7-2 Enrichment
Multiplying Polynomials
An architect is preparing to design an office building. She creates a sketch of the basic shape of the structure and
the relative dimensions of each piece shown. The angles in the sketch are all right angles. Find the volume of the structure using each approach. Draw lines on the sketches and record your expressions.

1. Divide the structure into two different rectangular
prisms. Add the volumes of the two prisms to create
an expression for the total
Volume

2. Divide the structure into three different rectangular
prisms. Add the volumes of the three prisms to create
an expression for the total volume.

3. View the structure as one prism with 2 equal sized prisms. Find the volume of the large prism and subtract the volumes of the smaller pieces to find the expression for the total volume!!

Thanks for helping!! (We are learning polynomials right now, like (2+x)^2 for clarification for the problems!!!) :)

Answer:

A

Step-by-step explanation:

since the 2 triangles are similar then the ratios of corresponding sides are in proportion.

let h represent the height of the tree , then

[tex]\frac{h}{5}[/tex] = [tex]\frac{24}{8}[/tex] = 3 ( multiply both sides by 5 )

h = 3 × 5 = 15

height of tree is 15 feet

Answer:

Slope is -5/3

Step-by-step explanation:

Let [tex](x_1,y_1)\rightarrow(-8,7)[/tex] and [tex](x_2,y_2)\rightarrow(-2,-3)[/tex]:

[tex]\displaystyle m=\frac{y_2-y_1}{x_2-x_1}\\\\m=\frac{-3-7}{-2-(-8)}\\ \\m=\frac{-10}{-2+8}\\\\m=-\frac{10}{6}\\ \\m=-\frac{5}{3}[/tex]

The depth of the submarine after 9 minutes will be -102 feet.

Expression in maths is defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.

Numbers (constants), variables, operations, functions, brackets, punctuation, and grouping can all be represented by mathematical symbols, which can also be used to indicate the logical syntax's order of operations and other features.

Given that the expression -210+ 12m represents a submarine that began at a depth of 210 feet below sea level and ascended at a rate of 12 feet per minute.

The depth of the submarine after 9 minutes will be calculated as:-

D =  -210+ 12m

D = -210 + ( 12 x 9 )

D = -210 + 108

D = -102 feet

Therefore, the submarine's depth will be -102 feet after 9 minutes.

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The lateral surface area of the cylinder would be 157.78π.

        LSA = 2πr(h + r)                               .... Eq { 1 }

       dV = πr² dh

       For complete cylinder -

       ∫dV = πr² ∫dh

       V - 0 =  πr² [h - 0]

       V = πr²h                                           .... Eq { 2 }

Given is a cylinder with a height of 11.2 cm and radius of 4.9 cm.

We can write the lateral surface area of a cylinder as -

LSA = 2πr(h + r)  

LSA = 2π x 4.9 x (11.2 + 4.9)

LSA = 2π x 78.89

LSA = 157.78π

Therefore, the lateral surface area of the cylinder would be 157.78π.

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Correct answer to the given question about Courtney's affordable car loan within her budget is option D. $27,425. 75.

To determine the maximum car loan that Courtney can afford, you need to calculate the total amount she will pay over 6 years and make sure it's less than the total amount she can afford to pay each month ($400) multiplied by the number of months (6 years x 12 months per year = 72 months).

First, calculate the total amount of interest she will pay over 6 years at 7.5% annual interest: $400 x 7.5% = $30

Next, calculate the monthly payment by dividing the annual interest amount by 12 months:$30 / 12 = $2.50

So, the monthly payment (not including the car loan amount) is $400 + $2.50 = $402.50.

Finally, you can calculate the maximum car loan amount by subtracting the monthly payment from the total amount she can afford to pay each month, and then dividing by the number of months:

$400 x 72 months = $28,800

$28,800 / $402.50 = 71.37

So, Courtney can afford a maximum car loan of $71,370.

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(1) The amount of interest he paid in the first year altogether is 302,000.

(2) The monthly installment is 25,166.67 per month

(1) The amount of interest he paid in the first year is 1.8 million x 9% = 162,000.

So, in the first year, he paid a total of 162,000 + 140,000 = 302,000.

(2) To calculate the monthly installment, we first need to find out the total amount he paid in a year and then divide it by 12:

302,000 ÷ 12 = 25,166.67

So, he paid 25,166.67 per month.

