I need help with this question

The measure of side AC is approximately 366.9 inches.

A triangle is a geometric shape that consists of three straight sides and three angles. It is one of the basic shapes in geometry and is often studied as a fundamental building block in mathematics, engineering, and physics. Triangles have several important properties, including:

The sum of the angles in a triangle is always 180 degrees.

The longest side of a triangle is opposite to the largest angle, and the shortest side is opposite to the smallest angle.

The area of a triangle can be calculated using the formula: area = 1/2 x base x height, where the base is one of the sides and the height is the perpendicular distance from the base to the opposite vertex.

Triangles can be classified based on the length of their sides and the measure of their angles. For example, a triangle with all three sides of equal length is called an equilateral triangle, while a triangle with two sides of equal length is called an isosceles triangle.

To find the measure of side AC, which is denoted by "b", we can use the Law of Cosines, which relates the lengths of the sides of a triangle to the cosine of one of its angles. Specifically, the Law of Cosines states that:

c² = a² + b² - 2ab cos(C)

where "a" and "b" are the lengths of the other two sides of the triangle, and "C" is the angle opposite to side "c".

In this case, we know that side AB has length "c" (290 inches), and side AC is denoted by "b". We also know that the angle opposite to side AB has measure 42 degrees. Therefore, we can write:

c² = a² + b² - 2ab cos(C)

290² = a² + b² - 2ab cos(42)

Simplifying this equation and solving for b, we get:

b² - 2ab cos(42) + (290² - a²) = 0

This is a quadratic equation in "b", which we can solve using the quadratic formula:

b = [2a cos(42) ± √((2a cos(42))² - 4(290² - a²)]/2

Simplifying this expression, we get:

b = a cos(42) ± √(a² cos²(42) - (290² - a²))

We don't know the value of "a", so we cannot find the exact value of "b". However, we can use the fact that "b" is a length of a triangle, so it must be positive. Therefore, we can discard the negative square root, and we get:

b = a cos(42) + sqrt(a² cos²(42) - (290² - a²))

Since "b" is positive, we can set the expression inside the square root to zero and solve for "a":

a² cos²(42) - (290² - a²) = 0

Simplifying and solving for "a", we get:

a = 290/sin(42)

Substituting this value of "a" back into the equation for "b", we get:

b = (290/sin(42)) cos(42) + sqrt((290/sin(42))² cos²(42) - 290² + (290/sin(42))²)

Simplifying this expression, we get:

b ≈ 366.9 inches

Therefore, the measure of side AC is approximately 366.9 inches.

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Bill's earnings are subject to deductions for Social Security, Medicare, and Federal Income Tax (FIT). Today, Bill's cumulative earnings are $110,000. This week, Bill earned $1,200. As Bill is married, is paid weekly, and claims three exemptions, the Social Security rate is 6.2% of $128,401.45, and the Medicare rate is 1.45%. Bill will not pay FIT for the week.

To calculate Bill's earnings for the week, we first subtract the applicable deductions from his $1,200 gross earnings. The Social Security rate of 6.2% on $128,401.45 is $7,966.98, and the Medicare rate of 1.45% is $181.80. This means that $9,148.78 is deducted from Bill's $1,200 gross earnings, leaving him with a net earnings of $286.22.

To calculate Bill's total cumulative earnings, we need to add the $1,200 that he earned this week to his existing $110,000 cumulative earnings. Thus, Bill's total cumulative earnings are $111,200.

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To find all the zeros of the function f(x) = x^3 - 3x^2 + 4x - 2 and write f(x) in factored form, we can use the Rational Root Theorem.

What is the Rational Root Theorem?

If we have a polynomial with integer coefficients, we can use the Rational Root Theorem to find all of the possible rational roots (solutions) of the polynomial. The Rational Root Theorem says that if a polynomial has a rational root p/q (where p and q are integers), then p must be a factor of the constant term of the polynomial (the term without x), and q must be a factor of the leading coefficient of the polynomial (the coefficient of the highest power of x).

