Simplify completely 12x+36/x^2-4x-21 and find the restrictions on the variable

Answer:

Answer is: A

Step-by-step explanation:

Because it is.

Answer:

A. $17.00

Step-by-step explanation:

$5.10 for 6 quarts.

Unit cost: ($5.10)/(6 qt) = $0.85/qt

Price for 5 gallons:

1 gallon = 4 quarts

5 gallons = 5 * 4 quarts = 20 quarts

Price for 5 gallons = unit price * number of quarts

Price for 5 gallons = $0.85/qt * 20 qt = $17.00

Answer: A. $17.00

Answer:

This is a direct variation where the constant of variation is 5/9

Step-by-step explanation:

The equation for direct variation is y = kx

5x-9y =0

Lets try to get this equation in the above form

Add 9y to each side

5x-9y +9y = 9y

5x = 9y

Divide each side by 9

5/9 x = 9y/9

5/9x = y

This is in the form y = k x where k =5/9

Answer:

7/10 x 2/5 = 14/50 = 7/25

Step-by-step explanation:

mutipily 7 and 2 that gives you 14 and mutilpily 10 and 5 that gives you 50 = 14/50. In simplest form the answer is 7/25.

Answer:

7/10 x 2/5 = 14/50 = 7/25

Step-by-step explanation:

mutipily 7 and 2 that gives you 14 and mutilpily 10 and 5 that gives you 50 = 14/50. In simplest form the answer is 7/25.

Answer:

0.49 m²

Step-by-step explanation:

The area (A) is calculated using A = πr² ( r is the radius )

A = external area - internal area

external radius = [tex]\frac{1.2}{2}[/tex] = 0.6

internal radius = [tex]\frac{0.9}{2}[/tex] = 0.45

A = π(0.6² - 0.45²)

   = π(0.36 - 0.2025) = 0.1575π ≈ 0.49 m²

Final answer:

The area of the washers is 0.49m².

Explanation:

To find the area of the washers, we need to find the areas of the outer and inner circles and then subtract the smaller area from the larger one. The area of a circle is given by the formula A = πr², where r is the radius of the circle. In this case, the external diameter is 1.2m, so the radius is half of that, or 0.6m. The area of the outer circle is A₁ = π(0.6)² = 1.13m². Similarly, the radius of the inner circle is 0.45m, and the area of the inner circle is A₂ = π(0.45)² = 0.64m². Finally, the area of the washer is the difference between the two areas: A = A₁ - A₂ = 1.13m² - 0.64m² = 0.49m².

Answer:

2/5 * pi radians = 72 degrees

Step-by-step explanation:

To convert radians to degrees, we multiply by 180/pi

2 * pi/5 * 180/pi =

The pi/pi cancels leaving

2/5 * 180

360/5

72

2/5 * pi radians = 72 degrees

Answer:

70, 105, 140

Step-by-step explanation:

Answer:

I Think It Is 0.12*2

Because I See Both Of The Shadings Colored Differently And Each One Has 12 That Is The Reason For Me Thinking This.

Answer:

2700 cookies.

Step-by-step explanation:

A dozen is a grouping of twelve, that is a dozen = 12.

If there are 225 dozen cookies in the bakery, then there are

[tex]225\cdot 12=2700[/tex] cookies in the bakery.

there are 2700 cookies in total

Since there are 225 dozen cookies in the bakery, we need to determine the total number of cookies.

To convert dozen to individual units, we multiply the number of dozens ( 225 )  by 12, as there are 12 cookies in each dozen :

225 dozen * 12 cookies / dozen = 2700 cookies

Therefore, there are 2700 cookies in total. By multiplying the number of dozens by 12, we account for the fact that each dozen contains 12 cookies. This calculation allows us to determine the precise quantity of cookies in the bakery.

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the cheap answer is, we divide 500 by (2+3+5+10) and then distribute accordingly.

