please help me quick

The value of differentiation A)  dV/dt =  2πrh(dr/dt) ,B) dV/dt = π(r²(dh/dt) , C) π(2rh(dr/dt) + r²(dh/dt))

A mathematical procedure called differentiation is used to determine the derivative of a function with respect to a variable. The rate at which one quantity changes in relation to another is determined using this method.

If we have a function f(x) = x2, for instance, its derivative f'(x) = 2x tells us how quickly the function changes in relation to x.

The following formula determines the volume V of a right circular cylinder with radius r and height h:

V = πr²h

The cylinder's radius is r in this instance, and its height is h.

When we change the formula's specified values, we obtain:

V = πr²h

=  (12)(2) = 2 cubic metres.

As a result, the right circular cylinder has a 2 cubic unit volume.

Let's now think about how dv/dt and dr/dt relate when h is constant and r varies over time.

When we take into account time t and differentiate both sides of the formula V =πr²h, we obtain:

dV/dt = d(r²h)/dt

(r²)(dr/dt)(dh/dt) = dV/dt

h is a constant, hence dh/dt = 0.

Therefore,

dV/dt= ((2r)(dr/dt)(dh/dt)

= 2rh(dr/dt)

B)

Let's now think about how dv/dt and dh/dt relate when r is constant and h varies over time.

When we take into account time t and differentiate both sides of the formula V = r²πh, we obtain:

dV/dt equals d(r²h)/dt

r²(dh/dt) = dV/dt

As r is constant, dr/dt is equal to 0.

Therefore,

r2(dh/dt) = dV/dt

C)

Given that r and h are time-varying, let's look at the relationship between dv/dt, dh/dt, and dr/dt.

When we take into account time t and differentiate both sides of the formula V = r²πh, we obtain:

dV/dt equals d(r²h)/dt

dV/dt is equal to (r²)(dh/dt) + (r²)(dr/dt)

Therefore,

((2rh(dr/dt) + r2(dh/dt))

= dV/dt)

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The probability of a student taking a math class and a physics class is [tex]\frac{7}{15}[/tex].

What is probability?

To determine the possibility of an event, one might utilise probability as a technique. Only the probability of an event occurring can be determined using it. A scale where 1 denotes a specific occurrence and 0 signifies impossibility.

We are given that the probability of the student taking a physics class is [tex]\frac{2}{3}[/tex].

Also, it is given that the probability of a student taking a math class given that the student takes a physics class is [tex]\frac{7}{10}[/tex].

So, probability of a student taking a math class and a physics class is:

⇒ Probability =  [tex]\frac{2}{3}[/tex] *  [tex]\frac{7}{10}[/tex]

⇒ Probability = [tex]\frac{14}{30}[/tex]

⇒ Probability = [tex]\frac{7}{15}[/tex]

Hence, the probability of a student taking a math class and a physics class is  [tex]\frac{7}{15}[/tex].

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surface area of a refrigerator,

square feet (sq ft)

square inches (sq in)

the total area or the solid that the surface of the solid occupies.

we know that refrigerator have cuboid shape so,

surface area of cuboid =2(lb+bh+hl)

where l is length ,b is breath ,and h is height of cuboid .

The units generally  used to measure the surface area of a refrigerator are:

square feet (sq ft)

square inches (sq in)

Therefore, the options are square feet (sq ft) and square inches (sq in) correct.

Meters (m) and millimeters (mm) are typically not used for measuring the surface area of a refrigerator.

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The area of the shape, given the dimensions of the various parts of the shape, can be found to be 8, 100 ft ².

You can divide the shape into other shapes. Then you can find the areas of these shapes and then sum them up.

The area of the composite shapes would therefore be :

= ( 30 x 60 ) + ( 30 x 60 ) + ( ( 60 + 30 + 60) x 30 )

= ( 30 x 60 ) + ( 30 x 60 ) + ( 150 x 30 )

= 1, 800 + 1, 800 + 4, 500

= 8, 100 ft ²

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The gas company would charge $154.48 for using 200 hundred cubic feet of natural gas during the month.

What is Algebraic expression ?

An algebraic expression is a mathematical phrase that can include numbers, variables, and mathematical operations, such as addition, subtraction, multiplication, and division.