To apply the Rational Root Theorem to the function f(x) = x^3 - 3x^2 + 4x - 2, we look at the constant term (-2) and the leading coefficient (1) of the polynomial to find all of the possible rational roots. The possible rational roots are: ±1, ±2.

First, let's try x = 1 as a possible root. We can use synthetic division to see if it's a zero:

1 | 1 -3 4 -2

| 1 -2 2

|__________

1 -2 6 0

There is no remainder, so x = 1 is a zero of the function. Now, we can use synthetic division again to factor out (x - 1):

(x - 1)(x^2 - 2x + 6)

To find the remaining zeros, we can solve the quadratic equation x^2 - 2x + 6 = 0 using the quadratic formula:

x = (2 ± √(2^2 - 4(1)(6))) / (2(1))

x = (2 ± √(-20)) / 2

x = 1 ± i√5

The zeros of f(x) are:

x = 1, x = 1 + i√5, x = 1 - i√5

So, we can write f(x) in factored form as:

f(x) = (x - 1)(x - (1 + i√5))(x - (1 - i√5))

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The required equation of the straight line is y = 3x - 8.

The formula for a straight line is y=mx+c where c is the height at which the line intersects the y-axis, also known as the y-intercept, and m is the gradient.

The equation of a straight line with gradient that passes through a point[tex]$(x_1, y_1)$[/tex] is given by:

[tex]$$y - y_1 = m(x - x_1)$$[/tex]

In this case, we are given that the gradient is 3 and the line passes through the point (5,7). So we have:

[tex]$\begin{align*}y - 7 &= 3(x - 5) \y - 7 &= 3x - 15 \y &= 3x - 8\end{align*}[/tex]

Therefore, the equation of the straight line is y = 3x - 8.

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Answer:

Step-by-step explanation:

The inequality is x > 45.

This is so because the dot is not shaded which tells us that the value of x is not equal to 45.

Also, the direction of the arrow is going to the right on the positive side of the number line which means it's greater than 45.

Lisa's situation could be represented by just reversing the arrow on the above graph.

Hope this helps! :)

This is an example of Stratified probability sampling, because here the samples are drawn at random and the conclusions based are  generalised to the population as a whole

A researcher conducting a study on streaming services divides the population of Netflix subscribers into four groups of users based on the type of material viewed and then sampling from within those groups. This is an example of what type of sampling?This is an example of Stratified probability sampling.What is stratified probability sampling?Stratified sampling is a type of probability sampling that divides the population into subgroups or strata based on relevant characteristics. The strata are then randomly chosen, and samples are obtained from within each stratum.

Stratified sampling may be useful in situations where researchers want to obtain a representative sample of a population while simultaneously ensuring that the sample is sufficiently representative of different subgroups within the population. It might be used, for example, in a study of people's attitudes about social media, with strata based on age or socioeconomic status.Stratified sampling, like all probability sampling methods, has the advantage of ensuring that the sample is representative of the population from which it was drawn. Because samples are drawn at random, the conclusions based on them can be generalised to the population as a whole.

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Answer:

£756

Step-by-step explanation:

To fill the tank completely, Mr. Patel needed:

500 liters initially + (total capacity of the tank - 500 liters) = 500 + (tank capacity - 500) = tank capacity

To find the tank capacity, we need to know how many liters of oil Mr. Patel bought initially:

Total cost of oil bought initially = 900

Cost per liter of oil = £0.60

Number of liters of oil bought initially = 900 / 0.60 = 1500

Therefore, the tank capacity is:

500 + (tank capacity - 500) = 1500

Tank capacity - 500 = 1000

Tank capacity = 1500 liters

At the beginning of February, Mr. Patel had 300 liters of oil left in his tank, so he needed to buy:

1500 - 300 = 1200 liters of oil

The cost of oil had increased by 5%, so the cost per liter was:

£0.60 + (£0.60 x 0.05) = £0.63

Therefore, the total cost of the oil in February was:

1200 x £0.63 = £756

Answer:

To make a the subject of the formula, we need to isolate it on one side of the equation.

