A rectangle has a length of (x+4)cm and a width of (3x-1)cm. It’s perimeter is 78cm
Calculate the value of x

The number we are looking for is N = 2 × 2^3 × 3^2 = 72.

Let's first recall some properties of the number of divisors of an integer. If we factorize an integer n as a product of prime powers, say

n = p_1^a_1 × p_2^a_2 × ... × p_k^a_k

then the number of divisors of n is given by

d(n) = (a_1 + 1) × (a_2 + 1) × ... × (a_k + 1).

Using this fact, we can deduce some information about the number we are looking for. Let's call it N. We know that N has 8 divisors, so it must be of the form

N = p_1^2 × p_2^2, or N = p_1^7,

where p_1 and p_2 are distinct prime numbers.

Now, we are told that each of N/2 and N/3 has four divisors. We can use the same fact about the number of divisors to conclude that

N/2 = q_1^3 × q_2, or N/2 = q_1^1 × q_2^3,

and

N/3 = r_1^3 × r_2, or N/3 = r_1^1 × r_2^3,

where q_1, q_2, r_1, and r_2 are distinct prime numbers.

To simplify the notation, let's introduce the variables a, b, c, d, e, and f, defined by

p_1 = q_1^a × q_2^b,

p_2 = r_1^c × r_2^d,

N/2 = q_1^e × q_2^f,

N/3 = r_1^g × r_2^h.

Using the information we have so far, we can write down equations for a, b, c, d, e, f, g, and h in terms of unknown exponents:

a + 1 × (b + 1) = e + 1 × (f + 1) = 4,

c + 1 × (d + 1) = g + 1 × (h + 1) = 4,

2a × 2b = ef,

2c × 2d = gh.

We can solve this system of equations by trial and error. For example, we can start by trying all possible values of a and b such that 2a × 2b = 4. This gives us two possibilities: a = 0, b = 2, or a = 1, b = 1. Using the first possibility, we get e = 3, f = 1, which leads to N/2 = q_1^3 × q_2, and hence N = 2 × q_1^3 × q_2^2. Substituting this into the equation for the sum of divisors, we get

(1 + q_1 + q_1^2 + q_1^3) × (1 + q_2 + q_2^2) = 216.

We can solve this equation by trial and error as well, or by observing that 216 = 2^3 × 3^3, and hence the two factors on the left-hand side must be equal to 2^3 and 3^3, respectively. This gives us the unique solution q_1 = 2 and q_2 = 3, and hence N = 2 × 2^3 × 3^2 = 72.

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Answer: It should be 470 cm^2

Step-by-step explanation:

The probability that the jury makes the correct decision is approximately 0.7063.

To solve this problem, we need to find the probability that the jury makes the correct decision if the defendant is guilty. Let's break down the problem into smaller steps.

We know that the probability of a single juror making the correct decision is 0.80. If the defendant is guilty, then the probability of a juror making the correct decision is still 0.80. Therefore, the probability that a single juror makes the correct decision if the defendant is guilty is 0.80.

We can use the binomial distribution formula to determine the probability of at least 10 out of 12 jurors making the correct decision. The formula is:

P(X ≥ k) = 1 - Σ(i=0 to k-1) [n!/(i!(n-i)!) x [tex]p^i \times (1-p)^{(n-i)}[/tex] ]

where:

P(X ≥ k) is the probability of at least k successes

n is the total number of trials (in this case, 12 jurors)

p is the probability of success in a single trial (in this case, 0.80)

k is the number of successes we want to find the probability of (in this case, 10)

Plugging in the values, we get:

P(X ≥ 10) = 1 - Σ(i=0 to 9) [12!/(i!(12-i)!) x [tex]0.80^i \times (1-0.80)^{(12-i)}[/tex]]

Using a calculator or software, we can calculate this to be approximately 0.7063.

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Answer:

Step-by-step explanation:

Given f(x) = 2x³ +x² +ax +b has a factor (x -2) and a remainder of 18 when divided by (x -1), you want to know a, b, and the factored form of f(x).