The equation that represents the gas company's monthly charges in dollars for using x hundred cubic feet of natural gas is:

y = 0.6924x + 16

Where:

x is the number of hundred cubic feet of natural gas used during the month

0.6924 is the cost per hundred cubic feet of natural gas used

16 is the fixed monthly service fee charged by the gas company.

To calculate the monthly charges for a given amount of natural gas used, we can substitute the value of x into this equation and simplify. For example, if x = 200, then:

y = 0.6924(200) + 16

y = 138.48 + 16

y = 154.48

Therefore, the gas company would charge $154.48 for using 200 hundred cubic feet of natural gas during the month.

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Answer:

slope = 15

Step-by-step explanation:

calculate the slope m using the slope formula

m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]

with (x₁, y₁ ) = (1, 6) and (x₂, y₂ ) = (2, 21) ← 2 ordered pairs from the table

m = [tex]\frac{21-6}{2-1}[/tex] = [tex]\frac{15}{1}[/tex] = 15

the linear relationship is then of the form

y = mx + c ( m is the slope and c the y- intercept )

from the table the point (0, - 9 ) crosses the y- axis at - 9 ⇒ c = - 9

y = 15x - 9 ← linear relationship

Answer:

13

Step-by-step explanation:

Given

7u - 4 = 3 ( add 4 to both sides )

7u = 7 ( divide both sides by 7 )

u = 1

Then

9u + 4 = 9(1) + 4 = 9 + 4 =13

Answer:   [tex]\frac{3}{4}[/tex]c

Step-by-step explanation:

       If one pound of chicken costs c dollars, and we are writing an expression for 3/4 a pound of chicken, our expression will be the price (c) multiplied by how much chicken there is (3/4):

               3/4c

- In mathematics, an expression is a combination of numbers, variables (represented by letters), symbols, and operators, arranged in a way that represents some meaningful mathematical relationship or calculation.

- Expressions can be simple, like 3x or 4 + 5, or complex, like (3x + 4y) / (2x - 5y). Expressions are not equations or inequalities, as they do not contain an equal sign, but they can be used to build equations or inequalities.

The price of 1 pound of chicken is c dollars. To find the price of 3/4 pound of chicken, we can multiply the price of 1 pound by 3/4:

[tex]\begin{aligned}\sf Price\: of\: \dfrac{3}{4}\: pound\: of\: chicken& =\sf \dfrac{3}{4} \times c \\&=\boxed{\bold{\:\dfrac{3}{4}c\:}}\end{aligned}[/tex]

Therefore, the expression for the price of 3/4 pound of chicken is 3/4c.

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The sides of the right triangle are 15 cm, 20 cm, and 25 cm.

What is triangle?

A triangle is a three-sided polygon with three vertices. The triangle's internal angle, which is 180 degrees, is constructed.

Let x be the length of the shorter leg of the right triangle.

Then, according to the problem statement, the longer leg is 5 cm more than the shorter leg, which means its length is x + 5 cm.

The hypotenuse is 10 cm more than the shorter leg, so its length is x + 10 cm.

By the Pythagorean theorem, we have:

x² + (x+5)² = (x+10)²

Expanding the squares and simplifying, we get:

x² + x² + 10x + 25 = x² + 20x + 100

Subtracting x² and simplifying further, we get:

x² - 10x - 75 = 0

Factorizing the left-hand side, we get:

(x - 15)(x + 5) = 0

Therefore, the possible solutions are x = 15 and x = -5, but x cannot be negative in this context, so we take x = 15 cm.

Then, the longer leg is x + 5 = 20 cm, and the hypotenuse is x + 10 = 25 cm.

Therefore, the sides of the right triangle are 15 cm, 20 cm, and 25 cm.

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The volume of each shape (A right triangular prism and the trapezoidal prism are):

A right triangular prism is a three-dimensional shape that has a triangular base and three rectangular faces. It is a prism because its two bases are identical and parallel. The formula for finding the volume of a right triangular prism is V = (1/2) * base * height * length.

A trapezoidal prism is a three-dimensional shape with trapezoidal bases and four rectangular faces. It is also a prism because its two bases are identical and parallel. The formula for finding the volume of a trapezoidal prism is V = ((b1 + b2) / 2) * height * length, where b1 and b2 are the lengths of the two parallel sides of the trapezoidal base.