[tex]\bf \cfrac{500}{2+3+5+10}\implies \cfrac{500}{20}\implies \cfrac{25}{1}\implies 25 \\\\\\ \stackrel{2\cdot 25}{2}~~:~~\stackrel{3\cdot 25}{3}~~:~~\stackrel{5\cdot 25}{5}~~:~~\stackrel{10\cdot 25}{10}\qquad \implies \qquad 50~~:~~75~~:~~125~~:~~250[/tex]

Answer:

[tex]8.05 * 10^{5}[/tex]

Step-by-step explanation:

Thinking process:

the population of Park Crest Middle School = 400 children

The population in the state = 806, 240 children

The difference:

[tex]806240 - 400\\= 805840[/tex]

Expressing the difference in standard form gives:

8.05 × 10⁵

Brian rented the book for 12 days.

Brian rented a book with the condition that the shop charges $2 for the first day and $3 from the second day onwards. He paid a total of $35 upon returning the book. To determine how many days he kept the book, we must calculate the cost for each day and subtract the initial $2 from the total amount paid.

Steps to Calculate the Number of Days:

Therefore, Brian kept the book with him for 12 days.

Answer:

is your answer $3250

Step-by-step explanation:

so we dont know our I (interest) but :

P (principal) = 10000

R (rate) = 6.5%  we'll have to turn it into the decimal which is 0.065

T (time) = 5 years

and our basic formula of finding I is I = PxRxT   so :

I = 10000 x 0.065 x 5

I = 3250

Answer:

8 daisies

Step-by-step explanation:

You have to multiply the amount of vases by daisies.

Answer:

Givens

We know that Morita arranges 4 cases, each of them has 10 flowers.

So, if there are 2/10 daisies per case, that means each can contains 2 daisies.

But, she did 4 arrangements. There are 2x4 = 8 daisies in total.

Now, last weekend, Morita used 10 daisies to make flower arrangements, and we know she tend to use 2/10 of the flowers as daisies, that means this fraction represents daisies,

[tex]\frac{2}{10}x=10\\ x=\frac{100}{2}\\ x=50[/tex]

Therefore, there are 50 flowers in total.

Answer:

180 degrees in radians is π

Step-by-step explanation:

0° = 0 radians

90°= π/2

180 °= π

270°= 3π/2

360° = 2π

and so on

Answer:

We have the following given

mileage = 70 miles per gallon

distance = 60 km

Required: amount of fuel in gallons

First, we must convert the distance to miles, so

60 km (1 mi/1.6 km) = 37.5 mi

So, the amount of fuel is

37.5 mi / (70 mi/gal) = 0.54 gallon

Answer:

B

Step-by-step explanation:

70 miles * [1.6 km/mile] = 112 km

Notice when you multiply by km/mile by miles the miles cancel out.

1 gallon will get 112 km

x gallon will get 60 km

Answer:

B is the right one i bet u 100 point s

Answer:

B is the correct answer.

The brackets ║ represent absolute value. Absolute value is ALWAYS positive. I hope this helps you!

-Mikayla

Answer:

The unit price is $2.33 per gallon

Step-by-step explanation:

To find the unit price, we take the price and divide by the amount

$6.99 / 3 gallons

$2.33 per gallon

The unit price is $2.33 per gallon

Answer:

Volume of tray =[tex]4x^{3} -60x^{2} +216x[/tex]

Step-by-step explanation:

The Dimensions of rectangular cardboard is given by 12 inches by 18 inches

Length of the rectangular cardboard = 18 inches

Width of the rectangular cardboard = 12 inches

if the square of sides x inches is cut from each corner of the rectangular cardboard and folded to make a tray, then we have

Length of the tray = length of cardboard - 2 (side of the square )

                               = [tex]18-2x[/tex]

Width of the tray = width of cardboard - 2( side of square)

                            = [tex]12-2x[/tex]

height of tray = sides of square

                       = x

volume of tray = Length × width × height

Volume of tray = [tex]x(18-2x)(12-2x)[/tex]

first we multiply (12-2x) and (18-2x)

Volume of tray =[tex]x(18(12-2x)-2x(12-2x))\\[/tex]

                         =[tex]x(216-36x-24x+4x^{2})[/tex]

                         =[tex]x(4x^{2}-60x+ 216)[/tex]

                         =[tex]4x^{3} -60x^{2} +216x[/tex]

Hence the volume of tray =[tex]4x^{3} -60x^{2} +216x[/tex]

The volume of the Tray is given by the equation [tex]4x^3-60x^2+216x[/tex] and this can be determined by using the given data.