If (x -2) is a factor, then the value of f(2) is zero:

  f(2) = 2·2³ +2² +2a +b = 0

  2a +b = -20 . . . . . . . subtract 20

If the remainder from division by (x +1) is 18, then f(-1) is 18:

  f(-1) = 2·(-1)³ +(-1)² +a·(-1) +b = 18

  -a +b = 19 . . . . . . . . . . add 1

Subtracting the second equation from the first gives ...

  (2a +b) -(-a +b) = (-20) -(19)

  3a = -39

  a = -13

  b = 19 +a = 6

The values of 'a' and 'b' are -13 and 6, respectively.

We can find the quadratic factor using synthetic division, given one root is x=2. The tableau for that is ...

  [tex]\begin{array}{c|cccc}2&2&1&-13&6\\&&4&10&-6\\\cline{1-5}&2&5&-3&0\end{array}[/tex]

The remainder is 0, as expected, and the quadratic factor of f(x) is 2x² +5x -3. Now, we know f(x) = (x -2)(2x² +5x -3).

To factor the quadratic, we need to find factors of (2)(-3) = -6 that have a sum of 5. Those would be 6 and -1. This lets us factor the quadratic as ...

  2x² +5x -3 = (2x +6)(2x -1)/2 = (x +3)(2x -1)

The factored form of f(x) is ...

  f(x) = (2x -1)(x -2)(x +3)

As a result, y has a mean of 203 and a variance of 85 as let x1 and x2 be two independent random variables both with mean 10 and variance 5.

A variable is a symbol or letter that is used to indicate a variable quantity in mathematics. The context or issue under consideration can alter the value of a variable. In order to express relationships between quantities, variables are frequently utilized in equations, formulae, and functions. For instance, x and y are variables in the equation y = mx + b, which depicts the linear relationship between x and y. Variables in statistics can reflect various traits or features of a population or sample, such as age, body mass index, or income.

given

To get the mean and variance of y, we can apply the characteristics of expected value and variance:

We can start by determining the expected value of y:

E[y] = 2E[x1] = E[2x1x2 + 3x1 + 2]

By the linearity of expectation, E[x2] + 3E[x1] + 2 is 2(10)(10) + 3(10) + 2 = 203.

Next, we may determine y's variance:

Var(y) = Var(3x1 + 2 + 2 + 3x1 ) = 4

Var(x1)

Var(x2) + 9

Var(x1) + Var(constant) = 4(5)(5) + 9(5) + 0 = 85 since x1 and x2 are independent.

As a result, y has a mean of 203 and a variance of 85 as let x1 and x2 be two independent random variables both with mean 10 and variance 5.

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The solution to the logarithmic equation [tex]2log12(-8x) = 6[/tex] is [tex]x = -9/32[/tex] .

What are logarithmic properties ?

Logarithmic properties are the rules that govern the behavior of logarithmic functions. These properties are important in simplifying logarithmic expressions and solving logarithmic equations. Some of the commonly used logarithmic properties include:

Product property: [tex]logb(xy) = logb(x) + logb(y)[/tex]

This property allows us to simplify the logarithm of a product of two numbers into the sum of logarithms of the individual numbers.

Quotient property: [tex]logb(x/y) = logb(x) - logb(y)[/tex]

This property allows us to simplify the logarithm of a quotient of two numbers into the difference of logarithms of the individual numbers.

Power property:[tex]logb(x^y) = ylogb(x)[/tex]

This property allows us to simplify the logarithm of a power of a number by bringing the exponent outside of the logarithm and multiplying it with the logarithm of the base.

Change of base formula: [tex]logb(x) = logc(x) / logc(b)[/tex]

This property allows us to change the base of a logarithm by dividing the logarithm of the number by the logarithm of the base in a different base.

Solving the given logarithmic equation :

The equation can be solved by using logarithmic properties and basic algebraic manipulation.

