The base and all three faces of a triangle pyramid are equilateral triangles with side lengths of 3ft. the height of each triangle is 2.6ft. what are the lateral area and the total surface area of the triangular pyramid?

The rate of change between August and October is

Rate is how a quantity changes over a period of time.

Therefore rate = change in quantity/change in time.

For example acceleration is defined as the rate of change of velocity with time. This means that acceleration = change in velocity/time

change in quantity = 85-81 = 4

change in time = 2 months

therefore rate of change = 4/2 = $2 per month

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To express the volume of D as an iterated triple integral in (a) spherical, (b) cylindrical coordinates, we can use the following formulas:

(a) Spherical Coordinates:

We know that the equation of a sphere with radius 6 units is given by:

x^2 + y^2 + z^2 = 6^2

The equation of the plane 3 units from the center of the sphere is given by:

z = 3

To find the equation of the smaller cap cut from the sphere by this plane, we need to find the upper limit of the radial coordinate. This can be found by solving the equation of the sphere for z, and substituting z = 3:

z = sqrt(6^2 - x^2 - y^2)
3 = sqrt(6^2 - x^2 - y^2)
x^2 + y^2 = 27

Thus, the smaller cap D is the region of the sphere bounded by the plane z = 3 and the surface x^2 + y^2 + z^2 = 6^2, with x^2 + y^2 <= 27.

To express the volume of D as an iterated triple integral in spherical coordinates, we can use the following limits:

0 <= r <= sqrt(27)
0 <= θ <= 2π
arcsin(3/6) <= φ <= π

The volume of D can be expressed as the triple integral:

∫∫∫ D  r^2 sin φ dr dφ dθ

(b) Cylindrical Coordinates:

To express the volume of D as an iterated triple integral in cylindrical coordinates, we can use the following limits:

0 <= r <= sqrt(27)
0 <= θ <= 2π
3 <= z <= sqrt(6^2 - r^2)

The volume of D can be expressed as the triple integral:

∫∫∫ D  r dz dr dθ
(a) In spherical coordinates, the volume element is given by dV = ρ²sin(φ)dρdθdφ. To find the limits of integration, note that the radius of the smaller cap ranges from 0 to 3 units, the angle θ ranges from 0 to 2π, and the angle φ ranges from 0 to φ₀, where φ₀ is the angle between the plane and the line from the sphere's center to the plane's intersection with the sphere.

Using the cosine rule, we can find φ₀ as follows:
cos(φ₀) = (6² + 3² - 3²) / (2 × 6 × 3) = 1/2
φ₀ = π/3

Now, we can express the volume of D as an iterated triple integral in spherical coordinates:
∫(ρ=0 to 3) ∫(θ=0 to 2π) ∫(φ=0 to π/3) ρ²sin(φ)dρdθdφ

(b) In cylindrical coordinates, the volume element is given by dV = rdzdrdθ. To find the limits of integration, note that the height z ranges from 3 to 6 units, the radius r ranges from 0 to √((6 - z)²), and the angle θ ranges from 0 to 2π.

Now, we can express the volume of D as an iterated triple integral in cylindrical coordinates:
∫(z=3 to 6) ∫(θ=0 to 2π) ∫(r=0 to √((6 - z)²)) rdzdrdθ

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John can afford approximately 6 magazines.

A) To write an equation that models the number of magazines John can afford, let's denote the number of magazines as 'm'. Since each magazine costs $2.95 and John has a total of $23.65 to spend, the equation can be expressed as:

2.95m + 5.95 = 23.65

B) To solve the equation, we can isolate the variable 'm' by subtracting 5.95 from both sides:

2.95m = 23.65 - 5.95

2.95m = 17.70

Then, divide both sides by 2.95:

m = 17.70 / 2.95

m ≈ 6

Therefore, John can afford approximately 6 magazines.

In conclusion, the equation 2.95m + 5.95 = 23.65 models the number of magazines John can afford, and by solving it, we find that he can purchase approximately 6 magazines with the given amount of money.

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Finding the answer requires finding the least common multiple (LCM) of 5, 3, and 4, making the correct answer B. ) Least Common Multiple.

To determine whether finding the answer requires finding a greatest common factor (GCF) or a least common multiple (LCM), we need to analyze the given information.

In this scenario, the red lights flash every fifth second, the blue lights flash every third second, and the green lights flash every fourth second. We want to find the first time when all three lights flash simultaneously.

To find this time, we need to find the smallest number that is divisible by all three given numbers (5, 3, and 4). This means we are looking for the least common multiple (LCM) of these numbers.

To calculate the LCM, we can use the formula:

LCM(a, b) = (a * b) / GCF(a, b),

where GCF(a, b) represents the greatest common factor of numbers a and b.

Therefore, in this case, finding the answer requires finding the least common multiple (LCM) of 5, 3, and 4, making the correct answer:

B. ) Least Common Multiple.

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The expanded form of the smallest bone in the human body located in the ear which is about 2. 8 millimeters in length is                                       2 × 1 millimeter + 8 × 0.1 millimeters.

Given that the smallest bone in the human body is the stapes bone. It is located in the ear and is about 2. 8 millimeters in length.

To write 2.8 millimeters in expanded form, we need to express each digit's place value in the number.

2.8 millimeters can be written as:

2 millimeters + 0.8 millimeters

or

2 millimeters + 8/10 millimeters

In expanded form, this is:

2 millimeters + 8 tenths of a millimeter

or

2 × 1 millimeter + 8 × 0.1 millimeters

Therefore, 2.8 millimeters in expanded form is:

2 × 1 millimeter + 8 × 0.1 millimeters = 2.0 + 0.8 = 2.8 millimeters

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The volume of the composite figure is 152 m³.

The volume of a rectangular prism can be found using the formula V = lwh, where l is the length, w is the width, and h is the height.

For the rectangular prism:

V_prism = lwh = 9m * 8m * 3m = 216 m³

For the cube:

V_cube = s^3 = 4m * 4m * 4m = 64 m³

Now, subtract the volume of the cube from the volume of the prism:

V_composite = V_prism - V_cube = 216 m³ - 64 m³ = 152 m³

The volume of the composite figure is 152 m³.

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The theoretical probability is: 12.5%. After 100 trials, the experimental probability is of: 20%. After 400 trials, the experimental probability is of: 11%. After more trials, the experimental probability is closer to the theoretical probability.

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

The dice has eight sides, hence the theoretical probability of rolling a six is given as follows:

1/8 = 0.125 = 12.5%.

(eight sides, each of them is equally as likely).

The experimental probabilities are obtained considering the trials, hence:

The more trials, the closer the experimental probability should be to the theoretical probability.

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Answer:

135.6 cubic centimeters

Step-by-step explanation:

To find the volume of soda in the cylindrical glass, we need to first find the volume of the glass and then multiply it by 60% to get the volume of soda. The formula for the volume of a cylinder is V = πr^2h, where r is the radius and h is the height.

Substituting the given values, we get:

V = π(3 cm)^2(8 cm) = 72π cm^3

To find the volume of soda, we multiply the volume of the glass by 60% or 0.6:

Volume of soda = 0.6 x 72π cm^3 = 43.2π cm^3

Rounding to the nearest tenth, we get:

Volume of soda ≈ 135.6 cm^3

Therefore, there are approximately 135.7 cubic centimeters of soda in the glass

Answer: 226.3

Step-by-step explanation:

Hello! Here is how to solve this problem.

The formula for volume of a cylinder is V(c) = (B)(h)

The height is 8 cm, and the radius is 3 cm. The base is a circle, so the base area will be 22/7(r)^2. Pi is represented as 3.14 or 22/7, but 22/7 is more accurate.

V(c) = 22/7((3)^2)(8)
V(c) = 22/7(9 x 8)
V(c) = 22/7(72)

V(c) = 226.285714.

So, rounded to the nearest tenth, the volume of the soda can is 226.3 cm^3.

Tips for you:

1. Round to the nearest tenth
2. Use 22/7 for pi unless it says to use 3.14 for pi or use the symbol for pi instead of solving further.

Hope this helps, have a great day!

The probability of the toy character wearing a polka-dotted shirt and a cowboy hat is P(polka-dot shirt and cowboy hat) = 2 / 12 = 1/6.

What is the probability of the toy character wearing a polka-dotted shirt and a cowboy hat in the number of outfits?

There are 3 choices for the shirt (striped, plain, or polka-dotted), 2 choices for the pants (denim or plaid), and 2 choices for the cap (cowboy or baseball). Therefore, there are a total of 3 x 2 x 2 = 12 different outfits that the toy character could be wearing.

The probability of the toy character wearing a polka-dotted shirt and a cowboy hat is the number of outfits with a polka-dotted shirt and a cowboy hat divided by the total number of outfits:

P(polka-dot shirt and cowboy hat) = number of outfits with a polka-dot shirt and cowboy hat / total number of outfits

To find the number of outfits with a polka-dotted shirt and a cowboy hat, we need to consider each clothing item separately. There is only 1 choice for the cowboy hat and only 1 choice for the polka-dotted shirt. There are 2 choices for the pants, but we don't care which pants the toy character is wearing. Therefore, the number of outfits with a polka-dotted shirt and a cowboy hat is:

1 (cowboy hat) x 1 (polka-dotted shirt) x 2 (pants) = 2

So the probability of the toy character wearing a polka-dotted shirt and a cowboy hat is:

P(polka-dot shirt and cowboy hat) = 2 / 12 = 1/6

Therefore, the answer is (B) 1/6.

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The system has no solution because the equations are parallel

Janelle is correct in saying that the system of equations has no solution. She can tell by looking at the coefficients of the variables in the two equations.

Both equations have the same coefficients for x and y, which means that they are parallel lines in the xy-plane.

Since parallel lines never intersect, there are no values of x and y that would satisfy both equations simultaneously, meaning that the system has no solution.

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The answer would be D

Since Lucy's number has a digit of 5 which is 10x greater than Ethan's number, 5 x 10 = 50; so 50 would be in the tenth place, and option d is the only one with 5 in the tenth place.

The value of a number moves one decimal place to the left for each position it moves to the right. So, the number that Lucy could have written where the number 5 has a value 10 times larger than Ethan's number is 4974.65.

In order for the value of the digit 5 to be 10 times larger in Lucy's number than it is in Ethan's, the '5' in Lucy's number should reside one place left compared to Ethan's. This means the 5 should be in the tens place, hundreds place, or beyond. So, we should look for a number having digit 5 at those places.

Let's examine the given options:

Therefore, the correct answer is C. 4974.65.

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Norma and David crawled for 37/7 minutes, or approximately 5.29 minutes.

Let x be the time (in minutes) that Norma and David spent crawling to the barn, and y be the time (in minutes) they spent hopping back to the house. We know that:

x + y = 7 (the total time they spent on the round trip was 7 minutes)

300x + 400y = distance to the barn and back (since their speeds are given in centimeters per minute, the product of their speeds and the time spent crawling or hopping gives the distance in centimeters)

We want to find x, the time they spent crawling to the barn. We can solve for x by rearranging the first equation as x = 7 - y, and substituting into the second equation:

300(7 - y) + 400y = distance to the barn and back

2100 - 300y + 400y = distance to the barn and back

100y = distance to the barn and back - 2100

y = (distance to the barn and back - 2100)/100

Now we need to find the distance to the barn and back. They crawled to the barn at 300 cm/min, so the distance they crawled is 300x cm. They hopped back to the house at 400 cm/min, so the distance they hopped is 400y cm. The total distance to the barn and back is:

distance to the barn and back = 300x + 400y

= 300x + 400[(distance to the barn and back - 2100)/100] (substituting the expression we found for y)

= 300x + 4(distance to the barn and back - 2100)

Simplifying and solving for distance to the barn and back, we get:

distance to the barn and back = 5400/7 cm

Finally, we can substitute this value into the expression we found for y, and solve for x:

y = (distance to the barn and back - 2100)/100

= (5400/7 - 2100)/100

= 24/7

x = 7 - y

= 7 - 24/7

= 37/7

Therefore, Norma and David crawled for 37/7 minutes, or approximately 5.29 minutes.

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The substitution is u = -x² + 9 and the evaluated value is -4.5(2/3)(-x² + 9)³/² + C.

To evaluate the given integral ∫9x√(-x² + 9dx),

we can use the substitution u = -x² + 9. This substitution will allow us to simplify the expression under the square root.

First, we can find du/dx by taking the derivative of u with respect to x: du/dx = -2x.

Next, we can solve for dx in terms of du by dividing both sides by -2x: dx = -du/(2x).

Using the substitution and the expression for dx in terms of du, we can rewrite the integral as:

∫9x√(-x² + 9dx) = -4.5∫√udu

Now, we can integrate the simplified expression √u using the power rule of integration:

-4.5∫√udu = -4.5(2/3)u³/² + C

Substituting back for u, we get:

-4.5(2/3)(-x² + 9)³/² + C

Therefore, the solution to the integral ∫9x√(-x^2 + 9dx) using the substitution u = -x^2 + 9 is:

-4.5(2/3)(-x² + 9)³/² + C

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Curved surface area of cone: 255π or approx. 801.41 sq units with radius 15 and height 8.

The curved surface area of a right circular cone can be calculated using the formula:

A = πrℓ

where A is the area of the curved surface,

r is the radius of the base of the cone, and

ℓ is the slant height of the cone.

To find the slant height, we can use the Pythagorean theorem:

ℓ² = r² + h²

where h is the height of the cone.

Substituting the given values, we get:

ℓ² = 15² + 8²

ℓ² = 225 + 64

ℓ² = 289

ℓ = √289

ℓ = 17

Now, substituting the values of r and ℓ in the formula for curved surface area, we get:

A = πrℓ

A = π(15)(17)

A = 255π

Therefore, the area of the curved surface of the cone is 255π square units, or approximately 801.41 square units.

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We may use the identity (a + b)³ = a³ + 3a²b + 3ab² + b³ to get 970299, 1061208, and 992016008 for (i), (ii), and (iii), respectively.

Using the identity (a + b)³ = a³ + 3a²b + 3ab² + b³ allows us to expand and simplify the expressions by distributing and collecting like terms. Any integer's cubes can be calculated using this method.

(i) Using the identity, we can write 99 as 100 - 1 and get:

= (99)³

= (100 - 1)³

= 100³ - 3(100²)(1) + 3(100)(1²) - 1³

= 970299

(ii) We can denote 102 as 100 + 2 and use the identity to obtain:

= (102)³

= (100 + 2)³

= 100³ + 3(100²)(2) + 3(100)(2²) + 2³

= 1061208

(iii) Using the identity, we may write 998 as 1000 - 2 and get:

= (998)³

= (1000 - 2)³

= 1000³ - 3(1000²)(2) + 3(1000)(2²) - 2³

= 992016008

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The mass of iron III sulfate is  300 g.

If 1 mole of iron III sulfate is produced when 3 moles of hydrogen is produced

x moles of iron III sulfate is produced when 2.25 moles of hydrogen is produced

x = 1 * 2.25/3

= 0.75 moles

Mass of the iron III sulfate = 0.75 moles * 400 g/mol

= 300 g

Number of moles of iron = 85 g/56 g/mol

= 1.5 moles

If 2 moles of iron produces 1 mole of   iron III sulfate

1.5 moles of iron produces 1.5 * 1/2

= 0.75 moles

Mass of the  iron III sulfate = 0.75 moles * 400 g/mol

= 300 g

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The function with the larger y-intercept is h, because it intersects the y-axis at a higher point than f.

To determine which function, f or h, has a larger y-intercept, we need to look at the graphs of the two functions. From the graph, we can see that function h has a larger y-intercept than function f.

The y-intercept of function h is approximately 4, while the y-intercept of function f is approximately 2. Therefore, we can conclude that the x-value of function h's y-intercept is larger than that of function f.

This is because the y-intercept of a function is the point at which it intersects with the y-axis, and the value of the x-coordinate at that point determines the x-value of the y-intercept.

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Answer:

Top

Step-by-step explanation:

The closer the data points come to forming a straight line when plotted, the higher the correlation between the two variables, or the stronger the relationship. Therefore, the top option has the most closest lines to a linear function.

(Ex. 1) Y=4x-2

This example shows the corresponding possibilities of a linear regression because of the way the line is represented as graphed, making a straight line as similar to the top option

(Ex. 2) f(x) = x^2-5+15

This example dialates to a similar option like the third option, which isnt linear regression because of it being a quadratic function.

In a nut shell, all data plotted on the graph that are formed closer together and corresponds to a linear equation is a linear regression

The problem involves finding the marginal cost of producing a certain number of DVD players, given the total cost function. Specifically, we are given the total cost function C(x) = 120 + 3x^2-3, where x is the number of DVD players produced, and we need to find the marginal cost when x = 3.

To find the marginal cost, we need to take the derivative of the total cost function with respect to x and evaluate it at x = 3. The marginal cost represents the cost of producing one additional unit of the product, and is an important concept in economics and business.

Understanding the relationship between marginal cost and production volume is crucial for optimizing production and pricing decisions for companies. Marginal cost analysis is used extensively in many fields, including manufacturing, finance, and marketing.

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Deloria is analyzing the relationships between relative frequencies of nuts.

She has found that the relative frequency of a specific type of nut is 15%, which is calculated by dividing the number of that nut (3) by the total number of nuts (30).

Deloria might be comparing this relative frequency to other types of nuts to determine their prevalence or importance in a given context.

To calculate the relative frequency, we can divide the number of occurrences of the specific nut (3) by the total number of nuts (30):

Relative frequency = (Number of occurrences of specific nut) / (Total number of nuts)

Relative frequency = 3 / 30 = 0.1 = 10%

Therefore, the relative frequency of the specific type of nut is 10%, not 15%. It seems there was an error in the initial statement.

By analyzing relative frequencies, Deloria can compare the prevalence or importance of different types of nuts in a given context.

By calculating and comparing the relative frequencies of different types of nuts, Deloria can gain insights into the distribution and significance of each nut type within the dataset or sample.

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Answer: 5/6

Step-by-step explanation:

probabilities = possibilities/outcome

To get an even you can possible roll 2, 4, 6 with 6 total outcomes

P(even) = probability of getting an even = 3/6

To get a divisor of 3: only 3 and 6 can be divided by 3

P(divisor of 3)=2/6

Because of the or in the statement, you add the 2 probabilities

P(even or divisor of 3) =  3/6+2/6 = 5/6

Answer:

-5z/v³ + 8 + 6v²z

Do you have to simplify it further or?

Step-by-step explanation:

[tex] \frac{ - 5z + 8v {}^{3} + 6v {}^{5} z}{v {}^{3} } = \frac{ - 5z}{v {}^{3} } + \frac{8v {}^{3} }{ {v}^{3} } + \frac{6v {}^{5} }{v {}^{3} } = \frac{ - 5z}{v {}^{3} } + 8 + 6v {}^{2}z[/tex]

The measure of arc a must be 2 times measure of inscribed angle, which is 90 degrees.

What is arc?

In geometry, an arc is a segment of a circle's circumference. It is defined by two endpoints and all the points on the circle's circumference between them.

What is inscribed angle?

