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Geometry

The completely factored form of the expression 4x² + 28x + 49 is (x + 7)(4x + 7) (option a).

In this case, we need to factor the expression 4x² + 28x + 49. One way to do this is by using the quadratic formula, which is a formula used to find the roots or solutions of a quadratic equation.

However, since we are only looking for the completely factored form, we can use a simpler method known as "factoring by grouping." This method involves grouping terms with common factors and factoring them separately.

Starting with the given expression:

4x² + 28x + 49

We notice that the first two terms have a common factor of 4x:

4x(x + 7) + 49

Now, we can factor out the common factor of (x + 7) from the first two terms:

4x(x + 7) + 49 = (x + 7)(4x + 49/7)

Simplifying the expression in the second set of parentheses:

4x + 49/7 = 4x + 7

Hence the correct option is (a).

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Complete Question:

Which is the completely factored form of 4x2 + 28x + 49?

a) (x + 7)(4x + 7)

b) 4(x + 7)(x + 7)

c) (2x + 7)(2x + 7)

d) 2(x+7)(x + 7)

Answer:

50 in

Step-by-step explanation:

Split into 4 shapes. One on each side with 5 in x 2 in, one on the top with 12 in x 2 in, and one in the middle with 3 in x 2 in. We add all together:

2(5×2)+(12×2)+(3×2) = 50 in

Adding and subtracting integrals is a fundamental technique in calculus that allows you to combine multiple integrals into a single expression. The process involves using the properties of integrals and the rules of algebra to simplify the expression.

How to add integrals?

To add two integrals, you simply add the integrands and integrate over the same limits of integration. For example, if you have two integrals, ∫f(x) dx and ∫g(x) dx

The sum of the two integrals would be

∫[f(x) + g(x)] dx

How to subtract integrals?

To subtract two integrals, you subtract the integrands and integrate over the same limits of integration. For example, if you have two integrals ∫f(x) dx and ∫g(x) dx

The difference of the two integrals would be ∫[f(x) - g(x)] dx

It's important to note that when adding or subtracting integrals, you must ensure that the limits of integration are the same. If they are not the same, you must first adjust one or both integrals to have the same limits of integration.

Another important point to keep in mind is that the properties of integrals allow you to simplify expressions before integrating. For example, if you have an integral of the form:

∫f(x) dx + ∫g(x) dx

You can simplify it as ∫[f(x) + g(x)] dx

before integrating.

In conclusion, adding and subtracting integrals is a straightforward process that involves using the rules of algebra and the properties of integrals. By following these techniques, you can simplify complex expressions and solve integration problems more efficiently.

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Correct question is "How to add and subtract integrals?"

Answer:

Step-by-step explanation:

Dilation by 3 means that you increase the length of each of the sides by 3. Since ED was 2 units long, the plotted dilation of ED is 6 units long. By multiplying each of the lengths of the sides by 3 units, you will be able to draw the whole image.

Answer:

see the attached image

Step-by-step explanation:

For a dilation of scale factor 3, we can multiply each side length by 3.

Answer:

77

Step-by-step explanation:

First, let's simplify the expression inside the innermost parentheses:

5x12 = 60

60/2 = 30

-14 + 30 - 6 = 10

Now, we have:

[64 + {23 - 10}] = [64 + 13] = 77

Answer:

Y-intercept: (0,4)

Slope: 2

Domain: input values

Range: output values

Increasing graph

x-intercept: (-2,0)

Step-by-step explanation:

According to the graph, the y-intercept is (0,4)

the rate of change every x-unit is 2

the x-intercept is (-2,0) as shown in the graph. The rest are just input and output values. Since the slope is positive, the graph is increasing

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ABCD is square, AC = 6, AO = 3√2, BO = 3√2, BO = 4, DC = 6, OC = 2√13 and area is  36 square units.

To solve the problems, we can use the following properties of squares:

The diagonals of a square bisect each other and are equal in length.

The diagonals of a square form two congruent right triangles.

The center of a square is equidistant from all four vertices of the square.

Since ABCD is a square, we have AC = AB = 6.