We can begin by using the property that states [tex]loga(b^n) = nloga(b)[/tex] for any base a and any positive real number b. Applying this property, we can rewrite the left side of the equation as:

[tex]log12((-8x)^2) = log12(64x^2)[/tex]

Next, we can use the property that states [tex]loga(b) = c[/tex] is equivalent to [tex]a^c = b[/tex]. Applying this property, we can rewrite the equation as:

[tex]12^{2log12(64x^2)} = 12^6[/tex]

Simplifying the left side, we get:

[tex]64x^2 = 12^6 / 12^2[/tex]

[tex]64x^2 = 144[/tex]

Dividing both sides by 64, we get:

[tex]x^2 = 144/64[/tex]

[tex]x^2 = 9/4[/tex]

Taking the square root of both sides, we get:

[tex]x=\pm 3/2[/tex]

However, we need to check the solutions for extraneous roots since the original equation has a logarithm with a negative argument. We can see that the solution x = 3/2 is extraneous since it results in a negative argument for the logarithm. Therefore, the only valid solution is x = -9/32.

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Answer:

Step-by-step explanation:

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Answer:

s= 6

Step-by-step explanation:

The speed s (in miles per hour) of a car can be given by s = √(30fg), where fis the

coefficient of friction and d is the stopping distance (in feet). The table shows the

coefficient of friction for different surfaces. You are driving 35 miles per hour on an

icy road when a deer jumps in front of your car. How far away must you begin to

brake to avoid hitting the deer? Round your answer to the nearest whole integer. NO

UNITS NEEDED

Answer:

D. The electrician charges $23 per hour.

Step-by-step explanation:

C(h)= 23h+30 is in the form y=mx +b

$30 is the initial fee (b)

$23 is the amount charged per hour (h)

The factorizations of these integers above represent their factorizations into their respective prime numbers.

(a) 756 = 2^2.3^3.7, (b) 819 = 3^2.7.11, (c) 9,075 = 3^2.5^2.7The unique factorization theorem refers to an essential theorem in standard algebraic theory that characterizes the unique factorization properties of integers. Standard factored form, on the other hand, refers to an expression in which an integer is factored into its standard, irreducible components.In view of this, the three provided integers, 756, 819, and 9,075 can be factored as follows:756 = 2^2.3^3.7 (in standard factored form)819 = 3^2.7.11 (in standard factored form)9,075 = 3^2.5^2.7 (in standard factored form)Note that the factorizations of these integers above represent their factorizations into their respective prime numbers.

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Answer:

C is your answer

Step-by-step explanation:

in my opinion, i think it would be the mode.

Answer:

28 is the ans of 252÷9here we go

252 divided by 9 equals 28.

To divide 252 by 9, you can use long division, which involves dividing the number in steps until there is no remainder left.

Here's the step-by-step process:

Write down the dividend (252) and the divisor (9), and set up the long division format:

9 | 252

Look at the leftmost digit of the dividend (2) and see if it's divisible by the divisor (9). Since 2 is less than 9, we bring down the next digit (5) to the right of 2, making it 25.

9 | 252

2

Divide the new number (25) by the divisor (9). The result is 2, which is the first digit of the quotient. Multiply this result by the divisor (2 x 9 = 18) and write it below the 25, then subtract it from 25:

9 | 252

25

18

--

7

Bring down the next digit (2) from the dividend to the right of the remainder (7), making it 72. Now, divide 72 by 9, which gives you 8. Multiply this result by the divisor (8 x 9 = 72) and write it below the 72, then subtract it from 72:

9 | 252

25

18

--

72

72

---

0

There is no remainder left, and the dividend has been completely divided. The quotient is the result of the division, which is 28.

Therefore, 252 divided by 9 equals 28.

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The given probability distribution "a teacher monitored the number of people texting during class each day and calculated the corresponding probability distribution." is a type of discrete probability distribution.

The probability distribution is used to describe the probability of each outcome in a series of possible outcomes. It is a mathematical representation of the outcomes of an experiment.

The teacher likely used a discrete probability distribution to calculate the probability of a certain number of people texting during class each day.