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Answer:

Step-by-step explanation:

You want to know the total paid during the first year of a $1.8M loan at 9% if the balance is reduced by $140,000 after 12 monthly payments. You also want to know the amount of the monthly payment.

The remaining balance of an amortized loan is given by the formula ...

  FV = PV(1 +r)^n - <future value of payments>

where r is the monthly interest rate, and n is the number of months.

Using the given loan values, we have ...

  1,800,000 -140,000 = 1,800,000(1 +0.09/12)^12 -<payment value>

Solving for the future value of the payments, we find ...

  <payment value> = 1,800,000(1.093806898) -1,660,000 = 308,852.42

This is the future value of the payments made, so some further calculation is required to find the amount of the payment.

  <payment future value> = P((1+r)^n -1)/r = P(1.0075^12 -1)/0.0075

  P = <payment future value>/12.5075864

  P = 308,852.42/12.5075864 = 24693.21

The amount of the monthly payment was $24,693.21.

12 monthly payments were made in the first year, so the total paid was ...

  12 × $24,693.21 = $296,318.52 . . . . total paid in the first year

__

Additional comment

The attachment confirms these numbers. It is the output of an online remaining balance calculator. The difference of $0.04 between the bottom two numbers is a consequence of rounding. The given payment actually reduces the balance by $140,000.04.

Answer:

602 .6

Step-by-step explanation:

∴14,400=160b

⇒b=90m

902m=45m

2πr=2π×45≈282.6m

320m+282.6m=602.6m

The value of P in this triangle is equal to 14.

In Mathematics, the basic proportionality theorem states that when any of the two (2) sides of a triangle is intersected by a straight line which is parallel to the third (3rd) side of the triangle, then, the two (2) sides that are intersected would be divided proportionally and in the same ratio.

By applying the basic proportionality theorem to the given triangle, we have the following:

24/p = 12/(21 - p)

By cross-multiplying, we have the following:

12p = 24(21 - p)

12p = 504 - 24p

36p = 504

p = 504/36

p = 14.

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Units of volume are cm³ and cubic meter.

Volume is the measure of the capacity that an object holds.

A unit of volume is a unit of measurement for measuring volume or capacity, the extent of an object or space in three dimensions.

For example, if a cup can hold 100 ml of water up to the brim, its volume is said to be 100 ml.

Given units of measurement,

G² - It is not unit of volume

cm³- It is unit of volume

m² - It is not unit of volume

kg³ - It is not unit of volume

cubic meter = meter³ - It is unit of volume

Hence, cm³ and cubic-meter are units of volume.

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The answers to questions 2, 3, and 4 can be used to assess the effectiveness of a technique after a procedure is finished.

Were the steps completed in order?

This is a crucial question to ask since it reveals the steps in the process and their chronological order, allowing the processes to be adjusted as necessary in the future.

What could be done differently in the future?

It is crucial since it can identify existing problems as well as those that can develop in the future, and it will aid in resolving them.

Was the expected outcome reached?

The most crucial question to ask is what result you want the process to produce; if not, you'll need to adjust the phases and the process.

Evaluation: Evaluation is the methodical conclusion of any topic, problem, or subject using predetermined standards or criteria. To judge their worth or value, the step and the methods are analyzed and assessed.

Therefore, the answers to questions 2, 3, and 4 can be used to assess the effectiveness of a technique after a procedure is finished.

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The volume of the sphere is 1437.3 cm³.

A sphere is a geometric shape with a perfectly round shape composed of an infinity of circles that have the same radius and center of the circle. Literally "ball" comes from the Greek word "globe" or "ball". Then in English the ball is called "sphere".

Properties of the Sphere

To find the volume of the sphere we use formula:

V=4/3. πr³

To calculate the sphere volume above with r=7cm

V=4/3. πr³

V=4/3.(22/7). 7³

V=1437.3 cm³

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The percentage of No purchase is 26%, the computer is 59.9% and the printer is 22%.

The percentage is defined as representing any number with respect to 100. It is denoted by the sign %. The percentage stands for "out of 100." Imagine any measurement or object being divided into 100 equal bits.

Given that he collected the following data during a recent week: 327 customers visited the store, 196 purchased computers, 72 purchased printers, and 87 made no purchase.