An inscribed angle is an angle formed by two chords in a circle that have a common endpoint.

For any inscribed angle in a circle, the measure of the angle is always half the measure of the arc that it intercepts. This is known as the inscribed angle theorem.

So, if we have an inscribed angle with a measure of 45 degrees, then the measure of its corresponding arc would be 2 times that, which is 90 degrees.

Therefore, if the inscribed angle is associated with arc a, and the measure of the corresponding angle is 45 degrees, then we know that the measure of arc a must be 2 times that, which is 90 degrees.

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The total inventory costs consist of two parts: the cost of storing the shoes and the cost of reordering the shoes. The cost of storing the shoes is given by the formula:

Cost of storing = $6 per shoe pair per year x number of shoe pairs

Since the shoes are stored for a year, this cost is incurred annually. The cost of reordering the shoes is given by the formula:

Cost of reordering = $14 per order + $7 per shoe pair x number of shoe pairs

This cost is incurred each time the store places an order with the manufacturer.

Let x be the number of shoe pairs in each order from the manufacturer. The number of orders needed to sell 1750 shoe pairs each year is given by:

Number of orders = 1750 shoe pairs / x shoe pairs per order

The total inventory costs can be expressed as:

Total cost = Cost of storing + Cost of reordering

Substituting the formulas for the two costs and the expression for the number of orders, we get:

Total cost = $6 per shoe pair per year x 1750 shoe pairs + ($14 per order + $7 per shoe pair x x shoe pairs) x (1750 shoe pairs / x shoe pairs per order)

Simplifying this expression, we get:

Total cost = $10,500 + $14(1750/x) + $7(1750)

Total cost = $10,500 + $24,500/x

Therefore, the function that models the total inventory costs as a function of x is:

Total cost(x) = $10,500 + $24,500/x

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The net worth is about $106.

This can be calculated by the addition of liabilities and assets amount

Assets        -               Amount

Checking amount -  $14,532

Retirement savings- $43,675

Car                          - $12,225

Therefore total assets equal to: $70,432

Now, for liabilities

Liabilities            -       Amount

Student loan             $1,225

Credit card debt       $3,876

Home loan balance  $65,225

Therefore total liabilities equals to: $70,326

The net worth can be find out by subtracting the total liabilities from the total assets:

Hence Net worth = Total assets- Total liabilities

                             = $70,432- $70,326

                             = $106

Therefore the total net worth is about $106.

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We can use an inequality to determine the number of biking outfits Shandra can purchase while staying within her budget.

Let o represent the number of outfits she can purchase. Here are the given terms and costs:
- Initial budget: $760
- Bicycle cost: $433.54
- 2 reflectors cost: 2 * $18.41 = $36.82
- Bike gloves cost: $10.76
- Outfit cost: $66.40 each

Now, we can set up the inequality:
760 >= 433.54 + 36.82 + 10.76 + 66.40 * o

First, combine the constants:
760 >= 481.12 + 66.40 * o

Now, subtract 481.12 from both sides:
278.88 >= 66.40 * o

Finally, divide both sides by 66.40:
o <= 4.2

Since Shandra can only purchase whole outfits, the maximum number of outfits she can buy is 4. So the inequality representing this situation is o <= 4.

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Solution:-

Here,

let, P=$1000

Money vested in IBM= x

Interest=(x×1×5)/100

              =5x/100

Money invested in Macintosh=1000-x

Interest=((1000-x)1×8)/100

              =(8000-8x)/100

Now,

Total Interest=5x/100 + (8000-8x)/100

or, 60.80=(5x+8000-8x)/100

or, 60.80×100=-3x+8000

or, 6080-8000=-3x

or, -1920/-3=x

x=$640

1000-x=1000-640

             =360 

Thus, Colton invested $640 in IBM, and $360 in Macintosh.

Answer:

List all the factors

6: 3, 2, 1 (6)

9:3,(9),1

5:(5),1

2:(2),1

Step-by-step explanation: