For George’s party, his dad bought an ice cream cake. After the party, 1/3 of the cake was left. The next day, George ate half of what was left. Choose the model that shows what fraction of cake George ate

Answer:

  A.  The volumes are the same, based on Cavalieri's principle.

Step-by-step explanation:

Cavalieri's principle tells us the volumes of solids will be identical if their cross sectional areas are identical at every height.

A right cylinder and an oblique cylinder of the same height and radius will both have circular cross sections of the given radius at any height. Since the radius is the same, the area of the circle is the same. Hence the requirements of Cavalieri's principle are met, and the cylinders have the same volume.

Based on the average monthly rainfall amounts as shown in the graph, the sum of the average monthly rainfall 11/12 inches.

The average can be found by adding up the monthly rainfall and dividing by 12 months.

Solving gives:

= (1/8 + 1/8 + 1/8 + 2/8+ 2/8 + 5/8 + 7/8 + 9/8 + 13/8 + 15/8 + 16/8 + 16/8) / 12

= (88/8) / 12

= 11/12 inches

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The number will represent a number line from negative 5 to 5 in increments of 1. An open circle is at 3 and a bold line starts at 3 and is pointing to the right.

Inequality are expressions not separated by an equal sign. Given the inequality expression:

3(8 – 4x) < 6(x – 5)

Expand

24 - 12x < 6x - 30

Collect the like terms

-12x-6x < -30 - 24

-18x <-54

x > 3

Hence the number will represent a number line from negative 5 to 5 in increments of 1. An open circle is at 3 and a bold line starts at 3 and is pointing to the right.

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The events are not independent

Say that there are n marbles in the bag

When the first red marble is drawn, the number of marbles in the bags becomes n - 1

This means that the first red marble has changed the probability of drawing the second red marble

This represents a dependent event

Hence, the events are not independent

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The type of transformations that maps triangle ABC onto Triangle prime ABC is a  Reflection across the line y=x ; translation 10 units to the right and 4 units up.

Note that: Triangle ABC has vertices at points  which are:

A(-6,2), B(-2,6) and C(-4,2).

Therefore, the reflection across the line y=x has the rule of"

(x, y) ---(y, x).

Hence:

A(-6,2) -- A''(2,-6);

B(-2,6)--- B''(6,-2);

C(-4,2)----C''(2,-4).

2. The translation 10 units to the right and 4 units up is:

(x, y)----(x+10,y+4).

Hence

A''(2,-6)----A'(12,-2);

B''(6,-2)----B'(16,2);

C''(2,-4)----C'(12,0).

Therefore, Points A'B'C' are said to be exactly of the vertices of that of triangle A'B'C'.

Hence, The type of transformations that maps triangle ABC onto Triangle prime ABC is a  Reflection across the line y=x ; translation 10 units to the right and 4 units up.

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Answer:

π & 4-π

Step-by-step explanation:

Answer: [tex]MP=10, PN=6[/tex]

Step-by-step explanation:

Corresponding sides of similar triangles are proportional, so:

[tex]\\\\frac{MP}{5}=\frac{8}{4}\\\\MP=10\\\\\frac{PN}{3}=\frac{8}{4}\\\\PN=6[/tex]

Answer:

there are 6 people that need to sit down so let's see how we can fill those seeds we will start while finding seats for Alice and bob they can't sit next to each other so will remove one of the seeds them both in the remaining 5 then at a seat between them to ensure that they are not next to each other they are 5 ways to choose seat and 4 ways from those that we make giving 20 ways to seats

Step-by-step explanation:

once a lies and verb are ct we can arrange the remaining people in 4 is equal to 24 west in each other four seats for the total of 20 into 24 4 80 ways to seat everyone

Answer:

6

Step-by-step explanation:

[tex]5(5+3)=4(x+4) \\ \\ 40=4(x+4) \\ \\ 10=x+4 \\ \\ x=6[/tex]

Answer:

18 students

Step-by-step explanation:

Answer:

the range between 18 and 24 is 6

Step-by-step explanation:

24-18=6

This can be proved by using identities of trigonometry.