The percentage of No purchase is:-

P = 87 / 327

P = 26%

The percentage of computer purchases:-

P = 196 / 327

P = 59.9%

The percentage of printer purchases is:-

P = 72 / 327

P = 22%

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The store pays $842.00 for the crystal sculpture and marks the price up by 29%,  then the amount of the markup is $243.98.

What is the arithmetic operation?

Arithmetic operations is a branch of mathematics, that involves the study of numbers, the operation of numbers that are useful in all the other branches of mathematics. It basically comprises operations such as Addition, Subtraction, Multiplication, and Division.

we have given,

The store pays $842.00 for the crystal sculpture and marks the price up by 29%,

so, the amount of the mark-up can be calculated as follows:

$842.00 * 29% = $842.00 * 0.29 = $243.98

Therefore, the amount of the markup is $243.98.

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Correct question:

A store pays $842.00 for a crystal sculpture and mark the price by 29%. What is the amount of the mark-up?

The store pays $842.00 for the crystal sculpture and marks the price up by 29%,  then the amount of the markup is $243.98.

A subfield of mathematics known as arithmetic operations deals with the study and manipulation of numbers, which are essential to all other subfields of mathematics. Basic operations like addition, subtraction, multiplication, and division are included in it.

The four mathematical operations are functions that take input values, or numbers, as inputs and turn one number into another. They are: multiplication, division, addition, and subtraction.

According to contemporary mathematicians, the Riemann Hypothesis is possibly the most significant open problem in all of mathematics. One million dollars will be awarded to the person who can solve it, which is one of the seven Millennium Prize Problems.

we have given,

The store pays $842.00 for the crystal sculpture and marks the price up by 29%,

so, the amount of the mark-up can be calculated as follows:

[tex]842.00 \times 29% \\= $ 842.00 \times 0.29 \\= $243.98[/tex]

Therefore, the amount of the markup is $243.98.

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Correct question:

A store pays $842.00 for a crystal sculpture and mark the price by 29%. What is the amount of the mark-up?

The pair of equivalent radical expression is option D. √180 and 3√20.

Radicals are defined as the nth root of a number using the symbol  [tex]\sqrt[n]{x}[/tex].

This is the opposite form of the exponential functions.

[tex]\sqrt[n]{x}[/tex] can be written in the exponential form as [tex]x^{1/n[/tex].

A.√45 can be written as the square root of (15 × 3).

√45 = √(15 × 3) ≠ 3√15

B. √75 cannot be written as 3√5, since, if we have to take 3 outside, inside he root there must be 3² = 9. But 75 is not a multiple of 9.

C. √54 = √(9 × 6) = 3√6 ≠ 9√6

D. √180 = √(9 × 20) = √9 × √20 = 3√20

Hence the correct option is D.

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The clear question is given below.

Which of the following is a pair of equivalent radical expressions?

A. √45 and 3√15

B. √ 75 and 3√5

C. √54 and 9√6

D. √180 and 3√20

Answer:

Below

Step-by-step explanation:

8 is the greatest common factor of 40 and 56....so factor it out to get

8 ( 5r-7)

Answer:

1/ At 10km, they will cost the same.

2/ It costs $199.50

Step-by-step explanation:

We know

Rental Company A, a rental car costs $55.50, and every kilometer costs $14.40.

Let Y represent the total cost, and x is the number of kilometers. We have the equation

y = 14.40x + 55.50

Rental Company B, a rental car costs $91.50, and every kilometer costs $10.80.

y = 10.80x + 91.50

How many kilometers will they cost the same?

14.40x + 55.50 = 10.80x + 91.50

3.6x = 36

x = 10 km

How much will it cost?

y = 14.40(10) + 55.50

y = 144 + 55.50

y = $199.50

Answer:

y'(9) = -7/3.

Step-by-step explanation:

We start by differentiating the given equation with respect to x:

d/dx(√x + √y) = d/dx(10)

1/(2√x) + (1/2)y'(x)/√y = 0

Simplifying, we get:

y'(x) = -√y/√x

To find y'(9), we plug in x = 9 and y = y(9) = 49:

y'(9) = -√49/√9 = -7/3

Therefore, y'(9) = -7/3.

The overall expected outcome for this game is $32.

Therefore the answer is $32.

The expected outcome for a game of chance can be calculated by multiplying the payoffs for each outcome by the probability of each outcome, and then summing these products.