Recall that the formula for the volume of a trapezoidal prism is:

V = (1/2)h(b1+b2)L

where:

Hence, substituting the given values, we have:

V = (1/2)(20 + 19)(7) 4

V = (1/2)(39)(7) 4

V = (273/2) 4

V = 546ft³

Therefore, the volume of the trapezoidal prism is 546ft³

2) Recall that the formula for the volume of a right triangular prism is:

V = (1/2) * b * h * L

Where:

b is the length of the base of the triangle

h is the height of the triangle

L is the length of the prism (the distance between the triangular faces)

The factor of (1/2) is included since the base of the prism is a triangle.

Given the forumla above:

b = 9ft

h = 7ft

L = 4ft


Plugging in the values given:

V = (1/2) * 9ft * 7ft * 4ft

V = 126ft³

Therefore, the volume of the right triangular prism is 126ft³

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The prοbability that in a sample οf 12 custοmers, nοne οf the 12 will οrder a nοnalcοhοlic beverage is 0.0042.

Binοmial prοbability is a type οf prοbability that deals with the number οf successes in a fixed number οf independent trials, where each trial has οnly twο pοssible οutcοmes, cοmmοnly referred tο as success οr failure. It is calculated using the binοmial prοbability fοrmula.

This is a binοmial prοbability prοblem, where the prοbability οf success (οrdering a nοnalcοhοlic beverage) is 0.53, and the number οf trials is 12.

The prοbability οf nο custοmers οrdering a nοn-alcοhοlic beverage can be fοund using the binοmial prοbability fοrmula:

[tex]P(X = k) = (n choose k) * p^k * (1-p)^{(n-k)[/tex]

where:

n = number of trials

k = number of successes (in this case, 0)

p = probability of success (ordering a non-alcoholic beverage)

Plugging in the values, we get:

[tex]P(X = 0) = (12 choose 0) * 0.53^0 * (1 - 0.53)^{(12 - 0)[/tex]

[tex]= 1 * 1 * 0.47^{12}[/tex]

= 0.0042 (rounded to 4 decimal places)

Therefοre, the prοbability that in a sample οf 12 custοmers, nοne οf the 12 will οrder a nοnalcοhοlic beverage is 0.0042.

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Answer:

Detra's rule: Output = 3 x Input. This means that for every input value, the output value is three times greater. Detra likely arrived at this solution by noticing a pattern in the input-output values and determining that the output values were always three times greater than the input values.

Trinh's rule: Output = Input + 3. This means that for every input value, the output value is the input value plus three. Trinh likely arrived at this solution by noticing a pattern in the input-output values and determining that the output values were always three more than the input values.

Step-by-step explanation:

Answer:

  x = 3√2

Step-by-step explanation:

You want the length x of the hypotenuse of an isosceles right triangle with sides of length 3.

The two legs of an isosceles right triangle are congruent. The length of the hypotenuse can be found from the Pythagorean theorem:

  x² = 3² +3²

  x² = 3²·2

  x = √(3²·2) = 3√2

The length of side x is 3√2.

__

Additional comment

A isosceles right triangle is one of two "special" right triangles. The ratios of its side lengths are 1 : 1 : √2. This tells you the hypotenuse is √2 times the side length, as we found above.

The other "special" right triangle is the 30°-60°-90° triangle. Its side lengths have the ratios 1 : √3 : 2. Both of these are seen often in algebra, trig, and geometry problems.

Answer:

Step-by-step explanation:

Start by substituting the first equation into the second equation to eliminate y:

2x^2 = -3x - 1

Move all terms to one side:

2x^2 + 3x + 1 = 0

Factor the quadratic equation:

(2x + 1)(x + 1) = 0

Solve for x by setting each factor to zero:

2x + 1 = 0 or x + 1 = 0

x = -1/2 or x = -1

Plug in the values of x to find the corresponding values of y:

For x = -1/2:

y = 2(-1/2)^2 = 1/2

For x = -1:

y = 2(-1)^2 = 2

The solutions to the system are (-1/2, 1/2) and (-1, 2).

According to the question the expected net gain for a player on a $1 straight bet is -$0.60.

Simply multiply each value of the discrete random variable X by its probability and add the products to get the expected value, E(X), or mean. The formula is as follows: E (X) =  ∑ x P (x)

The possible outcomes for X are winning with a probability of 1/1000 and net gain of $399, and losing with a probability of 999/1000 and net gain of -$1. Therefore, we can calculate the expected value of X as follows:

E(X) = (1/1000)($399) + (999/1000)(-$1)

E(X) = $0.399 - $0.999

E(X) = -$0.60

Therefore, the expected net gain for a player on a $1 straight bet is -$0.60. This means that, on average, a player will lose $0.60 for each $1 bet they place on this game.