Given that,  

sin x + sin²x = 1 ⇒ sin x = 1 - sin²x

⇒ sin x = cos²x   (∵sin²x + cos²x = 1)

We have to prove that,

cos¹²x + 3cos¹⁰x + 3cos⁸x + cos⁶x = 1

LHS = cos¹²x + 3cos¹⁰x + 3cos⁸x + cos⁶x

= sin⁶x + 3sin⁵x + 3sin⁴ + sin³x

= (sin²x)³ + 3(sin²x)²sin x + 3(sin²x) (sin x)² + (sin x)³

= (sin²x + sin x)³           (∵(a+b)³ = a³ + 3a²b + 3 ab² + b³)

= 1³ = 1 = RHS

LHS = RHS

Hence the equation is proved since both sides of the equation are equal.

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Answer:

m∠1 = 51°

m∠2 = 16°        

Step-by-step explanation:

First ,check the attached photo.

=================

In the right triangle ABC :

m∠1 = 90 - 39

       = 51°

……………………

In the right triangle ABD :

m∠2 = 90 - 74

       = 16°

Answer:

• ∠ 1 = 51°

• ∠ 2 = 16°

Step-by-step explanation:

•  ∠ 1, the 39° angle, and the right angle marked in red are angles of a triangle.

∴  ∠ 1 + 90° + 39° = 180°             [Angles in a triangle add up to 180°]

⇒ ∠ 1 = 180° - 90° - 39°

⇒ ∠ 1 = 51°

• The small triangle on the top-left of the image consists of ∠ 2, a right-angle (marked in red) and a 74° angle.

∴  ∠ 2 + 90° + 74° = 180°            [Angles in a triangle add up to 180°]

⇒ ∠ 2 = 180° - 90° - 74°

⇒ ∠ 2 = 16°

The value of r in the exponential decay function 7000 = 20000(1-r)^10 is 10%

The equation is given as:

7000 = 20000(1-r)^10

The above equation is an exponential decay function.

So, we have:

7000 = 20000(1-r)^10

Divide both sides of the equation by 20000

0.35 = (1-r)^10

Take the 10th root of both sides

0.35^(1/10) = 1 - r

Evaluate the exponent

0.9 = 1 - r

Add r to both sides

0.9 + r = 1

Subtract 0.9 from both sides

r = 1 - 0.9

Evaluate

r = 0.1

Express as percentage

r = 10%

Hence, the value of r in the exponential decay function 7000 = 20000(1-r)^10 is 10%

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The value of x when y =2 is 8

The inverse proportion to the square of x represented as:

k = yx^2

Where

k represents the constant of proportion

When y = 2, x = 8.

So, we have:

k =2 * 8^2

k =128

Substitute k = 128 in k = yx^2

So, we have:

yx^2 = 128

When y = 2, we have:

2x^2 = 128

Divide by 2

x^2 = 64

Take the square roots

x = 8

Hence, the value of x when y =2 is 8

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Answer:

d=(x^2-x^1)+(y^2-y^1)^2

Answer:

[tex]\mathfrak{Distance \: between \: two \:points}[/tex]

The distance between two points [tex]A(x_1,y_1)[/tex] and [tex]B(x_2,y_2)[/tex] is given by the formula,

[tex]AB= \sqrt{(x_2-x_1)^2+(y_2-y_1)^2 }[/tex]

Proof: Let X'OX and YOY' be the x-axis and y-axis respectively. Then, O is the origin.

Let[tex]A(x_1,y_1) [/tex] and [tex]B(x_2,y_2)[/tex]

be the given points.