For this game, the expected outcome would be calculated as follows:

Expected outcome

= (0.20 * $60) + (0.20 * $100) + (0.30 * $0)

= $12 + $20 + $0

= $32

So the overall expected outcome for this game is $32. This means that, on average, if you were to play this game many times, you could expect to win $32 each time. It's worth noting that expected outcome is just an average and does not guarantee a win, as the outcomes of each game are uncertain.

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The area of one rectangle is 21 square units.

Let's label the vertices of the large square as A, B, C, and D, with A in the top left corner and going clockwise. Let E be the center of the large square and also the center of the small square. Let F, G, H, and I be the midpoints of the sides AB, BC, CD, and DA, respectively, so that each of the four rectangles has two adjacent sides that are also sides of the large square.

Then, the side length of each rectangle is the same as the side length of the small square, which is 3. The height of each rectangle is the distance between two adjacent sides of the large square, which is 7. Therefore, the area of each rectangle is:

Area = base × height = 3 × 7 = 21 squared unit

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Answer:

x=5

Step-by-step explanation:

5x+2=8x-13

add the 13

5x+15=8x

subtract the 5x

15=3x

divide by 3

x=5

Simplifying the expression (6x⁻²)²(0.5x)⁴  by using mathematical operators gave us 2.25

Simplification of expression is the process of rearranging an expression so that it becomes simpler to understand and evaluate. This is done by combining terms, factoring, canceling out like terms, and other methods.

In the given expression;

(6x⁻²)²(0.5x)⁴ can be simplified using law of exponents which will become;

(6x⁻²)²(0.5x)⁴ = (36/x⁴) (0.0625x⁴)

This can be further simplified as;

(36/x⁴) (0.0625x⁴) = 2.25

The simplification of the original expression gave us 2.25

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Answer:

f(x) = -1 - 7/2 x*3

f(2) = -1 -7/2 (2)*3

f(2) = -1 -7/2 (8)

f(2) = -1 -56/2

f(2) = -1 -28

Answer:

x = 6

Step-by-step explanation:

the opposite sides in a rectangle are congruent.

since the rectangles are similar then the ratios of corresponding sides are in proportion, that is

[tex]\frac{5x}{18}[/tex] = [tex]\frac{10}{x}[/tex] ( cross- multiply )

5x² = 18 × 10 = 180 ( divide both sides by 5 )

x² = 36 ( take square root of both sides )

x = [tex]\sqrt{36}[/tex] = 6

The probability that a student who has skipped many classes has a GPA less than 2.0 is 0.5.

To calculate this probability, we can use the data from the table given. If a student has skipped many classes, they are in the "Many Skips" category. From the table, we can see that there are 200 students in the "Many Skips" category and 100 of them have a GPA of less than 2.0. Therefore, the probability of a student with many skips having a GPA less than 2.0 is given by:

P(GPA < 2.0 | Many Skips) = 100/200 = 0.5

This means that if we randomly select a student who has skipped many classes, there is a 50% chance that their GPA will be less than 2.0. This calculation provides insight into the relationship between attendance and GPA, and it suggests that skipping many classes is associated with a lower GPA.

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Answer:

141°

Step-by-step explanation:

In order to find the missing angle, we must first find the total sum of all the interior angles. To find this we use the following equation:

(n – 2) × 180°

Where n is equal to the number of sides

There are 7 sides therefore:

(7 – 2) × 180°

5 × 180°

900°

The sum of the total angles in the heptagon is 900°

We then must subtract all of the known angles

900 - 148 = 752

752 - 90 = 662

662 - 142 = 520

520 - 130 = 390

390 - 129 = 261

261 - 120 = 141

The missing angle is 141°

The left Riemann sum approximation is 14.892

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

f(x) = 3^x

The left Riemann sum can be expressed as:

[tex]R_n = h \sum_{i=1}^{n} \frac{4}{a + ih}[/tex]

Such that:

h = (b - a)/n

In this case, a = 2, b = 3.5, n = 3, and f(x) = 3^x

So, the left Riemann sum approximation is:

Approximation = (3.5-2)/3 * (f(2 + 1/3 * (3.5-2)) + f(2 + 2/3 * (3.5-2)) + f(2 + 3/3 * (3.5-2)))

Approximation = 0.5/3 * (3^(2 + 1/3 * (3.5-2)) + 3^(2 + 2/3 * (3.5-2)) + 3^(2 + 3/3 * (3.5-2)))

Approximation = 14.892

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