What is the minimum value of f(x) = x^2 + 6x + 4

The width of a rectangle is 2 units less than the length. The area of the rectangle is 24 square units. The length of the rectangle is 6 units.

Let's denote the length of the rectangle as L and the width as W. According to the given information, the width is 2 units less than the length, so we can express this relationship as:

W = L - 2

The formula for the area of a rectangle is A = L * W, and in this case, the area is given as 24 square units. Substituting the values, we have:

24 = L * (L - 2)

Expanding the equation, we get:

24 = L^2 - 2L

Rearranging the equation and setting it equal to zero, we have:

L^2 - 2L - 24 = 0

We can now factor the quadratic equation:

(L - 6)(L + 4) = 0

Setting each factor equal to zero, we have two possible solutions:

L - 6 = 0 or L + 4 = 0

Solving each equation, we find:

L = 6 or L = -4

Since the length cannot be negative, the length of the rectangle is 6 units.

The length of the rectangle is 6 units.

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The sοlutiοn οf the given prοblem οf unitary methοd cοmes οut tο be  there are 21 pοtential οutcοmes in the sample space.

One may use this cοmmοn cοnvenience, already-existing variables, οr all impοrtant elements frοm the οriginal Bishοp custοmizable questiοn that adhered tο a particular methοd tο achieve the gοal. If sο, there will be anοther chance tο get in tοuch with the οbject. If it dοesn't, all οf the crucial elements that make up an expressiοn prοοf οutcοme will be lοst.

Here,

Using the fοrmula fοr the number οf cοmbinatiοns, we can determine hοw many pοssibilities are pοssible in the sample space:

=> nCk = n!/(k!*(n-k))!

where k is the number οf οbjects being selected, and n denοtes the οverall number οf οbjects in the set.

As there are 7 marbles in the cοntainer, we want tο select 2 οf them, sο n = 7 and k = 2.

the sample space has the fοllοwing amοunt οf οutcοmes that cοuld οccur:

=> 7C2 = 7! / (2! * (7-2)!)

=>  21

Therefοre, when twο marbles are drawn frοm the given bag withοut a replacement, there are 21 pοtential οutcοmes in the sample space.

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Complete Question:

'You draw two marbles without replacement from bag containing three green marbles and four black marbles_ The number of possible outcomes in the sample space is'

To complete the table, we can use the expression 10 + 4h to calculate the total cost of renting the golf clubs for different values of h.

h Total cost

1 14

2 18

3 22

4 26

5 30

For example, if the golf clubs are rented for 1 hour, the total cost is $14 (10 + 4(1) = 14). If they are rented for 2 hours, the total cost is $18 (10 + 4(2) = 18), and so on.

The expression for the total cost of renting golf clubs for h hours at a golf course that charges $10 plus $4 for each hour rented is 10 + 4h.

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Answer:

42 /3 = x

Step-by-step explanation:

it is 14 for sure.....

[tex]\begin{array}{llll} f(x)~from\\\\ x_1 ~~ to ~~ x_2 \end{array}~\hfill slope = m \implies \cfrac{ \stackrel{rise}{f(x_2) - f(x_1)}}{ \underset{run}{x_2 - x_1}}\impliedby \begin{array}{llll} average~rate\\ of~change \end{array} \\\\[-0.35em] ~\dotfill\\\\ f(x)= 6x^2-7 \qquad \begin{cases} x_1=1\\ x_2=b \end{cases}\implies \cfrac{f(b)-f(1)}{b - 1}[/tex]

[tex]\cfrac{[6(b)^2-7]~~ - ~~[6(1)^2-7]}{b-1}\implies \cfrac{6b^2-6}{b-1}\implies \cfrac{6(b^2-1)}{b-1}\implies \cfrac{6(b^2-1^2)}{b-1} \\\\\\ \cfrac{6(b-1)(b+1)}{b-1}\implies 6(b+1)[/tex]

Answer:

16:24

Step-by-step explanation:

The ratio of boys to girls is 2 : 3

total of the ratio is 5

∴ fraction of the class are boys = 2/5

and  the girls=  3/5

Ratio of boys =( 2/5)×40=16

Ratio of girls=3/5  ×40 =24

The sum of two dice has a total of 36 possible outcomes. The sum can range from 2 (rolling a 1 on each die) to 12 (rolling a 6 on each die).

To find the probability distribution, we can make a table that lists all the possible sums, the number of ways to obtain each sum, and the corresponding probability.

