Every day, Austin packs his favorite sandwich in a plastic sandwich box. Austin's sandwich box is 13 centimeters long and 12 centimeters wide, but only 3 centimeters tall. What is the volume of Austin's sandwich box?

The volume of the solid is -5.54 cubic units.

The region is bounded by the parabola x = 1 + (y-2)^2 and the line x = 2. To use the method of cylindrical shells, we imagine rotating this region around the x-axis, which gives us a solid shape.

The shell has thickness Δx and height y, and is located at a distance x from the y-axis. To find the volume of the solid, we'll add up the volumes of all the cylindrical shells.

The volume of a cylindrical shell is given by the formula V = 2πrhΔx, where r is the radius of the shell and h is its height. In this case, the radius of the shell is x - 2 (the distance from the y-axis to the edge of the region), and the height is y. So we have:

V = 2π(x - 2)yΔx

To find the total volume of the solid, we integrate this expression over the range of x values that corresponds to the region we're interested in. The region is bounded by x = 1 + (y-2)^2 and x = 2, so we integrate with respect to x from x = 1 + (y-2)^2 to x = 2:

V = ∫[1+(y-2)²]^2 2π(x - 2)y dx

To evaluate this integral, we can use the substitution u = y - 2, which gives us:

V = ∫1^0 2π(u^2 + 3) (1 + u)^2 du

This integral can be evaluated using the power rule and the substitution v = 1 + u:

V = 2π/5 [(1 + u)^5 - (1 + u)^3] from 1 to 0

V = 2π/5 [1 - 32/5]

V = 2π/5 ( -22/5)

V = - 8.8π/5

So the volume of the solid is approximately -5.54 cubic units. Note that the negative sign means that the solid is oriented in the opposite direction from what we might expect - in other words, it has a hole in it.

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Answer:

Step-by-step explanation:

To simplify the expression, we need to divide -3x^9 + 2x^8 - 8 by x^4.

The procedure is as follows:

Divide -3x^9 by x^4: -3x^9 / x^4 = -3x^5

Divide 2x^8 by x^4: 2x^8 / x^4 = 2x^4

Divide -8 by x^4: -8 / x^4 = -8x^-4

Combine the results from steps 1-3: -3x^5 + 2x^4 - 8x^-4

Note that the exponent of x in each term is either positive or zero. We can't simplify the expression further since we can't simplify the exponents any more.

Answer:

c) [tex]10x<7z[/tex]

Step-by-step explanation:

[tex]5x<3y \implies 10x<6y \\ \\ \therefore 10x<6y<7z \implies 10x<7z[/tex]

Mr. Kazoo needs to buy 6 boards to complete his fencing as he can not buy exactly 5.71 boards.

A numerical expression is a mathematical statement written in the form of numbers and unknown variables.

We can form numerical expressions from statements.

Given, Mr. Kazoo is planning to build a fence.

The gate is 40 inches wide.

Therefore, He needs to buy (40/7) = 5.71 boards.

But, He can not buy a board 0.71 part of the whole part, So he needs to buy 6 boards to complete his fencing.

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52s represents the cost for the snowboard times s days.

What is an expression ?

An expression consists of one or more numbers or variables along with one more operation.

Given :

expression : 52s + 10

Since it represents , total cost to rent a snow board for s days plus the fee for the helmet

Therefore , 52 is the cost of the snowboard rental per day.

s represents the day(s) that the equipment is rented out.

Hence , 52s represents the cost for the snowboard times s days.

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Given question is incomplete , the complete question is given below:

At the ski shack customers can rent skis snowboards boots and poles by the day. josh rents a snowboard and a helmet. The expression 52s + 10 represents the toatak cost to rent a snow board for s days plus the fee for the helmet what dose the term 52s represent​

A structure may lack structural integrity if it experiences overloading and material degradation.

A structure may lack structural integrity if it experiences:

Overloading, where the loads placed on the structure exceed its design capacity, leading to failure of some structural components or members.

Material degradation, where the structural materials lose their strength and stiffness over time due to environmental factors such as corrosion or excessive temperature.

ii. The basic types of structural systems are:

Frame Structures, where the structural components are arranged in a series of interconnected frames that can resist lateral and gravity loads. Examples include steel frames and concrete frame structures.

Load-Bearing Wall Structures, where the structural components are arranged in a series of interconnected walls that bear the weight of the structure and transfer loads directly to the foundation. Examples include masonry and concrete walls.