Starting with:

a + p = f

Subtracting p from both sides, we get:

a = f - p

Therefore, the formula for a as the subject is:

a = f - p

Step-by-step explanation:

The coach brought a mean (average) of 5 footballs.

The sum of all values divided by the total number of values yields the mean (also known as the arithmetic mean, which differs from the geometric mean) of a dataset. The term "average" is frequently used to describe this central tendency measurement.

We must tally up all of the footballs the coach brought and divide that amount by the number of times he brought them in order to determine the mean (average) number of footballs:

Total number of footballs = 5 + 4 + 4 + 6 + 4 + 7 + 5 + 7 + 2 + 6

Total number of footballs = 50

Number of times the coach brought footballs = 10

Mean number of footballs = Total number of footballs / Number of times the coach brought footballs

Mean number of footballs = 50 / 10

Mean number of footballs = 5

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We can plot the point (0, 2) and then use the slope to find another point on the line. To do this, we can move 3 units horizontally (since b = 3) and -2 units vertically (since a = -2)

How to find?

The equation ax + by = c represents a straight line in a two-dimensional Cartesian coordinate system, where x and y are the variables and a, b, and c are constants.

a and b are the coefficients of x and y, respectively. They determine the slope or gradient of the line. Specifically, the slope of the line is given by -a/b.

c is the constant term, which represents the y-intercept of the line. It is the value of y when x = 0.

To solve an equation of this form, you can use standard algebraic techniques. One common method is to rearrange the equation to solve for y in terms of x:

ax + by = c

by = c - ax

y = (-a/b)x + (c/b)

This equation is now in slope-intercept form, where the slope is -a/b and the y-intercept is c/b.

To graph the line represented by the equation, you can plot the y-intercept (0, c/b) and then use the slope to find other points on the line. To do this, you can move b units horizontally and -a units vertically from the y-intercept to find another point on the line.

For example, let's say the equation is 2x + 3y = 6. To solve for y, we can rearrange the equation:

3y = 6 - 2x

y = (-2/3)x + 2

Now we know that the slope is -2/3 and the y-intercept is 2. We can plot the point (0, 2) and then use the slope to find another point on the line. To do this, we can move 3 units horizontally (since b = 3) and -2 units vertically (since a = -2):

(0, 2) --> (3, 0)

We can then connect the two points with a straight line to graph the equation.

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In conclusion, when looking at the relationship between the average price of a barrel of oil and the average retail price of a gallon of unleaded regular gasoline, you should expect to see a weak negative association between the two variables. However, there will also be some degree of variability in the relationship due to other factors that affect the price of gasoline.
The best way to see how well the price of oil predicts the price of gasoline is to make a scatterplot with the price of oil as the explanatory variable. The scatterplot will show the relationship between the average price of a barrel of oil and the average retail price of a gallon of unleaded regular gasoline.

When examining the scatterplot, you should expect to see a weak negative association between the two variables. This means that when the price of oil increases, the price of gasoline will tend to decrease. This is because when the price of oil goes up, the production costs of gasoline also go up, which in turn causes the retail price of gasoline to increase.

The scatterplot should also indicate that there is a certain degree of variability in the relationship between the two variables. This is because the price of gasoline is also affected by other factors, such as taxes, supply and demand, weather conditions, and availability of other fuels.

Therefore, while the price of oil is a good indicator of the price of gasoline, the relationship between the two is not always exact.