Sum Ways to obtain Probability

2 1 1/36

3 2 2/36 = 1/18

4 3 3/36 = 1/12

5 4 4/36 = 1/9

6 5 5/36

7 6 6/36 = 1/6

8 5 5/36

9 4 4/36 = 1/9

10 3 3/36 = 1/12

11 2 2/36 = 1/18

12 1 1/36

From the table, we can see that the sums 6 and 9 have the same probability distribution, and would be represented with equal bars on a bar graph

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Answer:16

Step-by-step explanation:

10/p=5/8

can be shown as

1/p=5/80

p=80/5=18

Answer:

[tex]P=16[/tex]

Step-by-step explanation:

Given

[tex]\frac{10}{P} =\frac{5}{8}[/tex]

We can use cross multiplication to find the value of [tex]P[/tex].

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

Following these steps we can evaluate [tex]P[/tex].

[tex]10*8=5P[/tex]

Multiply [tex]10[/tex] by [tex]8[/tex].

[tex]80=5P[/tex]

Divide both sides of the equation by [tex]5[/tex]. [tex]P[/tex] cancels out on the right side.

[tex]\frac{80}{5} =\frac{5P}{5}[/tex]

[tex]\frac{80}{5} =P[/tex]

Divide [tex]80[/tex] by [tex]5[/tex].

[tex]16=P[/tex]

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Answer:

Let's assume that a serving of Cereal B contains x grams of carbohydrates.

According to the problem, a serving of Cereal A contains 4 more grams of carbohydrates than Cereal B. So, the amount of carbohydrates in Cereal A is (x + 4) grams.

The total amount of carbohydrates consumed from both cereals is 66 grams:

x + (x + 4) = 66

Simplifying the equation:

2x + 4 = 66

Subtracting 4 from both sides:

2x = 62

Dividing by 2:

x = 31

So a serving of Cereal B contains 31 grams of carbohydrates.

And a serving of Cereal A contains 4 more grams:

31 + 4 = 35

Therefore, a serving of Cereal A contains 35 grams of carbohydrates.

The value of t is equal to 4.

The equation of PR is y = -x + 6.

The area of the rhombus is equal to 32 square units.

Since BPCR is a rhombus and all the side lengths of a rhombus are always equal in length, we would use the distance formula to determine the length of BP as follows:

Distance BP = √[(x₂ - x₁)² + (y₂ - y₁)²]

Distance BP = √[(0 - 2)² + (6 - 0)²]

Distance BP = √[4 + 36]

Distance BP = √40 units.

Also, side length BP is equal to side length PC:

√40 = √[(t - 6)² + (6 - 0)²]

√40 = √[(t - 6)² + 36]

40 = (t - 6)² + 36

(t - 6)² = 4

t - 6 = ±2

t = 6 ± 2

t = 8 or t = 4.

Therefore, C should be located at (6, 4).

Since Q is the midpoint PR and BC, we know that B is shifted 2 units right and 6 units down from P. Therefore, point R is represented by R(8, -2).

In order to determine the equation of PR, we would use the point-slope form:

[tex]y - y_1 = \frac{(y_2- y_1)}{(x_2 - x_1)}(x - x_1)\\\\y - 6 = \frac{(-2-6)}{(8 - 0)}(x - 0)[/tex]

y - 6 = -1(x)

y = -x + 6

In Mathematics, the area of a rhombus is calculated by using this formula:

Area = 1/2(d₁ × d₂)

Where:

d₁ and d₂ represent the diagonals of a rhombus.

PR = 2BC

8√2 = 2BC

BC = (8√2)/2

BC = 4√2

Area of rhombus = 1/2(PR × BC)

Area of rhombus = 1/2(8√2 × 4√2)

Area of rhombus = 1/2(64)

Area of rhombus = 32 square units.

