Geometry- Examine the diagrams below. What is the geometric relationship between the labeled angles? What is the relationship of their measures? Then, use the relationship to write an equation and solve for x.

The perimeter of the rectangle:

[tex]P_R=2(4x+2+2x)=2(6x+2)=(2)(6x)+(2)(2)=12x+4[/tex]

The perimeter of the square:

[tex]P_S=2(5x-3+3x+1)=2(8x-2)=(2)(8x)+(2)(-2)=16x-4[/tex]

The perimeter of the rectangle is equal to the perimeter of the square. We have the equation:

[tex]12x+4=16x-4\qquad|\text{subtract 4 from both sides}\\\\12x=16x-8\qquad|\text{subtract 16x from both sides}\\\\-4x=-8\qquad|\text{divide both sides by (-4)}\\\\x=2[/tex]

The side lengths of the square:

[tex]5x-3\to5(2)-3=10-3=7[/tex]

The side lenghts of the rectangle:

[tex]4x+2\to4(2)+2=8+2=10\\2x\to2(2)=4[/tex]

Other method.

If the first figure is a square, then the length of the sides are equal.

Therefore

[tex]5x-3=3x+1\qquad|\text{add 3 to both sides}\\\\5x=3x+4\qquad|\text{subtract 3x from both sides}\\\\2x=4\qquad|\text{divide both sides by 2}\\\\x=2[/tex]

Further part of the solution as above.

[tex]\displaystyle\\4^2+7^2+\ldots+(3n+1)^2=\\\\\sum_{k=1}^n(3k+1)^2=\\\\\sum_{k=1}^n(9k^2+6k+1)=\\\\\sum_{k=1}^n9k^2+\sum_{k=1}^n 6k+\sum_{k=1}^n1=\\\\9\sum_{k=1}^nk^2+6\sum_{k=1}^n k+n=\\\\9\left(\dfrac{n(n+1)(2n+1)}{6}\right)+6\left(\dfrac{n(n+1)}{2}\right)+n=\\\\\dfrac{3n(n+1)(2n+1)}{2}+3n(n+1)+n=\\\\\dfrac{3n(2n^2+n+2n+1)}{2}+3n^2+3n+n=\\\\\dfrac{3n(2n^2+3n+1)}{2}+3n^2+4n=\\\\\dfrac{6n^3+9n^2+3n}{2}+\dfrac{6n^2+8n}{2}=\\\\\dfrac{6n^3+15n^2+11n}{2}[/tex]

Given function : [tex]y=1.05(1.46)^x[/tex]

We need to identify a " initial amount", b "growth factor", r " rate of growth".

We know, exponential growth formula

[tex]y=a(b)^x[/tex], where a is initial amount, b is growth factor. On comparing with given function let us find values of a and b.

[tex]y=1.05(1.46)^x[/tex] ⇔ [tex]y=a(b)^x[/tex].

We can see a= 1.05 and b = 1.46.

Now, b=1+r.

Therefore, 1+r =1.46.

Subtracting 1 from both sides, we get

1+r-1 =1.46-1

r = 0.46.

On converting 0.46 into percentage, we get

0.46 × 100 = 46.

Therefore, intial amount a = a= 1.05 , growth factor b = 1.46, and the rate of growth r= 46%.

Final answer:

The initial amount, a, is 1.05, and the growth factor, b, is 1.46 in the exponential function y=1.05(1.46)ˣ. The growth rate, r, when expressed as (1+r), is 46%.

Explanation:

To identify the initial amount a and the growth factor b in the exponential function y=1.05(1.46)ˣ, we look at the standard form of an exponential function, which is y=a×bˣ. Comparing this with the given function, we can see that the initial amount a is 1.05 and the growth factor b is 1.46.

To express the growth factor in the form of (1+r), where r is the rate of growth, we can set b = 1 + r. This means that r would be b - 1, which is 1.46 - 1 = 0.46. To express r as a percentage, we multiply by 100, resulting in a growth rate of 46%.