Given :

If the square is cut from each corner of the cardboard and folded to make a tray, then according to the given data:

Tray length = 18 - 2x

Tray width = 12 - 2x

Tray height = x

Now, the volume of the tray is given by the expression:

The volume of Tray = [tex]x(12-2x)(18-2x)[/tex]

Simplify, the above expression.

[tex]\rm Volume = (12x - 2x^2)(18-2x)[/tex]

[tex]\rm Volume = 216x -24x^2-36x^2+4x^3[/tex]

[tex]\rm Volume = 4x^3-60x^2+216x[/tex]

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Answer:

g⁻¹(x) = ±√(x + 25)

Step-by-step explanation:

g(x) = x² - 25     Rename g(x) as y

  y = x² - 25

=====

Solve for x  

        y = x² - 25          Add 25 to each side

y + 25 = x²                  Take the square root of each side

       x = ±√(y + 25)     Switch x and y

       y = ±√(x + 25)     Rename y as g inverse

g⁻¹(x) = ±√(x + 25)

=====

See the graphs of g(x) and g⁻¹(x) below.

g(x) is the red line. g⁻¹(x) is the purple line.

Each graph reflects the other about the dashed line representing the function y = x.

Answer:

All explanations are correct

Step-by-step explanation:

In triangle ABC, CD is the perpendicular bisector of AB, thus using ΔADC and ΔBDC,

AC=BC ( Since CD is perpendicular to AB, therefore C is equidistant from both A and B)

∠ADC=∠BDC=90°(CD perpendicular AB)

CD=CD( Reflexive property)

therefore, by SAS rule of congruency,

ΔADC≅ ΔBDC,

Also, In the same triangles, AC=BC ( Since CD is perpendicular to AB, therefore C is equidistant from both A and B)

CD=CD( Reflexive property)

AD=BD (D is the midpoint and divides AB into two equal halves)

Thus, by SSS rule of congruency,

ΔADC≅ ΔBDC,

Thus, all the three explanations are correct.

In the given case, we can conclude that All explanations are correct

In triangle ABC, let CD be the perpendicular bisector of AB. We can use the properties of triangles to demonstrate that ΔADC is congruent to ΔBDC.

AC=BC: This holds true because CD is perpendicular to AB, making point C equidistant from both A and B.

∠ADC=∠BDC=90°: Since CD is perpendicular to AB, both angles ADC and BDC are right angles.

CD=CD: This is a reflexive property, stating that any line segment is equal to itself.

By applying the SAS (Side-Angle-Side) rule of congruency using the above properties, we can conclude that ΔADC is congruent to ΔBDC.

Furthermore, we can establish that in these congruent triangles:

AC=BC (since C is equidistant from A and B due to CD being perpendicular to AB).

CD=CD (reflexive property).

AD=BD (as D is the midpoint of AB).

Hence, using the SSS (Side-Side-Side) rule of congruency, we can also conclude that ΔADC is congruent to ΔBDC.

Therefore, all three explanations correctly demonstrate the congruence of these triangles.

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Answer: C < B < A

Step-by-step explanation:

First, let's evaluate the growth rate of A from [1, 3]

[tex]f(x) = 200\bigg(\dfrac{3}{2}\bigg)^1[/tex]

    = 300

coordinate is (1, 300)

[tex]f(x) = 200\bigg(\dfrac{3}{2}\bigg)^3[/tex]

    = 675

coordinate is (3, 675)

Average growth rate (AGR) is: [tex]\dfrac{675-300}{3-1} = \dfrac{375}{2} = 187.5[/tex]

AGR (A) = 187.5

*************************************************************************************

Next, let's evaluate the growth rate of B from [1, 3]

The coordinates are already provided as (1, 120) and (3, 480)

Average growth rate (AGR) is: [tex]\dfrac{480-120}{3-1} = \dfrac{360}{2} = 180[/tex]

AGR (B) = 180

*************************************************************************************

Lastly, let's evaluate the growth rate of C from [1, 3]

f(x) = 600(1.2)ˣ

f(x) = 600(1.2)¹

     = 720

coordinate is (1, 720)

f(x) = 600(1.2)³

     = 1036.8

coordinate is (3, 1036.8)

Average growth rate (AGR) is: [tex]\dfrac{1036.8-720}{3-1} = \dfrac{316.8}{2} = 158.4[/tex]

AGR (C) = 158.4

Answer:

2. m = b³ (= 216)

3. logp(x) = -4

Step-by-step explanation:

2. The given equation can be written using the change of base formula as ...

... log(m)/log(b) + 9·log(b)/log(m) = 6

If we define x = log(m)/log(b), then this becomes ...