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The point where the equation of the line touch the circle is (2.166, 3.083).

and

The equation of a circle with center (h, k) and radius r is:

(x - h)^2 + (y - k)^2 = r^2

So the equation of the circle with center (0, 2) and radius 5 is:

x^2 + (y - 2)^2 = 25

We can solve for the intersection of the circle and the line y = 0.5x + 2 by substituting y = 0.5x + 2 into the equation of the circle:

x^2 + (0.5x)^2 + 2x + 2^2 - 25 = 0

Simplifying and solving for x:

1.25x^2 + 2x - 9 = 0

Using the quadratic formula, we get:

x = (-2 ± √(2^2 - 41.25(-9))) / (2*1.25)

x = (-2 ± √(104)) / 2.5

x ≈ -6.166 or x ≈ 2.166

Since we are looking for a point in the first quadrant, we take the positive value of x:

x ≈ 2.166

Now we can use the equation of the line to find the corresponding y-coordinate:

y = 0.5x + 2

y ≈ 3.083

Therefore, the point where the line y = 0.5x + 2 intersects the circle with radius 5 and center (0, 2) in the first quadrant is approximately (2.166, 3.083).

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the total amount of rice in 8 packets of 11 kg will be 88 kg.

To solve this question, it is necessary to use the formula for finding the total amount of a given item when the number of packets and the mass of each packet are given. In this question, 8 packets of 11 kg is given. Therefore, the total amount of rice can be calculated using the formula, Amount = Number of Packets * Mass per Packet. This can be written as 88 kg = 8 * 11 kg. Hence, the total amount of rice in 8 packets of 11 kg will be 88 kg.

The formula to find the total amount of a given item is:

Amount = Number of Packets * Mass per Packet

We are given that there are 8 packets of 11 kg of rice. So we can substitute these values into the formula:

Amount = 8 * 11 kg

Now we can simplify by multiplying 8 and 11:

Amount = 88 kg

Therefore, the total amount of rice in 8 packets of 11 kg is 88 kg.

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The total number of cells after 9 hours is approximately 10671.

An exponential expression is a mathematical expression in which a number or variable is raised to a power, which is usually written as a superscript. The base is the number or variable being raised to a power, and the exponent is the number indicating how many times the base is multiplied by itself.

In the given question,

The initial growth of the bacterial cells can be modeled by an exponential function of the form:

N(t) = N0 * 3^(kt)

Where N(t) is the number of cells at time t, N0 is the initial number of cells, k is the growth rate, and t is the time elapsed.

Using the information given in the problem, we can find N0 and k as follows:

N0 = 1 (since the researchers started with one cell)

N(5) = 35

N(5) = N0 * 3^(5k) = 35

3^(5k) = 35

5k = log3(35)

k = log3(35)/5

Therefore, the function for the growth of bacterial cells is:

N(t) = 3^(t*log3(35)/5)

To find the total number of cells after 9 hours, we can substitute t=9 into the equation:

N(9) = 3^(9*log3(35)/5) ≈ 10671

Therefore, the total number of cells after 9 hours is approximately 10671.

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The area of the composite figure is calculate to be 96.07 square in

The area of the composite figure is solved by dividing the figure into simpler figures

We have to divide the given figure into 3 figures

2 rectangles and a quarter of a circle

Area of the first rectangle

= base * height

= 7 * 7

= 49 square in

Area of the second rectangle

= base * height

= 5 * 8

= 40 square in

Area of the quarter of a circle

= 1/4 * π r²

= 0.25 * π * 3²

= 7.07 square in

Area of the composite figure

= 49 square in + 40 square in + 7.07 square in

= 96.07 square in

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Among jordan's and jamie's boxes Jamie's box has a larger volume than Jordan's box.

To find the volume of a box, we need to multiply the width, length, and height (or depth) of the box together.

If the dimensions of the box are given in meters, then the volume will be in cubic meters.

For example, if a box has a width of 2 meters, a length of 3 meters, and a height of 4 meters, then its volume would be:

Volume = 2 meters x 3 meters x 4 meters = 24 cubic meters

Therefore, the volume of the box is 24 cubic meters.

To find the volume of each box, we can multiply the width, length, and depth together:

Jordan's box volume = 5.1 m x 6 m x 4 m = 122.4 cubic meters

Jamie's box volume = 4.7 m x 8 m x 4 m = 150.4 cubic meters

Therefore, Jamie's box has a larger volume than Jordan's box.

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The calculated value of x in the right triangle is 34.9 degrees

From the question, we have the following parameters that can be used in our computation:

The right triangle

The value of x can be calculated using

tan(x) = 4.0/5.8

Evaluate the quotient

So, we have

tan(x) = 0.6897

Take the arctan of both sides

So, we have

x = 34.9 degrees

Hence, the value of x is 34.9 degrees

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Answer:

Step-by-step explanation:

Which statement best compares and contrasts the two formats?

The infographic shows the movement of water visually, while the text describes it.

The infographic would be less appealing to visual learners than the text.

It is impossible to understand the infographic without the text.

The text is more accurate than the infographic.

The probability that the second interview will be given to a woman, given that the first interview was given to another woman is 18/323 and the probability of all three interviews being given to men is: (9/22) * (8/21) * (7/20) = 21/770

To find the probability that the second interview will be given to a woman, given that the first interview was given to another woman, we can use conditional probability. Let’s call the event that the first interview was given to a woman A, and the event that the second interview is given to a woman B.

Then we want to find P(B|A), the probability of B given A. Using a tree diagram, we can see that the probability of A is:(13/22) * (12/21) = 4/11The probability of B given A is: (9/20) * (8/19) = 18/323. Therefore, the probability that the second interview will be given to a woman, given that the first interview was given to another woman, is 18/323.(b) To find the probability that all three interviews will be given to men, we can use the multiplication rule of probability.

The probability of selecting a man on the first interview is 9/22. The probability of selecting a man on the second interview, given that the first interview was given to a man, is 8/21. The probability of selecting a man on the third interview, given that the first two interviews were given to men, is 7/20. So, the probability of all three interviews being given to men is:(9/22) * (8/21) * (7/20) = 21/770

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Answer:

5

Step-by-step explanation:

the diameter is 2 times the radius. So, 10=2r

r=5

Answer:

c ≈ 6.4 in

Step-by-step explanation:

using Pythagoras' identity in the right triangle.

the square on the hypotenuse (c) is equal to the sum of the squares on the other 2 sides, that is

c² = 4² + 5² = 16 + 25 = 41 ( take square root of both sides )

c = [tex]\sqrt{41}[/tex] ≈ 6.4 in ( to the nearest tenth )

Answer:

(3e+7) - (2/5)x

Step-by-step explanation:

Instead of the - 2/5x we can remove the - from it and convert the addition problem into a subtraction problem and they will be equivalent.

Mrs. Irfan needs to buy 10,000 tiles to cover the floor of her balcony.

To determine the number of tiles Mrs. Irfan needs to cover her balcony, we need to first calculate the area of each tile. The area of each tile can be determined by multiplying its length and breadth:

Area of each tile = Length × Breadth = 20 cm × 10 cm = 200 cm²

Next, we need to calculate the total area of the balcony, which is given as 200 cm². However, we need to convert this to square centimeters since we have determined the area of each tile in square centimeters. We can do this by multiplying 200 cm² by 10,000 (1 square meter is equal to 10,000 square centimeters):

Total area of balcony in square centimeters = 200 cm² × 10,000 = 2,000,000 cm²

To determine the number of tiles Mrs. Irfan needs, we can divide the total area of the balcony by the area of each tile:

Number of tiles needed = Total area of balcony ÷ Area of each tile

= 2,000,000 cm² ÷ 200 cm²

= 10,000 tiles

Complete question:

Mrs. Irfan wants to use tiles of a length of 20 cm and a breadth of 10 cm to cover the floor of her balcony. The area of the balcony is 200 cm^2. how many tiles does she need to buy(1sq m = 10,000 Sq cm)?​

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Answer:

2

Step-by-step explanation:

2

1.  Net income after deductions: $1,880.00 - $487.48 = $1,392.52. 2.  your total realized income for that month would be $2,574.00. 3. the total income for the second month would be $2,574.00 + $822.50 = $3,396.50.

Income refers to the money earned or received by an individual or an entity in exchange for goods or services provided or as compensation for work done. It can come from various sources such as employment, investments, rental properties, or business activities. Income is often used to pay for expenses, such as bills, food, housing, transportation, and other necessities. It is also commonly saved or invested to build wealth and achieve long-term financial goals.

1. Assuming a month has 4 weeks, your total realized income for one month would be:

40 hours/week x $11.75/hour x 4 weeks = $1,880.00

Your fixed expenses for one month would be:

Rent: $657.00

Telephone: $56.34

Groceries: $56.00 x 4 = $224.00

Clothing: $106.00

Water & Electric: $98.87

Total fixed expenses: $1,142.21

Your discretionary expenses for one month would be:

Weekly Dinner & Movie: $40.00 x 4 = $160.00

Total discretionary expenses: $160.00

Therefore, your total expenses for one month would be:

Fixed expenses + discretionary expenses = $1,142.21 + $160.00 = $1,302.21

To calculate your net income after deductions, we need to first calculate the amount of deductions:

FICA: 7.65% of $1,880.00 = $143.88

Federal tax withholding: 10.75% of $1,880.00 = $202.60

State tax withholding: 7.5% of $1,880.00 = $141.00

Total deductions: $487.48

Net income after deductions: $1,880.00 - $487.48 = $1,392.52

2. If you work 20 hours of overtime the next month and are paid 1.5 times your regular rate, your total realized income for that month would be:

(40 hours x $11.75/hour x 4 weeks) + (20 hours x $11.75/hour x 1.5 x 4 weeks) = $2,574.00

Your fixed expenses would remain the same, but your discretionary expenses may increase due to the additional income.

3. If you eliminated your discretionary expenses of $160.00 per month, you could put that amount towards savings each month. This would increase your savings by $1,920.00 per year.

To calculate the income for the first 40 hours of work, we simply multiply the hourly rate by the number of hours worked:

40 hours x $11.75/hour = $470.00

To calculate the income for the additional 20 hours of overtime at 1.5 times the regular rate, we first calculate the regular rate for the overtime hours (40 hours x $11.75/hour = $470.00), and then add 1.5 times that amount to the total:

40 hours x $11.75/hour = $470.00

20 hours x ($11.75/hour x 1.5) = $352.50

Total overtime pay = $470.00 + $352.50 = $822.50

Therefore, the total income for the second month would be $2,574.00 + $822.50 = $3,396.50.

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Water depth at 10:10 am is approximately 4.34 feet.

Given data:The low tide occurred at 5:40 am with a water depth of 1.25 feet. Six hours and 2 minutes later, the high tide occurred with a water depth of 6.82 feet.Tide is the periodic rise and fall of sea level due to the gravitational pull of the Moon and the Sun on the Earth. This motion can be modeled by sinusoidal functions.The general form of the sine function is given as f(x) = a sin bx + c, where a is the amplitude, b is the period, and c is the vertical shift.To find all components needed to write a model for this scenario since the first low tide:First, we need to calculate the amplitude and period of the sine wave, as follows:Given that the low tide occurred at 5:40 am with a water depth of 1.25 feet.Similarly, the high tide occurred with a water depth of 6.82 feet at 11:42 am, which is 6 hours and 2 minutes after the low tide. Therefore, the period of the wave is 12 hours and 4 minutes or 12.067 hours.Amplitude is given as half the difference between the maximum and minimum values of the wave, which is 2.285 feet.Writing the model for the scenario since the first low tide is given as:f(x) = 2.285 sin ((2π/12.067) x) + cSince the water depth at low tide is 1.25 feet, the vertical shift is given as 3.065 feet.f(x) = 2.285 sin ((2π/12.067) x) + 3.065Now, we need to find the water depth at 10:10 am.To do that, we substitute x = 4.5 (since 5:40 am to 10:10 am is a period of 4.5 hours) in the above equation, and we get:f(4.5) = 2.285 sin ((2π/12.067) × 4.5) + 3.065≈ 4.34 feetTherefore, the water depth at 10:10 am is approximately 4.34 feet.

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Volume of a cube is given by side × side × side. By using the equation the volume of the given figure can be determined as 28 cubic units.

In the given figure the whole figure is divided into unit cubes. A unit cube means a cube having a unit value as all its sides. To calculate the volume of such a figure, we have to determine the number of unit cubes and multiply it by the volume of the cube.

Volume of 1 unit cube = 1 cubic units

We have to count the number of unit cubes in it. From the front view, we can see that there are three rows and each having two cubes and there are four such columns.

So the number of cubes = 3× 4 = 12

In the back row there are 4×4 cubes = 16 cubes

Total number of cubes = 12 + 16 = 28

So the volume = 28 × 1 cubic unit = 28 cubic units.

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The area of the floor of the 15 ft convenience store is 225 ft² and the total cost of the floor is $450.

The tile cost is 82.00 per ft².

We have to determine the area of the floor of the 15 ft convenience store.

To calculate the cost of the new tile floor, you will need to know the area of the floor. To determine the area, you will need to measure the length and width of the store.

Once you have the measurements, you can multiply the length times the width to get the area of the floor. Then, multiply the area by $2.00 to get the total cost of the new tile floor.

The dimension of the store is 15 ft.

So the area of convenience store = (side)²

The area of convenience store = (15)²

The area of convenience store = 225 ft²

Total cost of floor = 225 × 2

Total cost of floor = 450

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The complete question is:

Kaleb's mom owns a convenience store. He is helping her replace the tile floor. The tile costs $2.00 per ft². What is the area of the floor of the 15 ft convenience store? What is the total cost of the floor?

ask your teacher. She/he might be able to help you.

Answer: From the graph, we see that Anna needs to sell 300 pre-sale tickets and 100 at-the-door tickets to reach her goal. Therefore, she needs to sell 100 at-the-door tickets to make her goal. Answer: 100.

Step-by-step explanation:

Answer:

5,040

Step-by-step explanation:

There are a total of 5,040 interesting numbers, which can be calculated using basic combinatorics. To calculate this, the first step is to count the total number of 10-digit numbers with distinct digits. Since there are 10 distinct digits (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9), there are 10! (10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) possible numbers with distinct digits. The next step is to count the number of 10-digit numbers divisible by 11,111. To do this, we need to find the number of 10-digit numbers divisible by 11. Since 11 is a prime number, there are 11 possible combinations of 10-digit numbers divisible by 11. For example, 0XXXXXXXXX, 1XXXXXXXXX, 2XXXXXXXXX, etc. So, there are 11 x 10! (110,000) possible 10-digit numbers divisible by 11. Finally, to find the number of interesting numbers, we need to subtract the number of 10-digit numbers divisible by 11,111 from the total number of 10-digit numbers with distinct digits. This gives us 10! - 11 x 10! = 5,040. Therefore, there are a total of 5,040 interesting numbers.

Answer:

To find the number of interesting numbers, we need to find the number of 10-digit numbers that are divisible by 11,111 and have distinct digits.

A number is divisible by 11,111 if and only if the alternating sum of its digits is divisible by 11. For example, 27,183,642 is divisible by 11,111 because 2 - 7 + 1 - 8 + 3 - 6 + 4 - 2 = -3, which is divisible by 11.

Since we want the number to have distinct digits, we can choose any 5 digits from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and arrange them in any order. There are 10 choices for the first digit, 9 choices for the second digit (since we can't repeat the first digit), 8 choices for the third digit, and so on, giving us a total of 10 x 9 x 8 x 7 x 6 = 30,240 possible 5-digit numbers with distinct digits.

To form a 10-digit number, we can repeat the 5 digits in any order. There are 5! = 120 ways to arrange the 5 digits, so there are 120 different 10-digit numbers that can be formed from each 5-digit number.

Therefore, the total number of interesting numbers is:

30,240 x 120 = 3,628,800

So there are 3,628,800 interesting numbers that are 10-digit numbers divisible by 11,111 with distinct digits.