Since O is the center of the square, AO is half the length of the diagonal of the square. Using the Pythagorean theorem, we can find the diagonal of the square:

diagonal² = AB² + BC²

= 6² + 6²

= 72

diagonal = √72 = 6√2

Therefore, AO = (1/2) diagonal = 3√2

Since O is the center of the square, BO is equal in length to AO, which we found in part (18). Therefore, BO = 3√2.

If AO = 4 and BO = AO, then BO = 4.

The area of ΔAOB is half the area of the square ABCD. The area of a square is given by side², so the area of ABCD is 6² = 36. Therefore, the area of ΔAOB is (1/2) × 36 = 18 square units.

The area of a square is given by side², so the area of ABCD is 6² = 36 square units.

Since the diagonals of a square bisect each other, DO is half the length of the diagonal of the square. Using the Pythagorean theorem, we can find the diagonal of the square:

diagonal² = AB² + BC²

= 6² + 6²

= 72

diagonal = √72 = 6√2

Therefore, DO = (1/2) diagonal = 3√2

Since ABCD is a square, DC = AB = 6.

The area of ΔDOC is half the area of the square ABCD. Using the Pythagorean theorem, we can find the length of OC:

OC² = AO² + AC²

= 4² + 6²

= 52

OC = √52 = 2√13

Therefore, the area of ΔDOC is (1/2) × 6 × 2√13 = 6√13 square units.

The area of a square is given by side², so the area of ABCD is 6² = 36 square units.

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Answer:

y = 4

Step-by-step explanation:

given equation of line is

y = - 3x - 2 and (- 2, y ) is a point on the line, then substitute x = - 2 into the equation for y , that is

y = - 3(- 2) - 2 = 6 - 2 = 4

The meter reading, in ccf, indicated by each of the gas meters would be  8.83 ccf.

Required to find the meter reading in ccf

Let x be [tex]m^{2}[/tex] - meter readings.

The readings in ccf is:

Reading = x/ 2.832

Now, Assume the value of x is 25; The readings will be:

Reading = 25/ 2.832 ccf

Reading = 8.83 ccf

The complete question is

Manuel works for the water company as a meter reader. The meter below is the Jansen's water meter. What is the reading, in ccf, on the meter shown?

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Answer: You will pay £1,912.54 each month.

Step-by-step explanation:

We can use the formula for the monthly payment of a loan with compound interest:

monthly payment = (principal * monthly interest rate) / (1 - (1 + monthly interest rate)^(-n))

where principal is the amount borrowed, monthly interest rate is the annual interest rate divided by 12, and n is the total number of monthly payments.

In this case, the principal is £64,950, the annual interest rate is 4%, so the monthly interest rate is 4%/12 = 0.00333, and the number of monthly payments is 36.

Plugging in these values, we get:

monthly payment = (64950 * 0.00333) / (1 - (1 + 0.00333)^(-36)) = £1,912.54

Therefore, you will pay £1,912.54 each month.

R is the set containing the two points (1, 0) and (-1, 0).

In mathematics, a set is a collection of distinct objects, called elements or members, that share a common property. The objects in a set can be anything, such as numbers, letters, shapes, or even other sets. Sets are denoted by braces {} and the elements are separated by commas.

Set theory is an important branch of mathematics and is used in various fields such as algebra, geometry, logic, and computer science. Sets provide a foundation for mathematical concepts such as functions, relations, and number systems. They are also used in the study of probability and statistics, and in the design and analysis of algorithms.

To find R, we need to determine the set of all points (x, y) that satisfy the conditions given:

x² + y² = 1 and |x - y| = 1

Let's consider the two cases of |x - y| = 1:

Case 1: x - y = 1

In this case, we can substitute y = x - 1 into the equation x² + y² = 1 to get:

x² + (x - 1)² = 1

2x² - 2x = 0

2x(x - 1) = 0

So x = 0 or x = 1. If x = 0, then y = -1, which does not satisfy the condition x² + y² = 1. If x = 1, then y = 0, which does satisfy the condition. Therefore, we have one point that satisfies the conditions in this case:

R1 = {(1, 0)}

Case 2: x - y = -1

In this case, we can substitute y = x + 1 into the equation x² + y² = 1 to get:

x² + (x + 1)² = 1

2x² + 2x = 0

2x(x + 1) = 0

So x = 0 or x = -1. If x = 0, then y = 1, which does not satisfy the condition x² + y² = 1. If x = -1, then y = 0, which does satisfy the condition. Therefore, we have one more point that satisfies the conditions in this case:

R2 = {(-1, 0)}

Putting both cases together, we have:

R = R1 ∪ R2 = {(1, 0), (-1, 0)}

Therefore, R is the set containing the two points (1, 0) and (-1, 0).

