A system of linear equation is said to be consistent if the graph of equation either intersect at a single point or two lines overlap each other.

that is a unique solution or infinite many solution.

2) equivalent :  When two system of equations have same solution then the two system are said to be equivalent.

3) Inconsistent : A system of linear equation is said to be inconsistent if the graph of equation are parallel to each other.

Thus, the given graph shows parallel lines,

Hence, The system is Inconsistent

The company could have mowed more than 54 lawns.

Let the number of lawns mowed be x.

Then the total earning of landscaper is 30x.

The total amount given to the employees in a week is 210*7=$1470.

Then the profit of landscaper would be 30x-1470 which is greater than $150.

30x-1470 > 150

30x > 1620

x > 54

Therefore, possible numbers of lawns the company could have mowed are greater than 54 lawns.

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Between 61 and 80 lawns could have been mowed by the business in a week in order to turn a profit of more than $150. So options (d) and (e) are correct.

To determine the possible numbers of lawns the company could have mowed, we need to calculate the profit for a 7-day week and compare it with the given condition that the profit exceeds $150.

Step 1: Determine daily and weekly expenses

The daily salary expense is $210.

For 7 days, the total salary expense is:

[tex]\[ 210 \times 7 = 1470 \text{ dollars} \][/tex]

Step 2: Calculate profit per lawn

The company earns $30 for each lawn mowed.

Let ( n ) be the number of lawns mowed in a week.

Step 3: Calculate total earnings from mowing lawns

Total earnings for ( n ) lawns is:

[tex]\[ 30n \text{ dollars} \][/tex]

Step 4: Determine the profit

Profit is total earnings minus total expenses:

[tex]\[ \text{Profit} = 30n - 1470 \][/tex]

Step 5: Apply the condition that profit is more than $150

The inequality to solve is:

[tex]\[ 30n - 1470 > 150 \][/tex]

Step 6: Solve the inequality

Add 1470 to both sides:

[tex]\[ 30n > 1620 \][/tex]

Divide both sides by 30:

[tex]\[ n > 54 \][/tex]

Step 7: Identify the possible number of lawns mowed

The number of lawns must be greater than 54.

From the given options (12, 37, 54, 61, 80), the possible numbers are:

[tex]\[ 61 \text{ and } 80 \][/tex]

Thus, the two possible numbers of lawns the company could have mowed to make a profit of more than $150 in a week are 61 and 80.

Complete Question:

A landscaper earns $30 for each lawn her company mows, but she pays $210 per day in salary to her employees. If her company made more than $150 profit from mowing lawns in a 7-day week, what are the possible numbers of lawns the company could have mowed? Select two options. 12

37

54

61

80

Answer with explanation:

Year                                                 Population

1992                                                   628

1994                                                   656→(628+28)

1996                                                  692→(656+36)

1998                                                  751→(692+59)

2000                                                 793→(751+42)

2002                                                 840→(793+47)

2004                                                 868→(840+28)

2006                                                 902→(868+34)

2008                                                 940→(902+38)

There is no fixed pattern the rate at which population is increasing.But if you will look at the way the population is increasing , the range of the number is from [28,59].

⇒Most appropriate prediction for the population in 2012 will be

Option A : 1000

Answer:

The answer is : $67,266.16

Step-by-step explanation:

This question is a future value question.

As given, Trent puts away 10% of his $32,000 every year and the company will put half amount of $1,600. So, total amount becomes =[tex]3200+1600=4800[/tex] each year.

Formula for future value or FV is Fv = Pmt (1 + r/n)^(nt) – 1 / (r/n)

Where, Fv = Future Value , Pmt = repeated payments , R = interest rate ,N = total number of payment periods

Putting the values in the formula,

Fv = 4800 (1 + 0.073/1)^(1x10) – 1 (0.073/1)

Fv = $67266.16

Hence, the answer is $67266.16

The amount saved in 10 years is [tex]\boxed{\$ 67266.20}.[/tex]

Further Explanation:

Annuity is a series of payment that is made after equal interval of time.

Future value of annuity of payment P for n year if the return is i can be expressed as,

[tex]\boxed{{\text{Future Value}} = {\text{P}} \times \frac{{{{\left( {1 + \frac{i}{n}} \right)}^{nt}}}}{{\frac{i}{n}}}}[/tex]

Given:

Total salary is [tex]\$ 32000.[/tex]

The interest rate is [tex]7.3\%.[/tex]

Trent save [tex]10\%[/tex] of his salary every year.

Company puts [tex]5\%[/tex] of salary every year.

Calculation:

The [tex]10\%[/tex] of [tex]\$32000[/tex] can be calculated as follows,

[tex]\begin{aligned}{\text{Amount}}&= \frac{{10}}{{100}} \times 32000\\&= \$ 3200\\\end{aligned}[/tex]

The amount that company put can be obtained as follows,

[tex]\begin{aligned}{\text{Company Amount}} &= \frac{5}{{100}} \times 32000\\&= \$ 1600\\\end{aligned}[/tex]

The total amount can be calculated as follows,

[tex]\begin{aligned}{\text{Amount}} &= 3200 + 1600\\&= \$ 4800\\\end{aligned}[/tex]

The future value can be obtained as follows,

[tex]\begin{aligned}{\text{FV}} &= 4800 \times \frac{{{{\left( {1 + 0.073} \right)}^{10}} - 1}}{{0.073}} \\ &= 4800 \times 14.014\\&= \$ 67266.20\\\end{aligned}[/tex]

Hence, the amount saved in 10 years is [tex]\boxed{\$ 67266.20}.[/tex]

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Answer details:

Grade: High School

Subject: Mathematics

Chapter: Investment and return

Keywords: Trent, 25 years old, works, company, matches, 401(k), 7.3%, 10%. $32000, salary, every year, 10 years, amount, 5%, nearest cent.