If the hypotenuse of a 30°-60°-90° triangle is 8 cm, what is the measure of the longer leg of the triangle?

246960 6-digit numbers contain exactly one 3, one 4, and one 5 as digits.

There are a total of 6 digits that make up any of the numbers in 0 to 999999 with the given condition

The total numbers are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

This can be represented as UVWXYZ

Digit 3 can occupy any of the 6 positions

Digit 4 can occupy any of the (6 - 1) positions

Digit 5 can occupy any of the (6 - 2) positions

The remaining 7 digits will occupy the last 3 positions in the following ways:

The first of the 7 digits can be selected from any of  7 ways

The second can be selected from any of 7 ways

The third can be selected from any of 7 ways

So, the total number of selection is:

= 6 × 6 × (6 - 1) × (6 - 2) × 7 × 7 × 7

= 36 × 5 × 4 × 343

= 246960

Therefore there are 246960 numbers.

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Note that the square feet of carpet that will be required for this area is:  42 Square Feet of Carpet.

This is solved using the knowledge of the area of rectangles and right-angled triangles.

A right-angled triangle is a triangle in which one of the angles is 90 degrees. The total of the other two angles is 90 degrees.

Note that the area  of a right-angled triangle is given as:
A = (½)×b×h square units

Where b is breath and h is height.

Note as well that the area of the rectangle is given as Lx B

To solve the above problem, we must create one more right-angled triangle beside the one already created, then solve for the areas of the respective shapes.

Now that the shape has been divided in to two Right-angled Triangles and one rectangle (See the attached) their respective dimensions are given as follows:

1) Right Angled triangled A

B = 2ft

H = 3ft

Area = 1/2 2 * 3 = 3ft

2) Right-Angled Triangle B

B = 4ft

H = 3ft

A = 1/2 * 4 * 3

A = 6ft

3) Rectangle - C

L = 11ft

B= 3ft

A = 11 * 3 = 33 Ft.

Thus total square feet of outdoor carpet required is:

3 + 6 + 33

= 42 Square Feet of Carpet.

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The number of bean plants should she plant will be 60

t = tomato plants

b = beans

To Solve by using system of equations

Clear any fractional coefficients if there are any. Alternate the coefficients of a single variable. Select the variable you want to remove. Multiply one or both equations to make the variable's coefficients opposite.

Total plants = 100

t + b = 100     {equation 1}

20 more tomatoes than beans

t = b + 20       {equation 2}

b + 20 + b = 100

2b + 20 = 100

Finish solving for b to find a number of beans.

Then tomatoes will be b+20

Putting this in equation 1

b+20+b=100

2b+20=100

b=40

then t= b+20  {equation 2}

t=60

Hence the number of bean plants should she plant will be 60

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The two missing positive numbers are; 50/3 and 200/3

The difference between two positive numbers is 50. Let the two numbers be x and y and as such we have;

x - y = 50   -----(1)

One number is four times the other number. Thus;

x = 4y  -----(2)

Put 4y for x in eq 1 to get;

4y - y = 50

3y = 50

y = 50/3

Thus;

x = 4(50/3)

x = 200/3

We conclude that x and y above are the two positive numbers.

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The volume of water that are left in the tank is equal to 3.2 gallons.

Mathematically, the volume of a cuboid tank is a measure of its capacity and it can be calculated by using this following formula:

Volume of a tank = L × W × H

Where:

In order to determine the quantity (volume) of water that are left in the tank, we would use this mathematical expression:

Volume of water left = capacity of tank A - volume of water poured

Volume of water left = 9.5 gallons - 6 1/3 gallons

Volume of water left = 9.5 gallons - 6.3 gallons

Volume of water left = 3.2 gallons.

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Answer: The perimeter of a parallelogram is the sum of all its sides.

In this case, since all sides of the parallelogram WXYZ are given as 34 units, the perimeter of parallelogram WXYZ is the sum of all its four sides.

So the perimeter of parallelogram WXYZ = 34 + 34 + 34 + 34 = 136 units

It's important to notice that the given value of perimeter = 60 is not correct.

The perimeter of parallelogram WXYZ is 136 units, not 60 units.