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Answer:

[tex]\large\boxed{\mathtt{Area = 586.2cm^{2}.}}[/tex]

Step-by-step explanation:

[tex]\textsf{We are asked to find the area of this figure.}[/tex]

[tex]\textsf{In order to make this simpler, draw a line between the rectangle, and the semi-circle.}[/tex]

[tex]\textsf{We should identify the formulas needed for the areas of the rectangle, and semi-circle.}[/tex]

[tex]\large\underline{\textsf{For the Semi-Circle:}}[/tex]

[tex]\mathtt{Area = 1/2( pi(radius)^{2}).}[/tex]

[tex]\large\underline{\textsf{For the Rectangle:}}[/tex]

[tex]\mathtt{Area = Length \ x \ Width.}[/tex]

[tex]\textsf{Let's find the area of the Rectangle first.}[/tex]

[tex]\mathtt{Area = 15 \ x \ 24}[/tex]

[tex]\mathtt{Area = 360cm^{2}.}[/tex]

[tex]\textsf{For the Semi-Circle, the radius is 1/2 of the Diameter. (24)}[/tex]

[tex]\mathtt{Area = 1/2( pi(12)^{2}).}[/tex]

[tex]\mathtt{Area = 1/2(144 \pi).}[/tex]

[tex]\mathtt{Area = 72 \pi.}[/tex]

[tex]\mathtt{Area \approx 226.2cm^{2}.}[/tex]

[tex]\textsf{Add the areas together to represent the area of the figure.}[/tex]

[tex]\large\underline{\textsf{Add:}}[/tex]

[tex]\mathtt{360cm^{2} + 226.2cm^{2} = 586.2cm^{2}. }[/tex]

[tex]\large\boxed{\mathtt{Area = 586.2cm^{2}.}}[/tex]

Answer:

1872 cards

Step-by-step explanation:

1 cards=52

36×52

=1872 cards

$1500

Step-by-step explanation:

if $1000 is equal to 2 parts then .....1 part is given by 1000÷2 =500

so John's share is given by 3x500

1500

Step-by-step explanation:

If the LOSE 27% per year, they RETAIN 100- 27 = 73%  (.73 in decimal) per year

after ten years :

 $  1128 ( .73)^10 = $48.48

Answer:

$48.48

Step-by-step explanation:

Using compound depreciation

time

amount × percentage^

10

1128 × 0.73^ =48.477

The quadratic equation is; x^2 -6x + 8 = 0

A quadratic equation is a mathematical equation of the form ax^2 + bx + c = 0, where x is the unknown variable, and a, b, and c are constants (numbers). The exponent of x in the equation is 2, which means it is a second-degree polynomial equation.

We know that the solution of  x^2- 3x + 2 = 0 is x = 2 and x = 1 hence;

α = 2, β = 1

We have that;

αβ^2 = 2

α^2β = (2)^2(1) = 4

The equation would now be;

x^2 - (2 + 4)x - (2 * 4) = 0

x^2 -6x + 8 = 0

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The number of hamburger sold on Tuesday is 136.

On Tuesday, a local hamburger shop sold a combined total of 408

hamburgers and cheeseburgers. The number of cheeseburgers sold was

two times the number of hamburgers sold.

Therefore, the number of hamburger sold on Tuesday can be calculated as follows:

Let

x = number of hamburger

y  =number of cheeseburger

Therefore,

x + y = 408

2x = y

x + 2x = 408

3x = 408

divide both sides by 3

x = 408 / 3

x = 136

Therefore,

number of hamburger = 136

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Below given are the steps to copy the original triangle using only two of its sides and the included angle.

It is a fundamental geometric shape in mathematics and has many applications in science, engineering, and everyday life. They can be classified based on the lengths of their sides and the measures of their angles.

Using point E as the center, draw a circle with a radius equal to the length of n×CA that was calculated in part B.

Using point E as the vertex, draw a ray DE in the direction of one of the sides of the angle to be copied (in this case, ∠BCA).

Create an angle at point E that is equal to the measure of ∠BCA. To do this, place the point of a compass at point E and adjust the compass to the length of segment CA. Then, draw an arc that intersects both sides of the ray DE. Without changing the width of the compass, place the compass at point C and draw an arc that intersects the previous arc. Draw a line connecting point E to the intersection of the two arcs to create an angle that is equal to ∠BCA.

Draw ray ED' from point E through the intersection of the two arcs.

Locate the intersection of ray ED' and the circle drawn in step 1. Label this point F.

Draw a polygon through points D, E, and F to complete the copied triangle, which is ΔDEF.

The resulting triangle ΔDEF will be a copy of the original triangle, with side EF corresponding to the included angle and sides DE and DF corresponding to the sides used to construct the copy.

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Answer:

36 miles

Step-by-step explanation:

speed = distance/time

distance = speed × time

distance = 12 miles/hour × 3 hours

distance = 36 miles

The multiplicatiοn οf the cοmplex number is 13+19i.

Cοmplex numbers are the numbers that are stated in the fοrm οf a+ib where, a,b are real numbers and I is an imaginary number termed "iοta".

Cοmplex numbers allοw sοlutiοns tο all pοlynοmial equatiοns, even thοse that have nο sοlutiοns in real numbers. Mοre precisely, the fundamental theοrem οf algebra asserts that every nοn-cοnstant pοlynοmial equatiοn with real οr cοmplex cοefficients has a sοlutiοn which is a cοmplex number.

Here the given cοmplex number is ,

=> (7-2i)(1+3i)

Nοw simplifying the expression then,

=> [tex]7\times1+7\times3i-2i \times1 -2i \times3i[/tex]

=> 7+21i-2i-[tex]6i^2[/tex]

=> 7+19i -6(-1) [[tex]i^2=1[/tex]]

=> 7+19i+6

=> 13+19i

Hence the multiplication of cοmplex number is 13+19i.

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Answer:

[tex]x = 2 , y = 6[/tex]

Step-by-step explanation:

The substitution method suggests that you will "substitute" a portion of one given equation to another in the system of equation:

In this case, substitute 3x for y in the 2nd equation:

[tex]y = 3x\\x + y = 8\\\\x + (3x) = 8[/tex]

Simplify. Combine like terms. Like terms are terms with the same amount of the same variable.

[tex]x + 3x = 8\\4x = 8[/tex]

Next, isolate the variable, x. Divide 4 from both sides of the equation:

[tex]4x = 8\\\frac{4x}{4} = \frac{8}{4} \\x = \frac{8}{4}\\ x = 2[/tex]

Next, plug in 2 for x in one of the equations, and simplify to solve for y:

[tex]y = 3x\\y = 3(2)\\y = 6[/tex]

Check: Plug in 2 for x and 6 for y in the second equation:

[tex]x + y = 8\\(2) + (6) = 8\\8 = 8[/tex](True)

~

[tex]x = 2 , y = 6[/tex] is your answer.

~

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The width of the book cover made by Meals for his brother is 40 cm.

An area is just a surface that is free of objects. With a shape's length and breadth, one may compute the area of the object. Measured in terms of feet (ft), yards (yd), inches (in), etc., length is a one-dimensional quantity. The size of a form, however, is a two-dimensional measurement. Because of this, it is measured in square units like square inches (in2), square feet (ft 2), square yards (yd2), etc. Corners and edges are present on the majority of items and forms. Calculating the area of a certain form takes into account the length and breadth of these edges.

The area of the rectangle is:

A = lw

Given that the area is 890 cm and the height is 22 1/4 cm.

890 = (22 1/4)w

22 1/4 = (4 * 22 + 1) / 4 = 89 / 4

890 = (89 / 4)w

w = 40

Hence, the width of the book cover is 40 cm.

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The complete question is:

The correct formula for the slope will be:

m = (100 - 50)/(4 - 2) = 50/2 = 25 . Rafael took the reciprocal of the values.

A line's slope is determined by how its y coordinate changes in relation to how its x coordinate changes. Δy and Δx are the net changes in the y and x coordinates, respectively.

So,

m = Δy / Δx

Using the slope-intercept form, as illustrated below, we can quickly get the slope for just a given line:

Y = mx + c can be used to represent a line.

For the stated question:

The slope calculated by Rachel is:

m = (4 - 2) / (100 - 50) = 2/50 = 1/25

Which is incorrect .

Thus, the correct formula for the slope will be:

m = (100 - 50)/(4 - 2) = 50/2 = 25

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The value of a in the given parallelogram such that 2a + 27 and 59 are opposite angles is 16

Given the attached parallelogram

Opposite angles in a parallelogram are equal in measure.

This property is known as the "Opposite Angles Property" or "Alternate Angles Property" of a parallelogram.

So, we have

2a + 27 = 59

Evaluating the like terms, we have

2a = 32

Divide both sides by 2

a = 16

Hence, the value of a is 16

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By trigonometric functions, the missing sides of the right triangles are:

Case 7: AC = 9.725, BC = 10.071

Case 8: AB = 21.301, BC = 18.807

Case 9: BC = 4.585, AC = 8.999

Case 10: AB = 10.461, BC = 7.774

In this problem we have four cases of right triangles, each of which has a known side and a known angle and we must determine all the missing lengths by trigonometric functions:

sin θ = y / r

cos θ = x / r

tan θ = y / x

Where:

Now we proceed to determine the values of the missing lengths:

Case 7

AC = 14 · sin 44°

AC = 9.725

BC = 14 · cos 44°

BC = 10.071

Case 8

AB = 10 / cos 62°

AB = 21.301

BC = 10 · tan 62°

BC = 18.807

Case 9

BC = 10.1 · sin 27°

BC = 4.585

AC = 10.1 · cos 27°

AC = 8.999

Case 10

AB = 7 / cos 48°

AB = 10.461

BC = 7 · tan 48°

BC = 7.774

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Answer:

38 in²

Step-by-step explanation:

We can solve for the area of this trapezoid by using the area formula:

[tex]A = \dfrac{b_1 + b_2}{2} \cdot h[/tex]

First, we need to identify the trapezoid's bases ([tex]b_1[/tex] and [tex]b_2[/tex]). We know that the bases of a trapezoid are parallel. Using this information, we can identify 8 in and 11 in as the bases.

Next, we need to identify the height ([tex]h[/tex]). We know that the height of a trapezoid is the length from [tex]b_1[/tex] and [tex]b_2[/tex] that is perpendicular (at a right angle) to both of the bases. We can identify this as 4 in because we can see that it adjoins at a right angle to both of them.

Finally, we can plug these values into the area formula and solve for A.

[tex]A = \dfrac{8 + 11}{2} \cdot 4[/tex]

[tex]A = \dfrac{19}{2} \cdot 4[/tex]

[tex]A = \dfrac{76}{2}[/tex]

[tex]\boxed{A = 38 \text{ in}^2}[/tex]

The amount that would be in the account after 20 years would be $33,277.48.

The doubling time of the emissions would be  23.45 years

The total emissions will be higher in 2030 than 2010 by 100.96%

To find the amount in the account after 20 years, we can use the formula:

A(t) = P(1 + r)^t

Take the 12th root of both sides:

(1 + r) = 2^(1/12)

Now find r:

r = 2^(1/12) - 1 ≈ 0.05946

Now we can find the amount in the account after 20 years with the investment of $10,000:

A(20) = 10000(1 + 0.05946)^20

A(20) = $33,277.48

To find the doubling time for emissions, we can use the formula:

Doubling time = ln(2) / ln(1 + r)

Doubling time = ln(2) / ln(1 + 0.03)

Doubling time = 23.45 years

To find how much higher total emissions will be in 2030 than in 2010, we can use the exponential growth formula:

E(t) = E₀(1 + r)^t

We want to find the ratio of emissions in 2030 to emissions in 2010:

E(20) / E(0) = (1 + 0.03)^20

E(20) / E(0) ≈ 2.0096

Higher emissions = E(20) / E(0) - 1

Higher emissions ≈ 1.0096

So, total emissions will be approximately 100.96% higher in 2030 than in 2010.

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The length of the diagonal AG = 7.56 cm.

A cuboid is a three-dimensional shape that is bounded by six rectangular faces. It is also known as a rectangular prism. A cuboid has six faces, twelve edges, and eight vertices.  

To find the Length of the side AG, find the length of AC by using the Pythagorean formula. Now using the trigonometric ratios formulas find the length of AG.

Here we have a cuboid

Where

AD = 23 cm

DC = 18 cm

∠GAC = 75°  

From the right angle triangle, ADC

=> AC² = AD² + DC²    [ Using the Pythagorean formula ]

=> AC² = (23)² + (18)²

=> AC² = 529 + 324

=> AC = √853 = 29.20    

From the right angle triangle, AGC

=> Cos B = AC/AG

=> Cos 75° = 29.20/AG

=> AG = Cos 75° (29.20)

=> AG = (0.26)(29.20)

=> AG = 7.56

Therefore

The length of the diagonal AG = 7.56 cm.

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Answer:

  94.3

Step-by-step explanation:

You want to know how to fill in the proportion associated with the question, "What is 115% of 82?"

The diagram is intended to help you match parts of the question to parts of the proportion. The attachment shows the intended relation.

The solution is found by multiplying both sides by 82:

  (x/82)·82 = (115/100)·82

  x = 94.3

The question can also be written as a statement:

  "what" is 115% of 82

When written as an equation, "is" means "equals", and "of" means "times":

  what = 115% × 82

Of course, you can use any variable name of your choosing in place of "what". In your diagram, that is "x". And, you know how to write a percentage as a fraction or decimal, so this becomes ...

  x = 115/100 × 82 = 1.15 × 82

All that remains is to evaluate this expression.

  x = 1.15 × 82 = 94.3

So, the answer to the question is ...

  94.3 is 115% of 82.