... x + 9/x = 6

Subtracting 6 and multiplying by x gives ...

... x² -6x +9 = 0

... (x -3)² = 0 . . . . . factored

... x = 3 . . . . . . . . . value of x that makes it true

Remembering that x = log(m)/log(b), this means

... 3 = log(m)/log(b)

... 3·log(b) = log(m) . . . . . multiply by the denominator; next, take the antilog

... m = b³ . . . . . . the expression you're looking for

___

3. Substituting the given expression for y, the equation becomes ...

... logp(x^2·(p^5)^3) = 7

... logp(x^2) + logp(p^15) = 7 . . . . . use the rule for log of a product

... 2logp(x) + 15 = 7 . . . . . . . . . . . . . use the definition of a logarithm

... 2logp(x) = -8 . . . . . . . . . . . . . . . . subtract 15

... logp(x) = -4 . . . . . . divide by 2

Answer:

For #3: [tex]\log_px=-4[/tex]

Step-by-step explanation:

I'm a little rusty on my logarithm rules for #2, but here's an explanation of #3.

In a sense, we can think of operations like subtraction and division as different ways of representing addition and multiplication. For instance, the same relationship described by the equation 2 + 3 = 5 is captured in the equation 5 - 3 = 2, and 5 × 2 = 10 can be restated as 10 ÷ 2 = 5 without any loss of meaning.

Logarithms do the same thing for exponents: the expression [tex]2^3=8[/tex] can be expressed in logarithms as [tex]\log_28=3[/tex]. Put another way, logarithms are a sort of way of pulling an exponent out onto its own side of the equals sign.

Our problem gives us two facts to start: that [tex]log_p(x^2y^3)=7[/tex] and [tex]p^5=y[/tex]. With that, we're expected to find the value of [tex]\log_px[/tex]. [tex]p^5=y[/tex] stands out as the odd-equation-out here; it's the only one not in terms of logarithms. We can fix that by rewriting it as the equivalent statement [tex]log_py=5[/tex]. Now, let's unpack that first logarithm.

For a refresher, let's talk about some of the rules logarithms follow and why they follow them:

Product Rule: [tex]\log_b(MN)=\log_bM+\log_bN[/tex]

The product rule turns multiplication in the argument (parentheses) of a logarithm into addition. For a proof of this, consider two numbers [tex]M=b^x[/tex] and [tex]N=b^y[/tex]. We could rewrite these two equations with logarithms as [tex]\log_bM=x[/tex] and [tex]\log_bN=y[/tex]. With those in mind, we could say the following:

And we have our proof.

Exponent Rule: [tex]\log_b(M^n)=n\log_bM[/tex]

Since exponents can be thought of as abbreviations for repeated multiplication, we can rewrite [tex]\log_b(M^n)[/tex] as [tex]\log_b(M\times M\cdots \times M)[/tex], where M is being multiplied by itself n times. From there, we can use the product rule to rewrite our logarithm as the sum [tex]\log_bM+\log_bM+\cdots+\log_bM[/tex], and since we have the term [tex]\log_bM[/tex] added n times, we can rewrite is as [tex]n\log_bM[/tex], proving the rule.

With those rules in hand, we're ready to solve the problem. Looking at the equation [tex]\log_p(x^2y^3)=7[/tex], we can use the product rule to split the logarithm into the sum [tex]\log_p(x^2)+\log_p(y^3)=7[/tex], and then use the product rule to turn the exponents in each logarithm's argument into coefficients, giving the equation [tex]2\log_px+3\log_py=7[/tex].