Simplifying

15 + -5(4x + -7) = 50

Reorder the terms:

15 + -5(-7 + 4x) = 50

15 + (-7 * -5 + 4x * -5) = 50

15 + (35 + -20x) = 50

Combine like terms: 15 + 35 = 50

50 + -20x = 50

Add '-50' to each side of the equation.

50 + -50 + -20x = 50 + -50

Combine like terms: 50 + -50 = 0

0 + -20x = 50 + -50

-20x = 50 + -50

Combine like terms: 50 + -50 = 0

-20x = 0

Solving

-20x = 0

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Divide each side by '-20'.

x = 0.0

Simplifying

x = 0.0

Answer is 71.1 because 0.9 x 079 in a calculator is 71.1

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Here you go the answers below

Zoe will have 210 beads left after making all six of the friendship bracelets.

Zoe has 1500 beads and wants to make 6 friendship bracelets, with 215 beads needed for each bracelet. To find the total number of beads needed for all the bracelets, multiply the beads required for one bracelet (215) by the number of bracelets (6). This equals 1290 beads. To find the number of beads Zoe will have left after making all the bracelets, subtract the total number of beads needed (1290) from the initial number of beads (1500). Therefore, Zoe will have 210 beads left.

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A) N +2 = Q which equals

A) N -Q = -2

B) .05N +.25Q = 3.50  multiplying A) by .25

A) .25N -.25Q = -.5  then adding A) and B)

.30N = 3

Nickels = 10 Quarters = 12

*************DOUBLE CHECK ***************

.05 nickels = $0.50  .25 Quarters = $3.00

Melinda has 12 quarters and 10 nickels in her bank, totaling $3.50. By setting up and solving equations based on the values of the coins, we were able to determine the exact number of each type of coin.

Little Melinda has nickels and quarters in her bank, with two fewer nickels than quarters. The total amount of money she has is $3.50. To determine how many coins of each type she has, we need to set up equations based on the values of the coins and the given conditions.

Let's define:
Q = number of quarters
N = number of nickels
Since each quarter is worth 25 cents and each nickel is worth 5 cents, we have the following equations:

1. N = Q - 2 (since she has two fewer nickels than quarters)
2. (5 × N) + (25 × Q) = 350 cents (because the total amount is $3.50)

Substitute the first equation into the second to find the number of each coin:
5(Q - 2) + 25Q = 350
5Q - 10 + 25Q = 350
30Q - 10 = 350
30Q = 360
Q = 12

Then, substitute Q = 12 into the first equation to find N:
N = 12 - 2
N = 10

Therefore, Melinda has 12 quarters and 10 nickels.

D

the volume (V ) of a cube = s³ ( where s is the length of the side )

here s = 20, hence

V = 20³ = 8000 cm³

3a + 4bc - d = 5a - 8j

Add d to both sides.

3a + 4bc = 5a - 8j + d

Subtract 4bc from each side.

3a = 5a - 8j + d - 4bc

Subtract 5a from both sides.

-2a = -8j + d - 4bc

Divide both sides by -2

a = [tex]\frac{(-8j -4bc + d)}{-2}[/tex]

A=2bc+-1/2d+4j would be your answer

The answer is  B. 525 square units.

The polygon pictured is a composite figure made up of several rectangles. To find the total area, we can find the area of each of the rectangles and sum them together.

From the image we can find the following areas:

Top left rectangle: 9 square units

Top right rectangle: 14 x 3 = 42 square units

Bottom left rectangle: 7 x 14 = 98 square units

Bottom right rectangle: 27 square units

Summing the areas of each rectangle together gives us a total area of  9 + 42 + 98 + 27 = 176 square units.

However the prompt specifies that this is not the total area. There is a small square removed from the bottom left rectangle.  We can find the area of this square by subtracting the length of the small rectangle’s missing side (3 units) from the length of the whole rectangle’s side (14 units) and squaring the result. Subtracting 3 from 14 gives us 11 and squaring 11 gives us 121.

So the final area of the polygon is 176 - 121 = 55 square units.

Of the answer choices provided, only  55 square units matches our calculation. So the answer is  B. 525 square units.