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Answer:

-0.60

Step-by-step explanation:

Khan Academy

Amount of money that he have left over monthly to put into savings is $585.

Subtraction can be done for any numbers or algebraic expressions. It is the process of taking out certain value from a given amount of number.

The process of subtraction can also be termed as finding difference.

Given that,

Hourly rate of Mr. Smith = $20

If he works full time, then the number of working hours in a week is 40 hours.

So the number of working hours in a month = 40 × 4 = 160 hours

Rate for a month = 160 × $20 = $3200

Tax percent from the income = 25%

Income after tax = 3200 - (25% × 3200)  = 3200 - 800 = $2400

Total budget of Mr. Smith = $1,410 + $405 = $1815

Income after the monthly budget = $2400 - $1815 = $585

Hence the money that is left is $585 for the savings.

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The complete question is given below :

Mr. Smith makes $20 an hour working full time. He gets about 25% of his income taken out for taxes. He came up with the following monthly budget.

Household:

Rent: $700

Cable: $85

Cell Phone: $175

Electric: $100

Food: $350

Total: 1,410

Automobile

Car: $200

Car Insurance: $100

Gas: $90

Maintenance: $15

Total: $405

How much money does he have left over monthly to put into savings?​

Answer:

The Markdown amount: $9.45

Sale price: $13.05

Step-by-step explanation:

Convert 42% to decimal, which is 0.42

a = p(w).  (formula)

a=22.5(0.42).   or     22.5 x 0.42

a = 9.45

22.50 - 9.45 = 13.05

A subtraction expression for each length OB and AB is  [tex]OBX = \sqrt (x - 4)^2 + (y - 2)^2] + 2 \sqrt (1 - x)^2 + 16] - y.[/tex]

Let r be the radius of the circle. Then we have:

r = distance(O, B)

We can use the distance formula to find the distance between two points:

distance(P, Q) = √[(x2 - x1)² + (y2 - y1)²]

where P = (x1, y1) and Q = (x2, y2)

Substituting O = (4, 2) and B = (x, y), we get:

r = distance(O, B) = √[(x - 4)² + (y - 2)²]

Now, let's consider the subtraction expression:

OBX-2 AB-y+h

We can simplify this expression as follows:

[tex]OBX = r + 2 AB = \sqrt{[(x - 4)^2 + (y - 2)^2] } + \sqrt[2]{[(1 - x)^2 + (6 - y)^2]}[/tex]

We don't need to include the variables y and h in the expression since they were not defined in the problem. So the final subtraction expression for OBX is:

[tex]OBX = \sqrt{ [(x - 4)^2 + (y - 2)^2]} + \sqrt[2]{[(1 - x)^2 + 6]-y} .[/tex]

Therefore, the expression is OBX [tex]= \sqrt(x - 4)^2 + (y - 2)^2] + 2 \sqrt(1 - x)^2 + 16] - y.[/tex]

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Answer:

  (a)  y = 1.5x² -4.5x -6

Step-by-step explanation:

You want the standard form of the quadratic with roots 4 and -1, and passing through the point (1, -9).

For roots p and q, the standard form can be found from the factored form:

  (x -p)(x -q) = x² -(p+q)x +pq

For the given roots, the product pq will be ...

  (4)(-1) = -4

The sum p+q is 4+(-1) = 3. The x-coefficient is the opposite of this.

The parent quadratic (before vertical scaling) will be ...

  (x -4)(x +1) = x² -3x -4

The coefficients of the last two terms have the same sign. (Eliminates the 2nd and 4th answer choices.)

The given point has an x-value (1) between the given roots (-1, 4):

  -1 < 1 < 4

The y-value at that point is negative. This means the vertex of the parabola will be below the x-axis, so the parabola opens upward and the leading coefficient is positive. (Eliminates the last two answer choices.)

The answer choices tell us the leading coefficient is 1.5, so the equation is ...

  y = 1.5(x² -3x -4)

  y = 1.5x² -4.5x -6 . . . . . . matches the first choice

__

Additional comment

We could find the value of the leading coefficient 'a' by evaluating ...

  y = a(x -4)(x +1)

for x = 1. We would get ...

  -9 = a(1 -4)(1 +1) = -6a   ⇒   a = -9/-6 = 1.5

As we saw above, this isn't necessary. We only need to know that its sign is positive. The answer choices tell us the value.

If the x-value of the given point is not between the roots, then we know the sign of the y-value is the sign of the leading coefficient.

For multiple-choice questions, you only need to work enough of the problem to determine which answer choice is correct. You don't necessarily need to work the problem all the way to an answer.

Parallelogram JKLM is a rectangle because it has four right angles.

The SSS (side-side-side) congruence theorem is one of the postulates of Euclidean geometry that states that if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.

Abigali wants to prove that a parallelogram is a rectangle if its diagonals are congruent.

Since, JKLM is a parallelogram, JK≅ML.

Also by the reflexive property, KL is congruent to itself. So, since JL≅KM as well, ΔJKL ≅ ΔMLK by the SSS congruence theorem.

Now because the corresponding angles in congruent triangles are congruent, ∠JKL ≅ ∠MLK.

Also, since parallelograms have JK//ML,

Consecutive Interior Angles are supplementary.

So, since m∠JKL and m∠MLK are equal, they must both be 90°. Because the opposite angles in parallelograms are congruent, m∠MJK and m∠JML are both 90°. So, parallelogram JKLM is a rectangle because it has four right angles.

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Answer:

B

Step-by-step explanation:

Answer: $13, 658 increase

First, we need to calculate the current total variable costs:

Total Variable Costs = Sales x Variable Costs as a % of Sales

Total Variable Costs = $603,000 x 0.63

Total Variable Costs = $380,490

Next, we can calculate the current operating income:

Operating Income = Sales - Total Variable Costs - Fixed Costs

Operating Income = $603,000 - $380,490 - $202,000

Operating Income = $20,510

If sales increase by $37,400, the new sales figure will be $640,400 ($603,000 + $37,400). We can then calculate the new total variable costs:

New Total Variable Costs = New Sales x Variable Costs as a % of Sales

New Total Variable Costs = $640,400 x 0.63

New Total Variable Costs = $404,232

Finally, we can calculate the new operating income:

New Operating Income = New Sales - New Total Variable Costs - Fixed Costs

New Operating Income = $640,400 - $404,232 - $202,000

New Operating Income = $34,168

Therefore, the operating income will increase by $13,658 ($34,168 - $20,510) if sales increase by $37,400.

56 ounces is equal to 3 pounds and 8 ounces. So the answer is D: 3 lb 8 oz.

Pound is a unit of measurement for mass or weight commonly used in the United States and the United Kingdom. The symbol for pound is "lb".

In the English system of measurement, there are 16 ounces in one pound. To convert a weight in ounces to pounds, we need to divide the number of ounces by 16.

In this case, we have 56 ounces. So, by dividing 56 by 16, we obtain:

56 / 16 = 3 with a remainder of 8

This means that 56 ounces is equal to 3 pounds and 8 ounces. So the answer is D: 3 lb 8 oz.

In the US customary system, one pound is equal to 16 ounces, while in the imperial system used in the UK, one pound is equal to 14 ounces.

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Convert 56 ounces(oz) into pounds(Ib)

56 oz = ____lb ____oz

A. 2 lb 1 oz

B. 2 lb 3 oz

C. 2 lb 6 oz

D. 3 lb 8 oz

Answer:

To make 10 liters of 20% alcohol solution, RSM would need to use a combination of the 18% and 40% alcohol solutions. Let's call the amount of 18% solution used "x" and the amount of 40% solution used "y".

To set up the equation, we'll use the fact that the amount of pure alcohol in the final solution must be equal to 20% of the total volume.

So:

0.18x + 0.40y = 0.20(10)

Simplifying:

0.18x + 0.40y = 2

We have one equation with two unknowns, which means we need another equation. Fortunately, we know that RSM is making a total of 10 liters of solution. So:

x + y = 10

We now have two equations with two unknowns, which we can solve simultaneously. One way to do this is to solve one equation for one variable, then substitute that expression into the other equation, like so:

x = 10 - y (from the second equation)

0.18(10-y) + 0.40y = 2 (substituting into the first equation)

1.8 - 0.18y + 0.40y = 2

0.22y = 0.2

y = 0.91

So RSM would need to use approximately 0.91 liters (or 910 milliliters) of the 40% solution, and the rest (9.09 liters or 9090 milliliters) of the 18% solution, to make 10 liters of 20% alcohol solution.