Remember how earlier we rewrote [tex]p^5=y[/tex] as [tex]log_py=5[/tex]? We can now use that fact to substitute 5 in for [tex]log_py[/tex], giving us

[tex]2\log_px+3(5)=7[/tex]

From here, we can simply solve the equation for [tex]\log_px[/tex]:

[tex]2\log_px+15=7\\2\log_px=-8\\\\\log_px=-4[/tex]

Answer:

The correct option is 1. The image of (2,4) is (4,-8).

Step-by-step explanation:

The given rule is

[tex]R_x{\circ}D_{o,2}:(2,4)[/tex]

The transformations perform from right to left. [tex]D_{o,2}[/tex] means dilation with scale factor 2 and center of dilation is origin.

The given rule defines the dilation with scale factor 2 and center of dilation is origin followed by reflection across x-axis.

If a figure dilated by scale factor k and the center of dilation is origin, then

[tex](x,y)\rightarrow (kx,ky)[/tex]

The scale factor is 2,

[tex](x,y)\rightarrow (2x,2y)[/tex]

[tex](2,4)\rightarrow (4,8)[/tex]

If a figure reflected across x-axis, then x-coordinate remains the same but the sign of y-coordinate is changed.

[tex](x,y)\rightarrow (x,-y)[/tex]

[tex](4,8)\rightarrow (4,-8)[/tex]

Therefore image of (2,4) is (4,-8) and option 1 is correct.

Answer: An x and y value, parenthesis around the values, and a comma in between them.

Examples of ordered pairs: (2, 5), (1872, 1683), (-3, 7), (7/8, -10/2)

Answer:

15%

Step-by-step explanation:

Percent means out of 100.  Lets get the denominator out of 100

3/20 * 5/5  = 15/100

This is 15%

Final answer:

To convert the fraction 3/20 to a percentage, first convert it to a decimal by dividing 3 by 20. Then, multiply the decimal by 100 to get the percentage.

Explanation:

To express 3/20 as a percentage, you can multiply the fraction by 100. The calculation is (3/20) * 100 = 15%. Therefore, 3/20 as a percentage is 15%. This means that 3 out of every 20 parts represent 15% of the whole. In percentage terms, it provides a convenient way to compare the fraction to 100, making it easier to understand its relative size or portion in relation to the total. In this case, 15% indicates the proportion of the whole that is represented by the fraction 3/20.

In combinatorics, we use the combination formula to calculate the number of ways Mrs. Wong can choose 2 books out of 14. The result is 91 ways.

The subject of this question is combinatorics, a topic in mathematics dealing with combinations of objects belonging to a finite set in accordance with certain constraints, such as those specified in this question. In this case, Mrs. Wong has 14 special objects (books) and wants to choose 2 out of them.

We use the combination formula in this scenario. The combination formula is given by C(n, r) = n! / [(n-r)!*r!], where n represents the total number of objects, r is the number of objects to choose, and '!' denotes factorial.

Substituting n as 14 and r as 2 into the formula, we get: C(14,2) = 14! / [(14-2)!*2!] = 91.

Therefore, Mrs. Wong can choose 2 books out of 14 in 91 different ways.

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The question is about combinations in mathematics. Using the formula for combinations, we find that Mrs. Wong can choose 2 books out of 14 in 91 different ways.

This is a problem of combinations in mathematics. Mrs. Wong can choose two out of 14 unread books in a certain number of ways, and we're tasked to find that number.

Considering that order of selection does not matter, we can use the formula for combination: nCr = n! / r!(n-r)!. Here, 'n' is the total number of items, 'r' is the items to be chosen.

So the combinations she can make, denoted as 14C2, can be calculated like this:

14C2 = 14! / 2!(14-2)! = (14*13) / (2*1) = 91 ways

Therefore, Mrs Wong can choose 2 books out of 14 in 91 ways.

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Answer:

4(3+x)

Step-by-step explanation:

If any thing is in the brackets you need to do it first

so if you do 3 + x × 4 it will give yo 4x + 3 or 3 +4x

The given expression 4 (x + 3) is equivalent to the expression 4x + 12.

The equivalent operations are those that have various forms but have the same outcome.

The expression is given below.

→ 4 (x + 3)

The expression can be written as

→ 4x + 12

The given expression 4 (x + 3) is equivalent to the expression 4x + 12.

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