Draw AL perpendicular to OX, BM perpendicular to OX and AN perpendicular to BM

Now,[tex]OL=x_1,OM=x_2,AL=y_1 \: and \: BM=y_2[/tex]

[tex] \therefore{AN=LM=(OM-OL)=(x_2-x_1)}[/tex]

[tex] \: \: \: BN=(BM-NM)=(BM-AL)=(y_2-y_1)[/tex]

In right angled triangle [tex] \triangle \: ANB[/tex],by Pythagorean theorem,

We have,

[tex] \: \: \: AB^2=AN^2+BN^2[/tex]

[tex]or,AB^2=(x_2-x_1)^2+(y_2-y_1)^2[/tex]

[tex] \therefore \: AB= \sqrt{(x_2-x_1)^2+(y_2-y_1)^2} [/tex]

Thus ,the distance between the points A(x_1,y_1) and B(x_2,y_2) is given by,

[tex] \implies \boxed{AB= \sqrt{(x_2-x_1)^2+(y_2-y_1)^2} }[/tex]

To calculate accurate vital statistics rates to represent the entire United States, the quotient is multiplied by an appropriate factor such as 1,000 or 100,000 to represent the larger population.

As it is given in the description itself, to compute accurate vital statistics rates for the full United States, multiply the quotient by a suitable factor, such as 1,000 or 100,000 to represent the greater population.

Therefore, to calculate accurate vital statistics rates to represent the entire United States, the quotient is multiplied by an appropriate factor such as 1,000 or 100,000 to represent the larger population.

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The expression of the equation 4(x - 8) + y = 9(x - 2) in slope intercept form is; y = 5x + 14

We want to express 4(x - 8) + y = 9(x - 2) in slope intercept form of y = mx + b.

Let us expand the equation to get;

4x - 32 + y = 9x - 18

Isolate y to get;

y = 9x - 4x - 18 + 32

y = 5x + 14

Thus;

Slope = 5

y - intercept = 14

x - intercept = -2.8

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Answer:

First, use the distributive property and distribute 8 to the [tex](x,2)[/tex] in the first part of the equation to get [tex]8x + 16[/tex].

Next, also use the distributive property to distribute the 4 to the [tex](2x+9)[/tex] in the second part of the equation to get [tex]8x+36[/tex].

The simplified equation looks like this: [tex]8x+16=8x+36[/tex].

Answer:

(-3, 1)

Step-by-step explanation:

the easiest thing to do is look at the graph and find the point where the two lines intersect

Answer:

(-1,-2)

Step-by-step explanation:

See attached graph.

Option C. The function that has the table here has  exactly one x-intercept.

As the signs of the functions are changing, we can see that there is a crossing to the x axis.

This tells us that the zero value that we see is the x intercept and there is no other x intercept in the question.

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Answer:

3x²y (2x²-3y²+ 4y)

Step-by-step explanation:

Factor the expression 6x^4y-9x²y³+12x²y².

First, check if there is a GCF. The GCF is 3x²y.

3x²y (2x²-3y²+ 4y).

This equation cannot be factored further. Therefore, the answer is
3x²y (2x²-3y²+ 4y).

Hope this helps!

Answer:

3x²y(-3y²+4y+2)

Step-by-step explanation:

6x⁴y-9x²y³+12x²y²

solution:

first common

3x²y(2x²-3y²+4y)

3x²y(-3y²+4y+2)

Answer:

$92,250 per town

Step-by-step explanation:

See attached image.

The two towns may be connected with a straight line, the road, that can be made into the hypotenuse of a right triangle.  The twio sides are calculated from the coordinates and shown in the image.  Then use the phythagorean theorem to calculate the hypotenuse.  The towns are 12.3 km apart.  The total road cost would be:

 ($150,000/km)(12.3 km) = $1,845,000, or $92,250 per town.

The value of the function is h(w - 5) = w^2 - 11w +25

The function is given as:

h(y) = y^2 - y - 5

Substitute w - 5 for y

h(w - 5) = (w - 5)^2 - (w - 5) - 5

Expand the equation

h(w - 5) = w^2 - 10w + 25 - w + 5 - 5

Evaluate the like terms

h(w - 5) = w^2 - 11w +25

Hence, the value of the function is h(w - 5) = w^2 - 11w +25

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Answer:

b.  -x + y = 0

Step-by-step explanation:

Direct variation:

Direct variation means "y varies directly as x”:

[tex]y \propto x \implies y=kx[/tex]

where k is the (non-zero) constant of variation.