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Answer:

Step-by-step explanation:

The three true statements about circle Q are:

1. The ratio of the measure of central angle PQR to the measure of the entire circle is One-eighth.

2. The area of the shaded sector is 4 units 2.

3. The ratio of the area of the shaded sector to the area of the circle is equal to the ratio of the length of the arc to the circumference of the circle.

The True statements are:

The ratio of the measure of central angle PQR to the measure of the entire circle is 1/8.

The area of the shaded sector depends on the length of the radius.

The area of the shaded sector depends on the area of the circle.

The measurement of the circle's perimeter, also known as its circumference, is called the circle's boundary. The region a circle occupies is determined by its area.  

First, The measure of the central angle is 45°.  

So, ratio of the central angle to the entire circle

= 45/360

= 9/72

= 1/8

Now, area of shaded sector

A = πr² = π(6²) = 36π

or, 1/8(36π)

= 36π/8

= 18π/4

= 9π/2

= 4.5π square units.  

Since, we use the circle's radius to determine area, the shaded sector's area is dependent on the circle's radius.

Given that the sector is a portion of the circle, its size is also influenced by the size of the circle.

and, The ratio of the shaded sector's area to the circle's area wouldn't be the same as the ratio of the arc's length to the circle's area.

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Answer:

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Answer:

A = $1853.96

Step-by-step explanation:

The formula for compound interest is

[tex]A(t)=P(1+r/n)^n^t[/tex], where A(t) is the amount, P is the principal (amount invested or deposited), r is the interest rate, n is the number of compound periods per year, and t is the time in years.

[tex]A(t)=1600(1+0.037/4)^(^4^*^4^)\\A(t)=1600(1.00925)^1^6\\A(t)=1853.958942\\A(t)=1853.96[/tex]

well, we know the sphere's "great circle" has a circumference of 13mm, so

[tex]\textit{circumference of a circle}\\\\ C=2\pi r ~~ \begin{cases} r=radius\\[-0.5em] \hrulefill\\ C=13 \end{cases}\implies 13=2\pi r\implies \cfrac{13}{2\pi }=r \\\\[-0.35em] ~\dotfill\\\\ \textit{surface area of a sphere}\\\\ SA=4\pi r^2 ~~ \begin{cases} r=radius\\[-0.5em] \hrulefill\\ r=\frac{13}{2\pi } \end{cases}\implies SA=4\pi \left( \cfrac{13}{2\pi } \right)^2 \\\\\\ SA=\cfrac{169}{\pi } \implies SA\approx 53.79~mm^2[/tex]

Answer:

The temperature in 5 hours will be -25°C.

The temperature 8 hours ago was 14°C.

Step-by-step explanation:

To find the temperature, y, in degrees Celsius at any given number of hours, x, from the current time, we can create a linear equation in the form y = mx + b, where m is the rate of change and b is the current temperature.

Given the current temperature is -10°C, then b = -10.

Given the temperature falls at a constant rate of 3°C per hour, then m = -3.

Substitute the values of m and b into the formula:

[tex]\boxed{y = -3x - 10}[/tex]

where:

To calculate the temperature in 5 hours time, substitute x = 5 into the equation:

[tex]\begin{aligned}\implies y&=-3(5)-10\\&=-15-10\\&=-25^{\circ}\text{C}\end{aligned}[/tex]

Therefore, the temperature in 5 hours will be -25°C.

To calculate the temperature 8 hours ago, substitute x = -8 into the equation. The value of x is negative since we want to find the temperature 8 hours before the current time.

[tex]\begin{aligned}\implies y&=-3(-8)-10\\&=24-10\\&=14^{\circ}\text{C}\end{aligned}[/tex]

Therefore, the temperature 8 hours ago was 14°C.

Answer: The length of the base of the ramp is approximately 27 inches when rounded to the nearest inch.

Step-by-step explanation:

We can use trigonometry to solve this problem.

