Use exponents to write the number 64 three different ways

Final answer:

Cynthia should calculate the dimensions of the rug by solving the quadratic equation derived from the area she can afford and the dimensions of her room, taking into consideration the width of the uniform strip she wants to leave around the rug.After finding [tex]\( w \), the dimensions of the rug will be \( (20 - 2w) \) ft by \( (31 - 2w) \) ft.[/tex]

Explanation:

To find the dimensions of the rug, we need to subtract the width of the strip left around the rug from the width and length of the room.

The area of the room is given by [tex]\( A_{\text{room}} = W \times L \).[/tex]

Cynthia wants to leave a uniform strip of floor around the rug. This means that the dimensions of the rug will be reduced by twice the width of the strip. So, the dimensions of the rug will be [tex]\( (W - 2w) \times (L - 2w) \).[/tex]

Given that she can afford to buy 350 square feet of carpeting, the area of the rug must be equal to 350 square feet.

So, we have the equation:

[tex]\[ (W - 2w) \times (L - 2w) = 350 \][/tex]

Substituting the given values:

[tex]\[ (20 - 2w) \times (31 - 2w) = 350 \][/tex]

Expanding the equation:

[tex]\[ 620 - 82w - 40w + 4w^2 = 350 \][/tex]

Simplifying:

[tex]\[ 4w^2 - 122w + 620 = 350 \][/tex]

[tex]\[ 4w^2 - 122w + 270 = 0 \][/tex]

Now, we can solve this quadratic equation for w . Once we find the value of w , we can calculate the dimensions of the rug.

After finding [tex]\( w \), the dimensions of the rug will be \( (20 - 2w) \) ft by \( (31 - 2w) \) ft.[/tex]

To compare and order rational and irrational numbers, convert rational numbers to decimals and compare them directly, and for irrational numbers, compare their approximate decimal values.

Comparing and ordering rational and irrational numbers involves understanding their magnitude on the number line.

Let's compare 3 rational and irrational numbers: [tex]\( \frac{5}{3} \), \( \sqrt{2} \), and \( -\frac{4}{7} \).[/tex]

1. [tex]\( \frac{5}{3} \)[/tex] is a rational number, as it can be expressed as a fraction.

2. [tex]\( \sqrt{2} \)[/tex] is an irrational number, as it cannot be expressed as a fraction.

3. [tex]\( -\frac{4}{7} \)[/tex] is a rational number, a negative fraction.

To compare them:

1. [tex]\( \frac{5}{3} = 1.67 \)[/tex]

2. [tex]\( \sqrt{2} = 1.414 \)[/tex]

3. [tex]\( -\frac{4}{7} = -0.57 \)[/tex]

Ordering them:

[tex]\( -\frac{4}{7} < \sqrt{2} < \frac{5}{3} \)[/tex]

Calculation:

1. [tex]\( \frac{5}{3} = 1.67 \)[/tex]

2. [tex]\( \sqrt{2} = 1.414 \)[/tex]

3. [tex]\( -\frac{4}{7} = -0.57 \)[/tex]

Order: [tex]\( -\frac{4}{7} < \sqrt{2} < \frac{5}{3} \)[/tex]

So, [tex]\( -\frac{4}{7} \)[/tex] is the smallest, followed by [tex]\( \sqrt{2} \)[/tex], and [tex]\( \frac{5}{3} \)[/tex] is the largest. This is because irrational numbers lie between rational numbers and have a greater magnitude.

Complete Question:

How can you compare and order rational and irrational numbers ?

Answer: Hello mate!

We want to distribute 24 muffins, 18 breakfast bars and 12 bottles of juice evenly into baskets, and we want to know the maximum amount of baskets that makes this possible.

This means that we are looking for the maximal common factor between 12, 18 and 24.

To find it, you can took the smallest number in the set, in this case, 12, and look at his divisors, then see if these divisors also divide the other numbers in the set.

Let's do it:

The biggest divisor of 12 is 6, because of 12/6 = 2.

Now 18/6 = 3, so 6 is also a divisor of 18.

24/6 = 4, then 6 is also a divisor of 24.

Then the maximal common factor of the numbers 12, 18 and 24 is 6.

This means that the maximum number of baskets that have an even distribution of the things is 6, where each basket has 2 bottles of juice, 3 breakfast bars, and 4 muffins.

Let us say that:

T = numbers from 2-9 
Z = numbers from 0-9 
X = all except 1 (0, 2-9)

If a number begins with 1: 
1 _ _ 
Then we can have 
1 1 Z (10 numbers) 
1 X 1 (9 numbers) 
So a total of 19. 

Now if a number starts with 2-9 
T _ _ 
The second digit must be 1 

T 1 _
but the third can be anything so
T 1 Z = 8 * 1 * 10 = 90 numbers 

Total = 99

The simplified form of the inequality expression is −4 ≤ F ≤ 5

Using the inequality expression given:

multiply through by 9

−180 ≤ 5 (F−32) ≤ -135

−180 ≤ 5F - 160 ≤ -135

Add 160 to all sides of the expression

−20 ≤ 5F ≤ 25

divide through by 5 to isolate F

−4 ≤ F ≤ 5

The simplified form of the inequality expression is −4 ≤ F ≤ 5

Complete Question:

Simplify the inequality: −20 ≤ 5/9(F−32) ≤ -15

Answer:

Find out the  how much is his overall budget .

To prove

Let us assume that the overall budget of Mr.Hollins be x.

As given

Mr. Hollins determined that he gives away $800 each month.

If he gives away 16% of his budget.

16% is written in the decimal form.

[tex]= \frac{16}{100}[/tex]

= 0.16

Than the equation becomes

x × 0.16 = 800

[tex]x = \frac{800}{0.16}[/tex]

x = $5000

Therefore the overall budget of Mr.Hollins be $5000 .

Mrs. Sanchez earns a total of $11,314 in commissions and bonus for the year, calculated by adding her annual net commission of $10,764 to her hiring bonus of $550.

How much Mrs. Sanchez earns in commissions and bonus for the year:

1: Calculate her weekly net commission.

She earns $275 per week in commission.

However, $68 is deducted for taxes and benefits, leaving her with:

Weekly net commission = $275 - $68 = $207

2: Calculate her annual net commission.

Assuming she works 52 weeks a year (standard for full-time employment):

Annual net commission = $207/week * 52 weeks/year = $10,764

3: Add the hiring bonus to her net commission.

Her total earnings from commissions and bonus are:

Total earnings = $10,764 + $550 = $11,314

Therefore, Mrs. Sanchez earns $11,314 in commissions and bonus for the year.