Step-by-step explanation:

The angle of depression of the airplane is 3.6⁰.

The angle of depression of the airplane is calculated by forming a triangle with base displacement of the airplane and the altitude.

base of the right triangle = 15 miles = 79,200 ft

height of the right triangle = 5,000 ft

The angle of depression is calculated as follows;

θ = arc tan ( h / b )

θ = arc tan ( 5000 / 79,200 )

θ = 3.6⁰

Thus, the angle of depression of the airplane is a function of the altitude and base displacement of the airplane.

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For the absolute value function, the vertex is at its maximum at (-4,0).

A function is an expression, rule, or law in mathematics that defines a relationship between one variable (the independent variable) and another variable (the dependent variable). Functions are common in mathematics and are required for the formulation of physical relationships in the sciences. A function is defined as a relationship between a set of inputs that each have one output. A function is a relationship between inputs in which each input is related to exactly one output.

Here,

The vertex for the absolute value function is at its maximum at (-4,0).

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The actual area of the garden without the scale factor is 6 square feet.

The ratio of the scale of an original thing to a new object that is a representation of it but of a different size is known as a scale factor.

Area is the entire amount of space occupied by a flat (2-D) surface or an object's form. Depending on the size and number of sides, different forms have varying areas.

Let the original area of the garden be x.

Given that the scale factor used is 1/3, thus we have:

x (1/3) = 2

x = (2)(3)

x = 6

Hence, the actual area of the garden without the scale factor is 6 square feet.

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Answer:

43 centigrams > 31.7 milligrams

Step-by-step explanation:

Since 43 centigrams is equal to 430 milligrams, it is therefore greater that 31.7 milligrams.

Answer: I am not able to create images such as tree diagrams. But I can explain to you how it could be done.

A tree diagram is a visual representation of the possible outcomes and their probabilities.

The tree diagram to show the above information would have two branches: one representing the event that the bus is late and the other representing the event that the bus is not late.

The first branch, "Bus is Late", would have a probability of 0.1 and would split into two further branches: one representing the event that Andrew is late to school (with a probability of 0.8) and the other representing the event that Andrew is not late to school (with a probability of 0.2).

The second branch, "Bus is not Late", would have a probability of 0.9 and would split into two further branches: one representing the event that Andrew is late to school (with a probability of 0.05) and the other representing the event that Andrew is not late to school (with a probability of 0.95).

This tree diagram will help you to understand the relationship between the different events and their probabilities, as well as the overall probability of Andrew being late to school.

Step-by-step explanation:

Proportionately, an average American household uses 100 BTU of energy per home per year while Californians use 75 BTU, which is 25% less than the national average.

Proportion refers to the ratio or relative size of a value or number compared to another.

Proportions are fractional values equated to each other.

Generally, proportions may be depicted as fractions, decimals, or percentages.

The level of energy per home per year in California = 75 BTU

The national average of BTU use = 100%

California's use = 75% (100 - 25)

Proportionately, if 75% = 75 BTU, 100% = 100 BTU (75/0.75)

Thus, the national average use of energy is expected to be 100 BTU.

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The average speed must you maintain in the remaining distance in order to have an overall average speed of 33.6 mph is 50.39 mph.

We have to find the average speed must you maintain in the remaining distance in order to have an overall average speed of 33.6 mph.

Let the speed be x.

The total average we want = 33.6 mph

The distance from your home to town = 90 miles

Time = Distance/Speed

Time = 90/33.6

At 45 miles (half way), you find you have averaged only 25.2 mph. So

Time first leg = 45/25.2

Time second leg= 45/x

Now the average speed;

45/25.2 + 45/x = 90/33.6

Subtract 45/25.2 on both side, we get

45/x = 90/33.6 - 45/25.2

After solving the ratio, we get

45/x = 0.893

Multiply by x on both side, we get

0.893x = 45

Divide by 0.893 on both side, we get

x = 45/0.893

x = 50.39 mph

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The complete question is:

On a drive from your home to town, you wish to average 33.6 mph. The distance from your home to town is 90 miles. However, at 45 miles (half way), you find you have averaged only 25.2 mph. What average speed must you maintain in the remaining distance in order to have an overall average speed of 33.6 mph?