To determine which of the given equations represents a direct variation, isolate y for each and compare with the direct variation equation.

Equation a

[tex]y=\dfrac{4}{3}x-2[/tex]

This is not a direct variation equation as there is an addition of -2.

Equation b

[tex]-x+y=0[/tex]

[tex]\implies y=x[/tex]

This is a direct variation equation where the constant of variation is 1.

Equation c

[tex]xy=8[/tex]

[tex]\implies y=\dfrac{8}{x}[/tex]

This is not a direct variation equation as y is inversely proportional to x.

Equation d

[tex]y=14[/tex]

This equation does not include the variable x, and so is therefore not a direct variation equation.

If y varies inversely with x then the line must be linear and increasing

And the y intercept must be 0 in y=mx+c

#1

c=-2

#2

Yes

#3

No

#4

Parallel to x axis no direct variation

The fraction m/n that lies between 0 and 1 is equivalent to the real number m · n⁻¹, where m, n > 0 and m < n.

By algebra, we know that fractions, a kind of rational numbers, are real numbers of the form m/n, where m and n are integers and n ≠ 0.

In addition, the operator "/" means division, an operator derived from the existence of multiplicative inverse and fundamental operation of multiplication in the real field.

Thus, the rational number m/n is equivalent to the number m · n⁻¹. If the rational number is between 0 and 1, then m, n > 0 and m < n.

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Answer:

a) is the correct answer

The quotient of 24x2y + 8xy2 + 8xy and -4xy is -6x - 2y - 2

The statement is given as:

Divide 24x2y + 8xy2 + 8xy by -4xy

The quotient is represented as:

Quotient = (24x^2y + 8xy^2 + 8xy)/(-4xy)

Factor out -4xy

Quotient = (-4xy)(-6x - 2y - 2)/(-4xy)

Evaluate the quotient

Quotient = -6x - 2y - 2

Hence, the quotient of 24x2y + 8xy2 + 8xy and -4xy is -6x - 2y - 2

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Option B is correct. 0.39 goldfish per cubic feet is the density of the tank given that there are 22 goldfishes and volume of the tank is 56 cubic feet. This can be obtained by using the formula of density.

Given that there are 22 goldfishes,

mass of tank = 22 goldfishes

volume of tank = 56 cubic feet

                      = 22/56

                      = 0.39 goldfish per cubic feet

Hence 0.39 goldfish per cubic feet is the density of the tank given that there are 22 goldfishes and volume of the tank is 56 cubic feet.

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Answer: 0.39 goldfish

Step-by-step explanation:

The correct option is A. m ≤ (8)(20)

The inequality which gives the possible values of 'm' is m ≤ (8)(20).

A declaration of an order relationship between two numbers or algebraic expressions, such as greater than, greater than or equal to, less than, or less than or equal to.

An inequality is a relation that compares two numbers or other mathematical expressions in an unequal way. The majority of the time, size comparisons between two numbers on the number line are made.

According to the question;

Terrence's car contains 8 gallons of fuel.

Terrence can drive the car 'm' miles using the fuel currently in the car.

The car can drive 20 miles per gallon of fuel,(which is maximum fuel capacity of the car to drive).

Then,

The total miles 'm' covered by the car is 8×20 which is maximum capacity of the car to travel.

Thus, total miles covered by the car are less than the maximum value which is given by the inequality-

m ≤ (8)(20)

Therefore, the inequality which gives the possible values of 'm' is m ≤ (8)(20).

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The complete question is-

Terrence's car contains 8 gallons of fuel. He plans to drive the car 'm' miles using the fuel currently in the car. If the car can drive 20 miles per gallon of fuel, which inequality gives the possible values of 'm'?

answer choices

A. m ≤ (8)(20)

B. m ≥ (8)(20)

C. 8 ≤ 20m

D. 8 ≥ 20 m