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Answer:

689.85 liters

Step-by-step explanation:

1 gallon =  about 3.78 liters

182.5 gallon = ? liter

We take  

182.5 times 3.78 = 689.85 liters

So, 182.5 gallons is about 689.85 liters

㏒Answer:

Step-by-step explanation:

The difference between 3.25 and 1.75 is done and marked on the number line as below.

What is meant by a number line?

A number line is a representation of a graduated straight line that is used to represent real numbers visually. It is assumed that every point on a number line corresponds to a real number and that every real number corresponds to a point. This line is used to compare integers on an endless line that extends on both sides, either horizontally or vertically, at equal intervals.

We are asked to plot the difference between 3 [tex]\frac{1}{4}[/tex] and 1 [tex]\frac{3}{4}[/tex] on the number line.

3 [tex]\frac{1}{4}[/tex] = 3 + 1/4 = 3 + 0.25 = 3.25

1 [tex]\frac{3}{4}[/tex] = 1 + 3/4 = 1 + 0.75 = 1.75

Difference = 3.25 - 1.75 = 1.5 = 1 [tex]\frac{1}{2}[/tex]

This can be shown on the number line below.

Each division has the value of 1/4 = 0.25.

We have marked 3[tex]\frac{1}{4}[/tex]  = 3.25 on the number line.

Since we are subtracting 1 [tex]\frac{3}{4}[/tex] = 1.75 from 3.25, we move 7 divisions backward from 3.25 as 7 * 0.25 = 1.75.

So after moving 7 divisions backwards, we land on 1.5, which is the difference we found earlier.

Therefore the difference between 3.25 and 1.75 is done and marked on the number line as below.

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Answer:

Step-by-step explanation:

To factorise the expression, we can use the factor theorem which states that if a polynomial p(x) has a factor (x - r), then p(r) = 0.

We can use this theorem to find the factors of the polynomial expression:

(a - 2b)^3 + (2a - b)^3

We can write (a - 2b)^3 as (a - 2b)(a^2 - 4ab + 4b^2). Using the factor theorem, we see that:

(a - 2b)(a^2 - 4ab + 4b^2) = 0 when a = 2b

Similarly, we can write (2a - b)^3 as (2a - b)(4a^2 - 4ab + b^2). Using the factor theorem, we see that:

(2a - b)(4a^2 - 4ab + b^2) = 0 when a = b/2

Now we have two equations: a = 2b and a = b/2, which we can use to solve for a and b in terms of each other. We see that:

a = 2b = b/2

Multiplying both sides by 2, we get:

2a = b

We can substitute this back into the original expression:

(a - 2b)^3 + (2a - b)^3 = (a - 2(2a))^3 + (2a - (2a))^3 = (-3a)^3 + (0)^3 = -27a^3

So the factorised form of the expression is:

(a - 2b)^3 + (2a - b)^3 = -27a^3.

Work Shown:

Use the sum of cubes factoring rule

x^3 + y^3 = (x+y)(x^2 - xy + y^2)

In this case,

So,

x^3 + y^3 = (x+y)(x^2 - xy + y^2)

(a-2b)^3 + (2a-b)^3 = (a-2b+2a-b)((a-2b)^2 - (a-2b)(2a-b) + (2a-b)^2)

(a-2b)^3 + (2a-b)^3 = (3a-3b)((a^2-4ab+4b^2) - (2a^2-ab-4ab+2b^2) + (4a^2-4ab+b^2))

(a-2b)^3 + (2a-b)^3 = (3a-3b)((a^2-4ab+4b^2) - (2a^2-5ab+2b^2) + (4a^2-4ab+b^2))

(a-2b)^3 + (2a-b)^3 = (3a-3b)(a^2-4ab+4b^2 - 2a^2+5ab-2b^2 + 4a^2-4ab+b^2)

(a-2b)^3 + (2a-b)^3 = (3a-3b)(3a^2-3ab+3b^2)

(a-2b)^3 + (2a-b)^3 = 3*3(a-b)(a^2-ab+b^2)

(a-2b)^3 + (2a-b)^3 = 9(a-b)(a^2-ab+b^2)

The answer has been fully confirmed with WolframAlpha.

If the two cars collide and couple, then their final velocity will be 1.91 meters per second.

According to the principle of momentum conservation, movements are only modified by the action of forces as they are outlined by Newton's equations of motion; momentum is never generated nor destroyed inside a problem domain.

The conservation of momentum is given as,

m₁u₁ + m₂u₂ = (m₁ + 2₂)v

Car 1 is moving at a speed of 7 m/s and weighs 3000 kg. Car 2 is not moving and weighs 8000 kg.

3,000 x 7 + 8,000 x (0) = (3,000 + 8,000) x v

21,000 + 0 = 11,000v

v = 21,000 / 11,000

v = 1.91 meters per second

If the two cars collide and couple, then their final velocity will be 1.91 meters per second.

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Answer:

m = -4

Step-by-step explanation:

-2 ( m + 1 ) = 10

( m + 1 ) = 10 / -2

m + 1 = -5

m = -4

Answer:

360

Step-by-step explanation:

Every 1/2pi is 90 degrees.

Going down the list...

1/2pi is 90

pi is 180

3/2pi is 270

2pi is 360

The probability of getting exactly one head after flipping a coin three times is equal to ( 3 / 8 ) .

Possible outcomes of flipping a coin three times are :

= { HHH, THH, HTH, HHT, TTH, THT, HTT, TTT }

Possible outcomes of getting exactly one head are:

= { TTH, THT, HTT }

Total number of outcomes are equal to 8

Favourable number of outcomes are 3

Probability of getting exactly one head is equal to

= ( Number of favourable outcomes ) / ( total number of outcomes)

= 3 / 8

Therefore, the probability of getting exactly one head by flipping a coin three times is equal to ( 3 / 8 ).

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a) Standard factored form is 2^4 × 3^2 × 5.

b) Standard factored form is 2^18 × 5^8.

c) There are 6 zeros at the end of (20!)^2

If n is an integer and n > 1, then  factorial of n is the product of n and every other positive integer that is less than n.

We have to find the standard factor form of 6!

6! = 6 × 5 × 4 × 3 × 2 × 1

Then, we can simplify by grouping the factors of 6! into pairs that multiply to 10:

6! = 2 × 3 × 2 × 2 × 5 × 1 × 3 × 2 × 1 = 2^4 × 3^2 × 5

Therefore, 6! in standard factored form is 2^4 × 3^2 × 5.

Part b

To write 20! in standard factored form, we start with 20 and multiply by every positive integer less than 20:

20! = 20 × 19 × 18 × ... × 3 × 2 × 1

To simplify this expression, we can group the factors into pairs that multiply to 10

20! = (2 × 10) × (2 × 9) × (2 × 8) × ... × (2 × 1) × (5 × 3) × (5 × 2) × 5

We can then combine the factors of 2 and the factors of 5:

20! = 2^18 × 5^8

Therefore, 20! in standard factored form is 2^18 × 5^8.

Part C

The number of zeros at the end of (20!)^2 in decimal form is equal to the number of factors of 10 in (20!)^2. Since 10 = 2 × 5, we need to count the number of factors of 2 and 5 in (20!)^2.

The number of factors of 2 in (20!)^2 is the number of even integers less than or equal to 20, which is 10.

The number of factors of 5 in (20!)^2 is the number of multiples of 5 less than or equal to 20, which is

5, 10, 15, and 20.

However, since we are squaring (20!), we also need to count the number of factors of 25, which is 2:

5^2 and 10^2.

There are 6 zeros at the end of (20!)^2 when it is written in decimal form.

Therefore, each part has been solved

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1.The athlete that did the most sit-up per minute is D'andrea with 44 sit up per minute

2. 22 Miles in 20 minutes < 38 Miles in 30 minutes

3. The rate that could be Jessica rate is 1 tile in 3 minutes

4. The best rate to find the best value of card is 6 greeting cards for $23.40

5. Elias can fold 3boxes in a minute and Dionne can fold 3.5 boxes in a minute.

A rate is a special ratio in which the two terms are in different units. For example, if a 12-ounce can of corn costs 69¢, the rate is 69¢ for 12 ounces. This is not a ratio of two like units, such as shirts. This is a ratio of two unlike units: cents and ounces.

Rate per minute is the quantity divide by no of minute.

For example Casey did 93 sit up in 3 minute, His rate = 93/3 = 31 sit up per minute.

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Answer:

Step-by-step explanation:

y = 2x - 6

Based on the recipe for making green slime, 3 cups of water is not enough to make a double recipe of green slime.

One recipe of green slime requires 1 pint of water, which is equal to 2 cups.

In order to make a double recipe, you would need to double the amounts of all the ingredients, including the water.

So, for a double recipe of green slime, you would need:

4 cups of water (2 cups per recipe)

1/2 cup of cornstarch (1/4 cup per recipe)

Green food coloring

Therefore, Rosa would need to measure out 4 cups of water to make a double recipe of green slime for the class project.

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Answer: To find the limit of the expression 3√x-1/x-1 as x approaches 1, we can use L'Hopital's Rule. L'Hopital's Rule is a method for finding the limit of indeterminate forms, such as 0/0 or ∞/∞, by taking the derivative of the numerator and denominator.

In this case, the expression is of the form 0/0, so we can use L'Hopital's Rule:

lim 3√x-1/x-1 = lim (3/2)((x-1)^(1/2)) / 1 = 3/2

So the limit of the expression 3√x-1/x-1 as x approaches 1 is equal to 3/2.

Step-by-step explanation:

Answer:

both Kareem and his parents added 4 each, to the ACC for five days

Guido can choose his Jolly Rancher in 78,125 different ways over the course of a week.

Guido chooses one sweet per day from a selection of five options; there are a total of five possible selections for a single day. Since there are seven days in a week, the sum of the ways Guido can choose candy over the course of a week equals the sum of the methods for each day.

Using the multiplication principle of counting

Total number of ways = 5 x 5 x 5 x 5 x 5 x 5 x 5

Simplifying this expression gives:

Total number of ways = 5^7

Using a calculator or by hand, we can compute:

Total number of ways = 78,125

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Reason: Split the angle 155 in half to get 155/2 = 77.5

Answer:

See the explanation below

Answer:

Step-by-step explanation:

18-5=13

18-9=9

Answer:

Step-by-step explanation:

12 x 3 =36

Answer:

1.75

Step-by-step explanation:

1. Algebraic equation: 8x - 9 = 5

2. Add 9 on both sides; 8x=14

3. Divide both sides by 8

4. x=1.75

Answer:

Number = 1/2

Step-by-step explanation:

9 - 8a = 5

9 - 5 = 8a

4 = 8a

4/8 = a

a = 1/2

Check:

9 - 8*1/2 = 5

9 - 4 = 5

The required scale factor for the first rectangle is 5.

What is a scale factor?

In mathematics, a scale factor is a numerical factor that describes the relationship between two similar shapes. It is the ratio of the length of a side or the measure of any other corresponding linear dimension of one shape to the length or corresponding dimension of another similar shape.

Given that,

A rectangle with a short side of 2.5,

And an arrow points to a smaller rectangle with a short side of 0.5

We have to find,

The scale factor was applied to the first rectangle.

According to the question,

The scale factor is the ratio of the length of a side of one figure to the length of the corresponding side of the other figure.

Then,

Scale factor = Dimensions of the short side ÷ Dimensions of the small rectangle with the same side.

Scale factor = 2.5/0.5

Scale factor = 5

Hence, The required scale factor for the first rectangle is 5.

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Step-by-step explanation:

The slashes through the lines indicate they're of the same length, which means this is an equilateral triangle. Therefore all three angles are 60 degrees.

The lines being parallel mean that angle x shares the same angle as the triangle in the bottom-right corner.

The parallel lines intersecting with the line segment that is consistent throughout the triangle makes the angle the same.

Step-by-step explanation:

We can see that the triangle in the picture is an equilateral triangle, because its sides are all marked with one dash, meaning they are equal lengths. We note that all the angles in an equilateral triangle are each 60°.

The little arrows on the top and bottom lines in the figure indicate that they are parallel, which means that the diagonal line between them creates a set of alternate angles (I've attached a pic of what alternate angles generally look like to make this clearer).

Alternate angles are equal, which means x is equal to the angle at the bottom right of the triangle, which is 60°.

The parameters that can help to raise the living standard are mentioned below.

Living standard is defined as the quality of housing, material comfort, and wealth experienced by an individual or group.

Given is that the standard of living is too low for many individuals in the united states.

The government can focus on the following factors -

Therefore, the parameters that can help to raise the living standard are mentioned above.

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The constant terms for the given expression are -7 and 2 and the variable terms for the given expression are  15 and 3a² respectively.

What is an expression?

Mathematical expressions consist of at least two numbers or variables, at least one maths operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation.

Given an expression 15a-7+ 3a²+2.

a) A constant term of an expression is a term that does not change its value.

So the constant terms are -7 and 2 as their value never changes in the expression.

b) The value of x changes in the expression because its value can change over time. So the terms containing x are variable terms.

In the given expression, the variable terms are 15a and 3a².

Therefore the constant and variable terms are found for the given expression.

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