Let's call the length of the base of the ramp "x". Then, we can use the tangent function to find x:

tan(22) = 11/x

To solve for x, we can multiply both sides by x and divide by tan(22):

x = 11 / tan(22)

Using a calculator, we can find that:

x ≈ 27 inches

Therefore, the length of the base of the ramp is approximately 27 inches when rounded to the nearest inch.

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The amount invested at 4% interest rate that is $3900 and

the amount invested at 3% interest rate is $4300.

Last year Sarah invested money in two accounts. The first account had an interest rate of 3% and the second account had an interest rate of 4%. If she invested $400 more in the first account than the second and her total interest income was $285,

the equations are:

       s= f + 400  (1)

      0.03s + 0.04f = 285   (2)

modifying (1) and (2) for simpler calculations

      3s - 3f = 3 * 400 (3)

      3s + 4f = 100 * 285  (4)

by solving eqns (3) and (4) we get

  we get f=3900 and s=4300

Here, f is the amount invested at 4% interest rate that is $3900 and

   s refers to the amount invested at 3% interest rate is $4300.

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the difference between the longest and shortest time Brandon spent reading the book was 3/4 of a day.

How to solve the problem?

To label and plot the data on a dot plot, we need to first determine the range of values. In this case, we have the following data:

1/4, 1/4, 1, 1/4, 1/2, 3/4, 1/2, 1/4

The range of values is from 1/4 to 1, and we can use a dot plot to represent the data. On the dot plot, we will place a dot above each value on the number line. For example, we will place four dots above the value 1/4 since it appears four times in the data.

The dot plot would look like this:

markdown

Copy code

     |

0.2   |   x

     |

0.4   |   x   x   x   x

     |

0.6   |   x   x

     |

0.8   |   x   x   x

     |

1.0   |   x

     |

From the dot plot, we can see that Brandon spent the most time reading the book when he read for the full day, which is represented by the dot at 1. The shortest time he spent reading was when he read for 1/4 of a day, which is represented by the dots at 1/4.

To find the difference between the longest and shortest time, we simply subtract the shortest time from the longest time. In this case, the longest time was 1 day, and the shortest time was 1/4 of a day. So the difference is:

1 - 1/4 = 3/4 of a day

Therefore, the difference between the longest and shortest time Brandon spent reading the book was 3/4 of a day.

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The percentage of buyers who paid between $150,000 and $156,000 if the standard deviation is $2000 is 49.9%

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

Mean = 150000

Standard deviation = 2000

Scores = from $150,000 and $156,000

So, we have

z = (150000 - 150000)/2000 = 0

z = (156000 - 150000)/2000 = 3

This means that it is between a z-score of 0 and a z-score of 3

This is represented as

Probability = (0 < z < 3)

Using a graphing calculator, we have

Probability =  0.49865

Express as percentage

Probability =  49.9%

Hence, the probability is 49.9%

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Answer:

V = 5747.02 ft^3

Step-by-step explanation:

Therefore, to solve for volume, we plug in 14 for r in the general formula:

[tex]V=2/3\pi (14)^3\\V=2/3\pi (14)(14)(14)\\V=2/3\pi *2744\\V=5747.020161\\V=5747.02[/tex]

Lastly, volume is always in units cubed (e.g., ft^3 or mi^3 since volume deals with the three-dimensional space of an object

The total interest that will accrue on the principal amount of $44,000 over a period of 2 years at a simple interest rate of 6% is $5,280.

The given problem involves calculating the amount of interest that will accrue on a principal amount of $44,000, which is invested at a simple interest rate of 6% for a period of 2 years.

To solve the problem, we can use the formula for simple interest, which is:

Interest = Principal x Rate x Time

In this case, the principal amount is $44,000, the rate of interest is 6%, and the time period is 2 years. So, plugging in these values, we get:

Interest = $44,000 x 0.06 x 2

Interest = $5,280

Unlike compound interest, simple interest is calculated only on the principal amount, and not on the accumulated interest. This means that the interest earned each year remains constant, which can make it easier to calculate and understand compared to compound interest.

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Answer:

Alex's gross pay for the month is $18/hour x 140 hours = $2520.