A discrete probability distribution is used to analyze data where the outcome is counted in whole numbers, such as the number of people texting in a given class period.

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The value of x in the right triangle when calculated is approximately 13.8 units

Given the right-angled triangle

The side length x can be calculated using the following sine ratio

So, we have

sin(39) = x/22

To find x, we can use the fact that sin(39 degrees) = x/22 and solve for x.

First, we can use a calculator to find the value of sin(39 degrees), which is approximately 0.6293.

Then, we can set up the equation:

0.6293 = x/22

To solve for x, we can multiply both sides by 22:

0.6293 * 22 = x

13.8446 = x

Rewrite as

x = 13.8446

Approximate the value of x

x = 13.8

Therefore, x is approximately 13.8 in the triangle

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The solution to the equation √(1 + 25/144) = 1 + x/12 is x = 5/6. This was achieved by simplifying the left side of the equation and isolating x on one side.

To solve the equation √(1 + 25/144) = 1 + x/12, we start by simplifying the left side of the equation. The expression inside the square root can be simplified to (144 + 25)/144 = 169/144. Taking the square root of this fraction gives us √(169/144) = (13/12).

Next, we subtract 1 from both sides of the equation to isolate x on one side: (13/12) - 1 = x/12. This simplifies to 1/12 = x/12.

Finally, we multiply both sides by 12 to solve for x: x = (1/12)*12 = 5/6.

So the solution to the equation √(1 + 25/144) = 1 + x/12 is x = 5/6.

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10/73 is the value of x in inequality.

What does the word "inequality" mean?

In mathematics, inequalities describe the connection between two values that are not equal. Equal does not imply inequality. The "not equal symbol ()" is typically used to indicate that two values are not equal.

                    However different inequalities are used to compare the values to determine if they are less than or higher than. The term "inequality" refers to a relationship between two expressions or values that is not equal to one another. Inequality originates from an imbalance, thus.

the inequality  12≥ 73x + 2

                    = 12 - 2  ≥  73x

                     = 10 ≥ 73x

                   = 10/73 ≥ x

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The given lengths of 3 feet, 3 feet, and 7 feet cannot form a triangle because they do not satisfy the Triangle Inequality Theorem, which is the sum of the lengths of any two sides is greater than the length of the third side.

To form a triangle, the sum of the lengths of any two sides of the triangle must be greater than the length of the third side. This is known as the Triangle Inequality Theorem.

Let's apply this theorem to the given lengths of 3 feet, 3 feet, and 7 feet:

The sum of the first two sides is 3 + 3 = 6 feet, which is less than the length of the third side of 7 feet. So, the first two sides cannot form a triangle.

The sum of the first and third sides is 3 + 7 = 10 feet, which is greater than the length of the second side of 3 feet. However, the sum of the second and third sides is 3 + 7 = 10 feet, which is also greater than the length of the first side of 3 feet.

Therefore, neither of the two combinations of sides satisfy the Triangle Inequality Theorem, and so it is impossible to form a triangle with sides of 3 feet, 3 feet, and 7 feet.

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Answer:

25 computers per hour

Step-by-step explanation:

look ate the point 2 hours corresponding to 50 computers

50/2 = 25/1 or 25 computers per hour

Answer: 30

Step-by-step explanation: So what you do is,

1) Add the 25 to the 8

2) Find the sum

3) Subtract 3 from the sum

4) Done!

I hope this helps!

Best,

Abigail H

8th Grade

The absolute maximum value is attained: (0, 0)