The mean could increase from 25 to 27.

The mean number of books in each box is 25. If Javier fills two more boxes, one with 26 comic books and one with 30 comic books, the mean could increase or stay the same. Let's calculate the mean with and without the additional boxes:

Mean without additional boxes: (4 boxes * 25 books per box) / 4 = 25

Mean with additional boxes: (6 boxes * 25 books per box + 26 books + 30 books) / 8 = (150 + 26 + 30) / 8 = 27

Therefore, the mean could increase from 25 to 27 if Javier fills two more boxes with 26 and 30 comic books.

A ratio is used to describe how two or more quantities are related. For example, we might say that a fruit drink calls for sugar crystals to be mixed with water in a ratio of 2:8. This means that for every 2 scoops sugar crystals, there will need to be 8 ounces of water. If there were 200 scoops of sugar crystals, there would need to be 800 ounces of water.

Sure, we'll start off by simplifying the ratio 2:8. Simplifying it gives us 1:4.

Our aim is to find a number, let's denote it by x, such that the ratio 2:x will be the same as 1:4.

To find such an x, we need to set up an equation using the principle of equivalence of ratios. This gives us 2/x = 1/4.

Now we proceed to solve this equation for x. We do this by cross-multiplying the fractions in the equation.

Cross-multiplication gives us 2*4 = x*1. Simplifying this yields 8 = x.

Hence, we can conclude that the number x that makes the ratio 2:x equivalent to 2:8 is 8.

Answer:

a) 87 double scoops and 113 single scoops.

b) 287 scoops of ice cream.

Step-by-step explanation:

1. You need to make a system of equations, where double scoops are represented by [tex]x[/tex] and the single scoops are represented by [tex]y[/tex]:

[tex]\left \{ {{x+y=200} \atop {1.25x+y=221.75}} \right.[/tex]

2. You can apply the Elimination Method:

- Multiply the first equation by -1 to cancel the variable [tex]y[/tex] from the system. Add both equations:

[tex]0.25x=21.75[/tex]

- Solve for [tex]x[/tex]:

[tex]x=\frac{21.75}{0.25}\\x=87[/tex]

- Substitute the value of [tex]x[/tex] into one of the original equations and solve for [tex]y[/tex]:

[tex]87+y=200\\y=113[/tex]

3. You can calculate the total number of scoops of ice cream that where sold as below:

[tex]2x+y[/tex]

4. Substitute values:

[tex]2(87)+113=287[/tex] scoops of ice cream

Answer:  The number of single scoop cones sold is 113, number of double scoop cones is 87 and the total number of cones sold is 287.

Step-by-step explanation:  Given that at a a summertime poolside snack bar, 200 ice cream scoops were sold in one day.

Double scoops were sold for $1.25 each and single scoops were sold for $1 each. The proceeds from the sale of the cones were $221.75.

We are to find the number of single scoop cones, double scoop cones and the number of scoops of ice cream that sold.

We Let x and y represents the number of cones of single scoop and double scoop ice cream that sold.

Then, according to the given information, we have

[tex]x+y=200\\\\\Rightarrow x=200-2y~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]

and

[tex]1\times x+1.25y=221.75\\\\\Rightarrow x=221.75-1.25y~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(ii)[/tex]

Comparing equations (i) and (ii), we get

[tex]200-y=221.75-1.25y\\\\\Rightarrow 1.25y-y=221.75-200\\\\\Rightarrow 0.25y=21.75\\\\\Rightarrow y=\dfrac{21.75}{0.25}\\\\\Rightarrow y=87.[/tex]

From equation (i), we get

[tex]x=200-87=113.[/tex]

Therefore, total number of scoops that were sold is

[tex]113+2\times87=113+174=287.[/tex]

Thus, the number of single scoop cones sold is 113, number of double scoop cones is 87 and the total number of cones sold is 287.

Answer:

x=1.3288 or -1.1288

Step-by-step explanation:

Given [tex]20x^{2}-30=4x[/tex]

    Subtract 4x from both sides

         [tex]20x^{2}-4x-30=0[/tex]

Since we can factor out 2 from whole equation, let's factor out 2.

         [tex]2(10x^{2}-2x-15)=0[/tex]

Divide with 2 on both sides

         [tex]10x^{2}-2x-15=0[/tex]

We got the quadratic equation, we can solve for x using quadratic formula.

[tex]x=\frac{-b\pm\sqrt{b^{2}-4ac} } {2a}[/tex]

[tex]x=\frac{2\pm\sqrt{2^{2}-4X10X(-15)} } {2X10}[/tex]

[tex]x=\frac{2\pm\sqrt{604} }{20}[/tex]

[tex]x=\frac{2\pm2\sqrt{151}}{20}[/tex]

[tex]x=1.3288or-1.1288[/tex]

The cube of the product of 3 and a number is expressed as (3x)³ or 27x³, which involves cubing the numeric part and multiplying the exponent of the variable by 3.

The question is asking about the volume of a cube with sides of length a and also about the process of cubing the product of a number and three. When we cube a product, such as 3 times an unknown number, we are calculating the result of multiplying that product by itself three times.

For example, if our number is x, we would express this as (3x)³ or 27x³. This is because when we cube a product, we cube the numeric part as usual, obtaining 3³ which is 27, and we multiply the exponent of the exponential term by 3, thus x becomes x³.

B.) Any temperature change is the original temperature plus the numerical value of the change. (temp + change) If it is an increase, the number will be positive. If it is a decrease, the number will be negative. 

Since there is a decrease of 11, c would be replaced with -11. When you plug this into your expression (85 + c: 85 + -11), the answer to this would be 74.

                                                 hope it helps

Final answer:

Explaining how to define a variable, write an expression, and evaluate the temperature after a decrease in Fahrenheit degrees.

Explanation:

Variable: Let c represent the temperature change.

Expression: To find the new temperature after a decrease of 11°F, the expression is 85 - c.

Evaluation: By substituting c with -11 (decrease of 11°F), 85 - (-11) equals 96°F.

Use the distance formula to find the distance between points C (5,6) and D (2,2).

d=[tex]\sqrt{(x2-x1)^2+(y2-y1)^2}[/tex]

Plug the corresponding values in to the equation.

d=[tex]\sqrt{(2-5)^2+(2-6)^2}[/tex]

Then simplify the rest of the equation.

d=[tex]\sqrt{(-3)^2+(-4)^2}[/tex]

d=[tex]\sqrt{9+16}[/tex]

d=[tex]\sqrt{25}[/tex]

d=[tex]5[/tex]

The distance of the two points CD is 5.

To find the distance between point C=(5,6) and point D=(2,2), we use the distance formula which results in d=5.0 units rounded to the nearest tenth.

To calculate the distance between point C and point D, we use the distance formula derived from the Pythagorean theorem. The coordinates given are C=(5,6) and D=(2,2). The distance formula is:

d = √{(x₂ - x₁)² + (y₂ - y₁)²}

Substituting the coordinates of points C and D into the formula gives us:

d = √{(2 - 5)² + (2 - 6)²}

This simplifies to:

d = √{(-3)² + (-4)²}

Further simplifying:

d = √{9 + 16}

d = √{25}

Finally, taking the square root of 25:

d = 5

Therefore, the distance between point C and point D, rounded to the nearest tenth, is 5.0 units.

Answer:

1. The fixed costs is $40 and the cost per mile is [tex]\$\frac{3}{2}[/tex].

2.Therefore, the coffees did they sell is, 85 coffees

Step-by-step explanation:

1. To find the fixed cost and Cost per mile.

Since, a shipping company charges a fixed amount for their services and a certain amount per mile , they must travel.

Let the number of fixed amount be x and the certain amount per mile be y

then, the general linear equation for this we have:

[tex]x+y=C[/tex] where C is the charges cost by the shipping company.

It is given that for a 50 mile trip company costs $115 and an 80 mile trip costs $157.

we have an equation in linear form from the above data:

[tex]x+50y=\$115[/tex]            .....(1)

[tex]x+80y=\$157[/tex]           ....(2)

Now, solving these equation simultaneously we get,

[tex]y=\frac{3}{2}[/tex]

Substitute the value of y in equation (1),

[tex]x+50\times\frac{3}{2}=\$115[/tex]

[tex]x+75=\$ 115[/tex]

⇒[tex]x=\$40[/tex]

Therefore, the fixed costs is $40 and the cost per mile is [tex]\$\frac{3}{2}[/tex].

2.

Let the hot chocolate sells be x and that of coffee be y.

At one game they sold, $200 worth of drinks and used 295 cups. if hot chocolate sells for $0.75 and coffee sell for $0.50.

An equation from the given condition are given by:

[tex]x+y=295[/tex]

[tex]0.75x+0.50y=200[/tex]

Solving these equation simultaneously we get the value of y=85

Therefore, the coffees did they sell is, 85 coffees

2) angles are vertical so they are congruent- 7x-27= 4x+12 --> subtract 4x from both sides so 3x-27=12 --> add 12 to both sides so 3x=39 (divide both sides by 3) so then x= 13

3) again, vertical angles so 8x-120= 4x+16 --> subtract 4x so that 4x-120=16 --> add 120 to both sides and then 4x=136, divide both by 4 and x= 34

hope this helps:))

Answer:

-29, -30, -31, -32, -33

Step-by-step explanation:

add them together

what do you mean ¨ what is the distance between¨ what is the rest of the question?

It must be 1.

Factors of prime numbers are 1 or themselves, the only factor common to all prime numbers is 1.

Echy~

The Greatest Common Factor (GCF) of two prime numbers is always 1.

The Greatest Common Factor (GCF) of two prime numbers is always 1 because prime numbers are numbers with only two factors: 1 and themselves.

For example, the GCF of 3 and 5, both prime numbers, is 1 because they do not have any common factors other than 1.

Therefore, when dealing with two prime numbers, the GCF will always be 1.

Answer:

y = 3x^2+15x-18

Step-by-step explanation:

Parabola has an equation with one variable in degree 2 and other in degree 1.

Given that y=3(x-1)(x+6) is the function.

y is of degree 1 and x of degree 2

Standard form of these types of parabolas would be

y =ax^2+bx+c.

To make the given equation in standard form, we multiply all factors on  the right side

y = 3(x-1)(x+6) = 3(x^2+5x-6)

= 3x^2+15x-18

a=3 b = 15 and c =-18

y = 3x^2+15x-18 is the standard form of the parabola

Answer:

y = 3x^2 + 15x - 18

Step-by-step explanation:

To represent the function y = 3 (x - 1) (x + 6), start by by expanding the equation by multiplying the common factor with each term inside the brackets to get:

y = 3 (x - 1) (x + 6)

y = (3x - 3) (x + 6)

Now multiply both the terms with each other to get:

y = 3x^2 + 18x - 3x - 18

Arrange the like terms together and add them:

y = 3x^2 + 15x  - 18

Hence, y = 3x^2 + 15x - 18 is the standard form of the given function.

These are the polygons

The transformation of Polygon A to Polygon B could involve translation, rotation, or reflection. Observe differences in position and orientation between the two polygons to identify the type of transformation.

The transformation of a polygon to another can involve a series of moves such as translation (sliding), rotation (turning around a point), or reflection (flipping over a line). By closely comparing the two polyhgons A and B, you can determine which type of transformation has taken place.

For instance, if every point on polygon A has moved in the same direction and distance to form polygon B, a translation has taken place. If polygon B can be obtained by turning polygon A around a point, then a rotation has occurred. If polygon B is a mirror image of polygon A, then it has been reflected.

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By substituting x=1 into the function f(x)=3x5^x, we find that the point (1,15) is on the graph.

To find a point that lies on this graph, we can select a value for x, substitute it into the function, and calculate the corresponding y value. Let's try x=1:

f(1) = 3(1)51 = 3(5) = 15.

Therefore, the point (1,15) is on the graph of f(x)=3x5^x.