The total deductions from his paycheck are:

$180.00 (FICA) + $215.00 (Federal income tax) + $62.00 (State income tax) + $72.89 (Health insurance) + $50.00 (Retirement savings) = $579.89

So Alex's net pay (take-home pay) is:

$2520 (gross pay) - $579.89 (deductions) = $1940.11

To find the percentage of his gross pay that he took home, we can divide his net pay by his gross pay and multiply by 100:

($1940.11 / $2520) x 100 = 77.06%

Therefore, Alex took home 77.06% of his gross pay after deductions.

This polynomial cannot be factored any further using integer coefficients, so our final factored form is: 40w²11 + 16w²6 = 8w²6(5w²5 + 2)

How to solve?

To factor the polynomial 40w²11 + 16w²6, we need to look for common factors in the two terms. We notice that both terms have a factor of 8w²6, so we can factor that out:

40w²11 + 16w²6 = 8w²6(5w²5 + 2)

Now, we have factored out the greatest common factor of the two terms, leaving us with the remaining factor of 5w²5 + 2. This polynomial cannot be factored any further using integer coefficients, so our final factored form is:

40w²11 + 16w²6 = 8w²6(5w²5 + 2)

This form of the polynomial is useful for simplifying expressions, finding roots, and solving equations.

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Answer:

45 Minutes

Step-by-step explanation:

1.5 Hours = 90 Minutes (1.5*60)

90/2 = 45

45 Minutes on the trampoline.

The balance A for $1900 invested at a rate of 8% for 8 years and compounded n times per year is approximately

$3,538.62, when compounded annually.

$3,601.77 when compounded semi-annually

$3,663.13 when compounded quarterly

$3,710.58 when compounded monthly

$3,748.26 when compounded daily

$3,754.04 when compounded continuously.

We have,

The balance A for $1900 was invested at a rate of 8% for 8 years and compounded n times per year.

A = P(1 + r/n)^(n*t)

where P is the principal amount ($1900 in this case), r is the annual interest rate (8%), t is the number of years (8), and n is the number of times compounded per year.

When compounded annually (n = 1):

A = $1900(1 + 0.08/1)^(1*8) = $1900(1.08)^8 ≈ $3,538.62

When compounded semi-annually (n = 2):

A = $1900(1 + 0.08/2)^(2*8) = $1900(1.04)^16 ≈ $3,601.77

When compounded quarterly (n = 4):

A = $1900(1 + 0.08/4)^(4*8) = $1900(1.02)^32 ≈ $3,663.13

When compounded monthly (n = 12):

A = $1900(1 + 0.08/12)^(12*8) = $1900(1.00666667)^96 ≈ $3,710.58

When compounded daily (n = 365):

A = $1900(1 + 0.08/365)^(365*8) = $1900(1.00021918)^2920 ≈ $3,748.26

When compounded continuously:

A = Pe^(rt) = $1900e^(0.08*8) ≈ $3,754.04

Now,

n     1                2             4           12                365        continuous

A    $3,538.6    $3,601.77  $3,663.13   $3,710.55     $3,748.26   $3,754.04

Therefore,

The balance A for $1900 invested at a rate of 8% for 8 years and compounded n times per year is approximately $3,538.62.

When compounded annually, $3,601.77 when compounded semi-annually, $3,663.13 when compounded quarterly, $3,710.58 when compounded monthly, $3,748.26 when compounded daily, and $3,754.04 when compounded continuously.

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The surface area of the cylinder is 471 mm².

Given is a cylinder of height 5 mm and radius 10 mm, we need to find the surface area of the cylinder,

So, the surface area of the cylinder is = 2π × r (h+r)

= 2 × 3.14 × 5 (5+10)

= 2 × 3.14 × 75

= 471 mm²

Hence, the surface area of the cylinder is 471 mm².

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Step-by-step explanation:

The height of the coin, after t seconds, is given by the following equation:

How long will it take the coin to reach the stream?

The stream is the ground level.

So the coin reaches the stream when h(t) = 0.

Multiplying by (-1)

Solving a quadratic equation:

Given a second order polynomial expressed by the following equation:

This polynomial has roots  such that , given by the following formulas

In this question:

So

Time is a positive measure, so we take the positive value.

It will take 5.61 seconds for the coin to reach the stream.The height of the coin, after t seconds, is given by the following equation: