A 6-foot-wide hallway is painted as shown, using equal amounts of white and black paint.
a. How long is the hallway?
b. Can this same hallway be painted with the same pattern, but using twice as much black paint as white paint? Explain.

The question pertains to exponential notation in our decimal numbering system, where each place is worth ten times the next right place. It creates a more convenient way to express very large or small numbers. The rules of multiplication in this system involve multiplying the numbers out front and adding the exponents.

The question refers to place value in a number, where we are told how much each digit is worth based on their position. Here, '3' is worth 1/10 as much and '2' is worth 10 times as much as the number 431, 621. The power (exponent) of 10 is equal to the number of places the decimal is shifted to give the digit number.

This is part of exponential notation, which is especially useful for very large and very small numbers. For instance, 1,230,000,000 can be written as 1.23 × 10⁹, and 0.00000000036 can be articulated as 3.6 × 10-¹0. It's called powers-of-ten notation because each place in our numbering system is ten times greater than the place on its right.

To compute this, we use a method where we multiply the numbers out front and add the exponents. For instance, (3 × 10⁵) × (2 × 10º) equals 6 × 10⁵. This notation not only simplifies arithmetic but is also compact and convenient.

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Answer:

The range of possible values for the length of the third side of the triangle is [tex]\( 6 < x < 26 \)[/tex], meaning that the third side must be greater than 6 cm and less than 26 cm.

Explanation:

To determine the range of possible values for the third side of the triangle, we can use the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Let's denote the length of the third side as [tex]\( x \)[/tex]. According to the triangle inequality theorem:

1. The sum of the lengths of the two given sides (10 cm and 16 cm) must be greater than the length of the third side:

[tex]\[ 10 + 16 > x \][/tex]

[tex]\[ 26 > x \][/tex]

2. The length of the third side must be greater than the difference of the lengths of the two given sides:

[tex]\[ 16 - 10 < x \][/tex]

[tex]\[ 6 < x \][/tex]

Combining both inequalities, we have:

[tex]\[ 6 < x < 26 \][/tex]

Answer:

x = [tex]\frac{819}{598}[/tex].

Step-by-step explanation:

Given  :  4.1363636…

To find : Convert it as fraction .

Solution : We have given   4.1363636…

Let x =   4.1363636…

On multiplying both sides by 100

100 x = 100 *   4.1363636…

100 x = 413 .63636...

We can convert it in term of x

100 x = 409 .5 + 4.1363636...

100 x = 409.5 + x

On subtracting both sides by x

100 x -x = 409.5

99 x = 409. 5

On dividing both sides by 99

x = [tex]\frac{409.5}{99}[/tex].

x =  [tex]\frac{4095}{990}[/tex].

On dividing both sides by 5

x = [tex]\frac{819}{598}[/tex].

Therefore, x = [tex]\frac{819}{598}[/tex].

Ivan needs to take out the trash at least 5 times to earn enough money to buy a double-scoop ice cream cone.

To buy a double-scoop ice cream cone, Ivan needs to earn enough money to cover the cost of the double-scoop cone, which is 75 cents for the first scoop and an additional 50 cents for the second scoop. Since Ivan earns 25 cents each time he takes out the trash, we can calculate how many times he needs to take out the trash to earn enough money.

Let's assume Ivan needs to take out the trash 'x' number of times. Each time he takes out the trash, he earns 25 cents, so the total amount he earns will be 25 cents multiplied by 'x'.

Setting up an equation, 25x >= 75 + 50. Simplifying, we have 25x >= 125. Dividing both sides of the inequality by 25, we get x >= 5. Therefore, Ivan needs to take out the trash at least 5 times to earn enough money to buy a double-scoop ice cream cone.

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Answer:

its 3 and the dollar one is non-portianal and the table to the left the second row is porpianal and the table to the right

Step-by-step explanation:

Final answer:

To find the costs, we set up two equations based on the given data. Using elimination, we first determine the cost of 1 kg of carrots and then the cost of 1 kg of potatoes. The final costs are $2 per kg for potatoes and $2.40 per kg for carrots.

Explanation:

The student's question involves solving a system of linear equations to determine the cost of potatoes and carrots per kilogram. We can set up two equations from the given information:

8 kg of potatoes (8P) + 5 kg of carrots (5C) = $28

2 kg of potatoes (2P) + 3 kg of carrots (3C) = $11.20

These equations can be solved using either substitution or elimination methods to find the price per kilogram (P and C) for potatoes and carrots, respectively.

Let's use the elimination method:

Multiply the second equation by 4 to align the potato coefficient with the first equation:

8 kg of potatoes (8P) + 5 kg of carrots (5C) = $28

8 kg of potatoes (8P) + 12 kg of carrots (12C) = $44.80

Now subtract the first equation from the modified second equation:

(8P + 12C) - (8P + 5C) = $44.80 - $28

7C = $16.80

Divide by 7 to find the price per kilogram of carrots:

C = $2.40

Substitute the value of C into the first equation and solve for P:

8P + 5($2.40) = $28

8P + $12 = $28

8P = $16

P = $2

The cost of 1 kg of potatoes is $2 and the cost of 1 kg of carrots is $2.40.

Answer: The rate of both are 0.23 and 0.796 respectively.

Step-by-step explanation:

Since we have given that

Rate of 2.50/10.50

We need to find the rate which is given by

[tex]\dfrac{2.50}{10.50}\\\\=0.23[/tex]

Similarly,

Rate of 1.99/2.50

We need to find the rate which is given by

[tex]\dfrac{1.99}{2.50}=0.796[/tex]

Hence, the rate of both are 0.23 and 0.796 respectively.

To find the width and length of a rectangle given an area of 36 square meters and a perimeter of 30 meters, a system of equations can be set up and solved, showing the rectangle could have dimensions 9m x 4m or 4m x 9m.

To find the width and length of a rectangle with a known area and perimeter, we can use the following formulas: Area = length x width and Perimeter = 2(length + width). We know the area is 36 square meters and the perimeter is 30 meters. Let's denote the length as L and the width as W. Therefore, our system of equations will be:

L x W = 36

2(L + W) = 30

Dividing the perimeter equation by 2 gives us L + W = 15. We can express the width in terms of the length using this equation: W = 15 - L. Now we can substitute this expression for W into the area equation: L x (15 - L) = 36. Expanding this gives us a quadratic equation: [tex]L^2[/tex] - 15L + 36 = 0. We can solve this equation for L to find possible values for the length.

To find the length L, we look for factors of 36 that add up to 15. It turns out that 9 and 4 are such factors, so our lengths are 9 meters and 4 meters. We then use the expressions for W to find that W is 4 meters if L is 9 meters, and W is 9 meters if L is 4 meters. Hence, the rectangle could have dimensions 9m x 4m or 4m x 9m.

Each number provided is a rational number, as they can all be expressed as a ratio of two integers. Therefore, they all qualify as rational numbers.

In Mathematics, a rational number is a number that can be expressed as a ratio of two integers. An integer is a whole number that can be positive, negative, or zero. Let's express each number given as a ratio of two integers:

As you can see, each of these numbers can be written as a ratio of two integers, thus they are all rational numbers.

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Each given number (27, 0.075, 4 2/3, -9, 0.43, and -18) is shown to be a rational number by expressing it as a ratio of two integers, such as 27/1, 3/40, 14/3, -9/1, 43/100, and -18/1, respectively.

To show that a number is a rational number, we must express it as a ratio of two integers.