Answer:

For y=2x+1

The line y=2x+1 has a y-intercept of +1, so begin with a point at (0,1)

and its slope is 2/1, which means from there you go Up 2 and Right 1.

Follow that pattern until you can draw a straight line through all the points.

For -2x=y=-2

In this point it only passes (0,2) and (1.0)

Have a wonderful day! :-)

The value of 'x' in the equation 8 - x/x = 3/2 is x equal to [tex]3\frac{1}{5}[/tex].

An equation is written in the form of variables and constants separated by the operation of multiplication and division,

An equation states that terms in different forms on both sides of the equality sign are equal.

Multiplication and division do not separate the terms of an equation.

Given, An equation in 8 - x/x = 3/2 fractions.

Now, The LCM of both the denominators is 2x, So we'll multiply each term by 2x.

2(8 - x) = 3x.

16 - 2x = 3x.

- 2x - 3x = - 16.

- 5x = - 16.

5x = 16.

x = 16/5.

x = [tex]3\frac{1}{5}[/tex].

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The translation rule (x + 7, y - 19) has the meaning given as follows:

The four translation rules are defined as follows:

The meaning of each translation in this problem is given as follows:

The problem asks for the meaning of the translation (x + 7, y - 19).

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Option A, (x,y) → (2x,2y) This rule describes an origin-centered stretch with a factor of 2. That is, the image doubles its original size in both the x and y directions.

The new rectangle is similar to the original, but with larger dimensions, so the correct rule is A) (x, y) → (2x, 2y).

In this case, the rule described in option A) (x, y) → (2x, 2y) specifies that the image is twice the size of the original rectangle in both the x and y directions increase. This means that the scale factor is 2. This will result in a similar shape but with larger dimensions. This is the correct rule describing stretching the rectangle URST to the rectangle U'R'S'T'.

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The approximate probability that less than 10 out of 200 U.S. citizens are working more than 50 hours a week on average is 0.019

We can use the normal distribution to approximate the probability that less than 10 out of 200 U.S. citizens are working more than 50 hours a week on average.

To do this, we have to standardize the variable by subtracting the mean and dividing by the standard deviation.

In this case, X = (50 - 37) / hours = 1.83

We know that for a standard normal distribution, the probability of X being less than a certain value can be found using a standard normal table or a calculator with a built-in standard normal cumulative distribution function (CDF).

Now, we will use the cumulative distribution function to find the probability of less than 10 out of 200 working more than 50 hours a week on average. Using a standard normal table or calculator, we can find the probability of having X < -2.04.

P(X < -2.04) = 0.0188

So the approximate probability that less than 10 out of 200 U.S. citizens are working more than 50 hours a week on average is 0.0188 which rounding it off to the third decimal place we get 0.019.

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Answer:

BD = 36

Step-by-step explanation:

• the diagonals of a parallelogram bisect each other , so

BE = DE , that is

15x - 12 = 9x ( subtract 9x from both sides )

6x - 12 = 0 ( add 12 to both sides )

6x = 12 ( divide both sides by 6 )

x = 2

Then

BD = BE + ED

     = 15x - 12 + 9x

     = 24x - 12 ( substitute x = 2 )

    = 24(2) - 12

    = 48 - 12

    = 36

The table is filled considering the numeric values of the function, as follows:

To find the numeric value of a function or of an expression, we replace each instance of the variable in the function or in the expression by the value at which we want to find the numeric value.

The numeric value in this problem is found replacing the lone instance of x by it's value, hence the numeric value at x = -6 is given as follows:

y = 2(-6) - 3

y = -12 - 3

y = -15.

The same procedure is applied for x = -3, x = 0 and x = 3.

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The probability that neither of the people have their ears pierced is 0.6157

What is the probability?

Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, with 0 indicating that an event is impossible and 1 indicating that an event is certain.

The probability of A happening is the number of people who do not have their ears pierced divided by the total number of people in the sample, which is:

P(A) = (2000 - 430) / 2000 = 1570 / 2000 = 0.785

The conditional probability of B given A is the number of people who do not have their ears pierced and are still available to be selected as the second person divided by the total number of people available to be selected as the second person, which is:

P(B|A) = (2000 - 430 - 1) / (2000 - 1) = 1569 / 1999 = 0.7845

Therefore, the probability that neither of the people have their ears pierced is:

P(A) * P(B|A) = 0.785 * 0.7845 = 0.6157

Hence, the probability that neither of the people have their ears pierced is 0.6157

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The factor 3 does not occur in pair. Thus, ‘3’ is the smallest number by which 8112 should be divided to get a perfect square.

The LCM of 8, 15, and 20 is 120. Prime factors 2, 3, and 5 do not have a pair in this case. To construct a perfect square, 120 must be multiplied by 2 x 3 x 5. As a result, the least square number divisible by 8, 15, and 20 is 3600.

The smallest square matrix is 2 2 2 2 ; there is no theoretical limit to the highest size square matrix, however anything larger than 10 10 10 10 becomes impossible to manage numerically.4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, and 961

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Answer:

S(t) is a function, because there is a unique output value (number of students) for each input value (number of years after 2000)

S(t) can be represented by a graph, where the horizontal axis represents the number of years after 2000, and the vertical axis represents the number of students

Step-by-step explanation:

The statements regarding function S are true are:

S(5) = 2024 means there were 2024 students in 2005.

S(10)= S(5) there same number of students in 2005 and 2010.

We have function,

S(t)= 2000t

First, S(0) in 2000 means that there were 2000 students in starting.

Second, S(5) = 2024 means there were 2024 students in 2005.

Third, S(10)= S(5) there same number of students in 2005 and 2010.

Fourth, S(15)- S(10)= -40 means that there were 40 less students in 2015 than in 2010.

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Becca is not correct because her probability of 2/14 is not equal to the true experimental probability of 2/16.

We are given the tools in the box as;

Hammer = 2 nos

Pliers = 5 nos

Screw drivers = 6 nos

Wrench = 3 nos

Thus;

Total number of tools in the box = 2 + 5 + 6 + 3

= 16

Probability of selecting a hammer = 2/16

Now, Becca says that   for every 14 tools in a toolbox, there are 2 hammers. This means she is suggesting a probability of 2/14.

This is wrong because it is different from the experimental probability that we got.

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The measure of each interior angle is calculated to be 172° and each exterior angle is calculated to be 8°.

Each interior angle in a polygon can be calculated using the formula, (n - 2) ×180°/n

where, n is the polygon's number of vertices.

So, each exterior angle can be calculated as, 180 - (n - 2) 180°/n

Now, let us put the number of vertices into these formulae,

Each interior angle = (45 - 2)×180°/45 = 43×180°/45 = 172°

Each exterior angle = 180 - (45 - 2) 180°/45 = 8°

So, the interior angle is 172° and the exterior angle is 8°.

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Here is a simple framework to model the problem as an Integer Programming (IP) problem and solve it using Excel Solver:

Let x(i,j) be the number of full-time employees assigned to work on day i (1 <= i <= 7) and given day off on day j (1 <= j <= 7).

a. Work days constraint: For each day i, the number of employees working must meet the demand:

∑(j=1 to 7) x(i,j) = demand(i) (1 <= i <= 7)

b. Availability constraint: Each employee can work only 5 consecutive days, so for each day j (1 <= j <= 7), the number of employees given day off on day j must be less than or equal to the number of employees working on day j-5:

∑(i=1 to 7) x(i,j) <= ∑(i=j-5 to j-1) x(i,j-5) (1 <= j <= 7)

The objective is to minimize the total number of full-time employees hired, so the objective function is:

minimize: ∑(i=1 to 7) ∑(j=1 to 7) x(i,j)

Excel Solver can be used to find the optimal solution to the IP problem. The constraints and objective function should be set up in the spreadsheet, and Solver should be configured to use the integer constraint option. The solution will give the optimal number of full-time employees hired and their assignment on each day.

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Answer:

1: 5/10, square root of 2, 2.5, square root of 25 3: 4.5

Step-by-step explanation:

2.5

5 divided by 10 comes to 0.5

25 square rooted is 5

2 square rooted is 1.4

the biggest number is 5 and the smallest is 0.5

subtract 0.5 from 5 and you get 4.5