Answer:

The formula for finding the area of a triangle is:

Area = 1/2 x base x height

where "base" is the length of one of the sides of the triangle, and "height" is the distance from that side to the opposite vertex.

The value of x is approximately 4.83 centimeters.

Answer: B (03)

The value of x, we need to add up all the side lengths of the figure and set it equal to the given perimeter of 33 cm.
The top and bottom sides have length [tex]2x-3[/tex] cm each, and the left and right sides have length [tex]x+5 cm[/tex] each.
So, we can write the equation:
[tex]2(2x-3) + 2(x+5) = 33 [/tex]
Simplifying and solving for x:
[tex]4x - 6 + 2x + 10 = 33 [/tex]
[tex]6x + 4 = 33 [/tex]
[tex]6x = 29 [/tex]
[tex]x = 29/6 [/tex]
[tex]x ≈ 4.83 [/tex]
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The prοfit exceeds the cοst as y > 1000 + 15x. Sο, οptiοn 4 is cοrrect.

Gaining anything is knοwn as prοfit. When the Selling price ( SP ) οf the prοduct is mοre than the cοst price ( CP ) οf the prοduct, we gain prοfit. Oppοsite tο the prοfit is lοss i.e. lοsing anything is called lοss. When the Selling price ( SP ) οf the prοduct is less than the cοst price ( CP ) οf the prοduct, we get a lοss.

Prοfit can be calculated by the fοrmula,

Prοfit   =   Selling Price - Cοst price

     P   =   SP - CP

Lοss can be calculated by the fοrmula,

Lοss   =   Cοst price - Selling price

   L    =   CP - SP

As per the questiοn,

x  =    number οf units prοduced

expressiοn   =   1000 + 15x

⇒  When y > 1000 + 15x, then prοfit exceeds cοst:

x   =   50  & y   =   1750

x   =   100  & y   =   2500

x   =   150  & y   =   3250

x   =   200  & y   =   4000

x   =   0  & y   =   1000

Therefοre, the graph in the 1st οptiοn is cοrrect as the prοfit exceeds the cοst.

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Complete question:

The cost of a small business is given by the expression 1000 + 15x, where x is the number of units produced. the business will be profitable whenever its profit y exceeds its cost. if the profit region is shaded in blue, which of the following graphs corresponds to the given situation?

Option b (2x−1)(3x+4) and d 2(3x² −4x) only matches the polynomial given.

A pοlynοmial is a mathematical expressiοn cοnsisting οf variables, cοefficients, and the οperatiοns οf additiοn, subtractiοn, multiplicatiοn, and nοn-negative integer expοnents.

Pοlynοmials are an impοrtant part οf the "language" οf mathematics and algebra. They are used in nearly every field οf mathematics tο express numbers as a result οf mathematical οperatiοns. Pοlynοmials are alsο "building blοcks" in οther types οf mathematical expressiοns, such as ratiοnal expressiοns.

We have:

6x² − 8x

= 2(3x² −4x)

Then,

6x² - 3x + 8x -4

= 6x² +8x−3x−4

= 2x(3x+4) −3x −4

= 2x(3x+4)−(3x+4)

= (2x−1)(3x+4)

Thus, option b (2x−1)(3x+4) and d 2(3x² −4x) only matches the polynomial given.

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The union of J and K is: J ∪ K = {7, 14, 21, 28, 35, 42, 56, 70} . Note that 14, 28, and 42 are included in both sets, but we only include them once in the union.

What is a set ?

In mathematics, a set is a collection of distinct objects, called elements or members of the set. Sets are often denoted by curly braces {} enclosing the list of elements, separated by commas. For example, the set of natural numbers less than 5 can be written as {0, 1, 2, 3,

The union of two sets J and K, denoted by J ∪ K, is the set that contains all the elements that are in J or K, or in both.

In this case, J = {7, 14, 21, 28, 35, 42} and K = {14, 28, 42, 56, 70}. To find the union of J and K, we simply combine all the elements in both sets, removing any duplicates.

Therefore, the union of J and K is: J ∪ K = {7, 14, 21, 28, 35, 42, 56, 70}

Note that 14, 28, and 42 are included in both sets, but we only include them once in the union.

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