The given function is, f(x, y) = xy - 7y - 49x + 343The region is on or above y = x^2 and on or below y = 50. To find the absolute minimum and absolute maximum value of the function, f(x, y), first we will find the critical points of the function.f(x, y) = xy - 7y - 49x + 343 ⇒ ∂f/∂x = y - 49 = 0 ⇒ y = 49 ⇒ ∂f/∂y = x - 7 = 0 ⇒ x = 7Thus, the critical point is (7, 49).Next, we will check for the boundary points. The boundary of the region is y = x^2 and y = 50. The points of intersection are:x^2 = 50 ⇒ x = ±√50 (not in the region)x = ±1.58 ⇒ y = x^2 = 2.50 (not in the region)Also, x = 0 ⇒ y = 0, and x = 0 ⇒ y = 50Thus, the critical points are (7, 49) and (0, 0).f(7, 49) = 7(49) - 7(49) - 49(7) + 343 = -7f(0, 0) = 0 - 7(0) - 49(0) + 343 = 343f(0, 50) = 0 - 7(50) - 49(0) + 343 = -357f(±1.58, 2.50) = ±1.58(2.50) - 7(2.50) - 49(±1.58) + 343 = ∓36.97The absolute minimum value is -7. The points at which the absolute minimum value is attained are (7, 49) and (0, 50).The absolute maximum value is 343. The point at which the absolute maximum value is attained is (0, 0).Hence, the required points are as follows:Points at which the absolute minimum value is attained: (7, 49) and (0, 50)Points at which the absolute maximum value is attained: (0, 0)

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There will be a total of 28 handshakes when the 3 late members arrive.

The first n-1 positive integers are added together using the formula n(n-1)/2, where n is the total number of terms being added together.

The number of combinations or arrangements of a given collection of items may be determined using this formula, which can be obtained using the process of mathematical induction. It is often used in combinatorics and discrete mathematics. For instance, the formula was used to determine how many times a group of individuals shook hands in the scenario above. It might also be used to determine the number of pathways in a network of nodes or vertices or the number of ways to choose a portion of an object set from a bigger set in other settings.

Given that, 3 members arrive late, and 5 members are already present.

The total members are 5 + 3 = 8.

Now, using the sum of positive integer formula:

n(n-1)/2

We can determine the number of handshakes.

8(8-1)/2 = 28

Hence, there will be a total of 28 handshakes when the 3 late members arrive.

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Answer:

3m

Step-by-step explanation:

red wire = 27

black wire= 12

so, we take HCF (highest common factor)

which would be 3 so all the wires would be cut into 3m long.

I hope it helps.

Answer:2m

Step-by-step explanation:

as the factors of 12 are:1, 2, 3, 4, 6 and 12

and the factors of 26 are:1, 2, 13 and 26

so if you are talking meters 2 would be the longest

i hope you get this right x

Answer:

x=1

Step-by-step explanation:

2.5+x=3.5

3.5-2.5=x

1=x

x=1

Answer:

-12 is the y intercept while your slope is -4

Step-by-step explanation:

Answer:51

Step-by-step explanation:

The scale factor of the following pair of similar polygons after the dilation is 0.7

Given

The pair of similar polygons

From the pair of similar polygons, we have the following corresponding side lengths

Pre-image of the polygon = 30

Image of the polygon = 21

The scale factor of the similar polygons is then calculated as

Scale factor = 21/30

Evaluate the quotient

Scale factor = 0.7

Hence, the scale factor is 0.7

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Step-by-step explanation:

remember, the sum of all angles in a triangle is always 180°.

the law of sine :

a/sin(A) = b/sin(B) = c/sin(C)

with a, b, c being the sides, and A, B, C being the three corresponding opposite angles.

so, the angle at Q is

180 = 48 + 48 + angle Q = 96 + angle Q

84° = angle Q

5mm/sin(48) = PR/sin(84)

PR = 5×sin(84)/sin(48) = 6.691306064... mm

The length of PR is approximately 10.33 mm.

what is isosceles triangle ?

An isosceles triangle is a triangle with at least two sides that have equal length, and thus two corresponding angles that are also equal in measure. The third side and angle of an isosceles triangle may or may not be of different length or measure. The two sides that are equal in length are called the legs, and the third side is called the base. The angle opposite the base is called the vertex angle, while the angles adjacent to the legs are called the base angles. In an isosceles triangle, the two base angles are equal in measure.

Since the sum of the angles in a triangle is 180 degrees, we can find the measure of angle PQR as follows:

PQR = 180 - QPR - QRP

PQR = 180 - 48 - 48

PQR = 84 degrees

Since angles QRP and QPR have the same measure, we know that sides OP and OR have equal length (they are opposite those angles). Therefore, triangle POR is an isosceles triangle.

To find the length of PR, we can use the Law of Cosines:

PR^2 = OP^2 + OR^2 - 2(OP)(OR)cos(POR)

Since OP and OR are equal in length, we can simplify this equation to:

PR^2 = 2(OP^2) - 2(OP^2)cos(POR)

We know that POR is 180 - PQR = 96 degrees. We also know that OP = OR, and that QP = 5 mm. Using the Law of Cosines, we can find the length of OP:

OP^2 = QP^2 + OR^2 - 2(QP)(OR)cos(QPR)

OP^2 = 5^2 + OR^2 - 2(5)(OR)cos(48)

OP^2 = OR^2 - 5ORcos(48) + 25

Since OP = OR, we can substitute OP for OR in the above equation:

OP^2 = OP^2 - 5OPcos(48) + 25

5OPcos(48) = 25

OP = 25/(5cos(48))

OP ≈ 6.25 mm

Now we can substitute this value into the equation we derived earlier to find PR:

PR^2 = 2(OP^2) - 2(OP^2)cos(POR)

PR^2 = 2(6.25^2) - 2(6.25^2)cos(96)

PR ≈ 10.33 mm

Therefore, the length of PR is approximately 10.33 mm.  

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We can graph this solution on the number line by placing a point at 7.6.

Graphs are visual representations of data or information. They are used to show relationships between different pieces of data and display numerical data in a more meaningful way. Graphs are made up of individual elements called nodes which are connected by edges. Nodes represent individual data points or pieces of information. Edges represent the relationship between two nodes and can represent either a physical or a logical link. Graphs can be used to represent many different types of data or information, such as social networks, transportation networks, and even biological relationships. Graphs are a powerful tool for understanding complex data sets, making them an essential tool for data analysis.

80∠ 10x + 4 = 20

80 10x + 4 = 20

80 - 4 = 10x

76 = 10x

7.6 = x

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We can graph this solution on the number line by placing a point at 7.6.

Graphs are visual representations of data or information. They are used to show relationships between different pieces of data and display numerical data in a more meaningful way. Graphs are made up of individual elements called nodes which are connected by edges. Nodes represent individual data points or pieces of information. Edges represent the relationship between two nodes and can represent either a physical or a logical link. Graphs can be used to represent many different types of data or information, such as social networks, transportation networks, and even biological relationships. Graphs are a powerful tool for understanding complex data sets, making them an essential tool for data analysis.

80∠ 10x + 4 = 20
80 10x + 4 = 20
80 - 4 = 10x
76 = 10x
7.6 = x

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We have that, using Euclid's algorithm, we find the inverse of 200 modules 1001 is -5 (or 1001+5).

To find the inverse of a module m using Euclid's algorithm, the steps are as follows:

1. Calculate the greatest common divisor (GCD) of a and m using the Euclidean algorithm.

2. Let a = GCD * s + m*t, where s is the inverse of a module m.

3. The GCD in terms of a and my is written as 1 = m-s*a.

4. Find s = -a, so the inverse of a module m is -a (or m+s).

For example, a = 2, m=17, so GCD = 1 = 17-8*2 and the inverse of 2 modulo 17 is -8 (or 17+8). Similarly, for a = 34, m= 89, the GCD = 1 = 89-34*2 and the inverse of 34 modulo 89 is -34 (or 89+34). Finally, for a = 200, m= 1001, the GCD = 1 = 1001-5*200 and the inverse of 200 modulo 1001 is -5 (or 1001+5).

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Answer:

30 minutes

Step-by-step explanation:

Use ratios. This is an inverse function, as speeding up makes the time traveling go down. So, when dividing the speed by 3 (done so 15 can get to 5), we multiply the time traveled by 3.

10 minutes * 3 = 30 minutes