Is 4n+12 and 4(n+3) equivalent

Robin can clean 108 rooms in 9 days, given she cleans 72 rooms in 6 days; this is determined by calculating her daily cleaning rate and extending it over the 9-day period.

The student asks how many rooms Robin can clean in 9 days if she can clean 72 rooms in 6 days. To solve this problem, we first need to determine the rate at which Robin cleans rooms per day, which is done by dividing the total number of rooms by the total number of days. Once we have the daily rate, we can then multiply this rate by 9 to find out how many rooms can be cleaned in 9 days.

First, we calculate the daily rate:

Rate = Total Rooms / Total Days

Rate = 72 rooms / 6 days

Rate = 12 rooms/day.

Now, we multiply the daily rate by 9 days:

Rooms in 9 days = Rate times Number of Days

Rooms in 9 days = 12 rooms/day times 9 days

Rooms in 9 days = 108 rooms.

Therefore, Robin can clean 108 rooms in 9 days.

The amount of wax needed to create one set of candles is 84.75 cubic inches.

A cylinder is a 3-D figure that has a radius and a height.

The volume of a cylinder is given as πr²h.

Example:

The volume of a cup with a height of 5 cm and a radius of 2 cm is

Volume.

= 3.14 x 2 x 2 x 5

= 62.8 cm³

We have,

Let's calculate the volume of each candle and then add them together to find the total wax needed for one set of candles.

For the smallest candle, the radius is 0.5 inches and the height is 3 inches, so the volume is:

V1 = π x (0.5)² x (3) = 0.75π cubic inches

For the medium-sized candle, the radius is twice as big, or 1 inch, and the height is twice as big, or 6 inches. So the volume is:

V2 = π x (1)² x (6) = 6π cubic inches

For the largest candle, the radius is three times as big, or 1.5 inches, and the height is three times as big, or 9 inches.

V3 = π x (1.5)² x (9) = 20.25π cubic inches

To find the total wax needed for one set of candles, we add up the volumes:

V (total) = V1 + V2 + V3

= 0.75π + 6π + 20.25π

= 27π cubic inches

= 27 x 3.14

= 84.78 cubic inches

Therefore,

The amount of wax needed to create one set of candles is 84.75 cubic inches.

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Answer:

The amount invested at 4% was $3,000 and the amount invested at 7% was $8,000

Step-by-step explanation:

we know that

The simple interest formula is equal to

[tex]I=P(rt)[/tex]

where

I is the Final Interest Value

P is the Principal amount of money to be invested

r is the rate of interest  

t is Number of Time Periods

in this problem we have

[tex]t=1\ year\\ P_1=x\\P_2=11,000-x\\I=\$680\\r_1=4\%=4/100=0.04\\r_2=7\%=7/100=0.07[/tex]

substitute in the formula above

[tex]I=P_1(r_1t)+P_2(r_2t)[/tex]

[tex]680=0.04x+(11,000-x)0.07[/tex]      

solve for x

[tex]680=0.04x+770-0.07x[/tex]

[tex]0.07x-0.04x=770-680[/tex]

[tex]0.03x=90[/tex]

[tex]x=\$3,000[/tex]

so

[tex]\$11,000-x=\$11,000-\$3,000=\$8,000[/tex]

therefore

The amount invested at 4% was $3,000 and the amount invested at 7% was $8,000

Answer:

x^2-5x+4

Step-by-step explanation:

(x-3)^2+(x-3)-2

x^2-3x-3x+9+x-3-2

x^2-6x+9+x-5

x^2-5x+9-5

x^2-5x+4

Answer:

40%

Step-by-step explanation:

Percentage Passed = 36 ÷ 90 × 100 = 40%

Answer: 8 ft and 4 in, 8 1/3 ft

Step-by-step explanation:

100/12=50/6=25/3=8.3333333...

1/3 of a foot is 1/3x12 which is 12/3 which is 4 in.

Final answer:

100 inches is equal to 8 feet and 4 inches or 8.33 feet.

Explanation:

To solve this problem, you can convert 100 inches to feet using the unit equivalence:

1 foot = 12 inches

Multiply the number of inches by the unit equivalence:

100 inches ÷ 12 inches per foot = 8 feet and 4 inches

Answer:

The length of prism is  9 centimetres.

Step-by-step explanation:

Given:

The volume of a rectangular prism is 72 cubic centimeters.

The prism is 2 centimeters wide and 4 centimeters high.

Now, to find the length of prism:

Let the length be [tex]l[/tex].

Width = 2 centimeters.

Height = 4 centimeters.

Volume = 72 cubic centimeters.

So, putting the formula to get the length of prism:

[tex]Volume = length\times width\times height[/tex]

[tex]72=l\times 2\times 4[/tex]

[tex]72=l\times 8[/tex]

Dividing both sides by 8 we get:

[tex]9=l[/tex]

Length = 9 centimetres.

Therefore, the length of prism is  9 centimetres.

Answer:

4x+12

Step-by-step explanation:

An expression is a mathematical sentence that doesn't end in an equal sign. This question is asking for you to find four times as much as 12, but doesn't explain what number or phrase you need to find it for. We would replace the unknown number with the variable X.

I may be confusing this with another subject, so sorry if I am! Hope this helps!

The expression  which represents 4 times as much as 12 is,  [tex]4x=12[/tex]

Let us consider that number is x.

Since, 4 times of number is equal to 12.

So that,       [tex]4*x=12\\\\4x=12[/tex]

Hence, the expression  which represents 4 times as much as 12 is,  [tex]4x=12[/tex]

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Answer:

[tex]L=\frac{15}{16}\ yd[/tex]

Step-by-step explanation:

The correct question is

Devon needs a wood board with an area of 5/16 square yard to complete a project. If the wood board is 1/3 yard wide, how long must the board be?

we know that

The area of a rectangle is equal to

[tex]A=LW[/tex]

where

L is the length

W is the width

we have

[tex]A=\frac{5}{16}\ yd^2\\\\W=\frac{1}{3}\ yd[/tex]

substitute

[tex]\frac{5}{16}=L(\frac{1}{3})[/tex]

solve for L

[tex]L=\frac{5}{16}(3)[/tex]

[tex]L=\frac{15}{16}\ yd[/tex]

The length of the wood board must be approximately 39.69 yards.

To find the length of the board, we need to divide the total area by the width of the board. The area of the board is 516 square yards and the width is 13 yards.

Therefore, the length of the board can be found by dividing 516 by 13.

516 ÷ 13 = 39.69 yards.

So, the board must be approximately 39.69 yards long to complete the project.

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Answer:

50

Step-by-step explanation:

150 divided by 3 is 50 and since you shared only one third the answer will be 50

Answer:50

Step-by-step explanation:

One third equals 1/3 of 150

1/3 ×150 =50

Answer: y = -3x - 5

Step-by-step explanation: since the two lines are parallel, it implies that

m1 = m2. From the equation given,

m1 = -3 , m2 = -3

Substantial for m2 in y = mx + c

We now have

y = -3x - 5 since the new line is not passing through any coordinate.

Answer:

The answer is y=-3x-2

I hope it's not too late;)

Step-by-step explanation:

Answer:

- 59.4

Step-by-step explanation:

The given function is [tex]f(x) = 120(0.1)^{x}[/tex]

Therefore, at x = 0, [tex]f(0) = 120(0.1)^{0} = 120[/tex] and  

at x = 2, [tex]f(2) = 120(0.1)^{2} = 1.2[/tex]

Hence, the average rate of change of the given function from x = 0 to x = 2 will be  

[tex]\frac{f(2) - f(0)}{2 - 0}[/tex]

= [tex]\frac{1.2 - 120}{2 - 0}[/tex]

= - 59.4 (Answer)

Answer:

2K + 12L ≥ 87 and 1.5K + 8L > 63

Step-by-step explanation:

It takes 2 wooden sticks and 1.5 square feet of paper to make a kite, and it takes 12 wooden sticks and 8 square  feet of paper to make a lamp.

If K denotes the number of kites that Min-Young made and L denotes the number of lamps she made then  

2K + 12L is the number of wooden sticks that she used which is greater than equal to 87

i.e. 2K + 12L ≥ 87 ........ (1)

Again, 1.5K + 8L is the number of square feet of paper that she used which is greater than 63 sq. feet

i.e. 1.5K + 8L > 63 ........... (2)

Hence, inequalities (1) and (2) are the required answer. (Answer)

Answer:

1) 2K + 12L ≥ 87

2) 1.5K + 8L > 63

Step-by-step explanation:

Answer:

Degree = 1

Step-by-step explanation:

Given:

The differential equation is given as:

[tex]\frac{d^2y}{dx^2}+(\frac{dy}{dx})^2+6y=0[/tex]

The given differential equation is of the order 2 as the derivative is done 2 times as evident from the first term of the differential equation.

The degree of a differential equation is the exponent of the term which is the order of the differential equation. The terms which represents the differential equation must satisfy the following points:

The above differential equation doesn't involve any of the above conditions. The exponent to which the first term is raised is 1.

Therefore, the degree of the given differential equation is 1.

Answer:

0.9375

Step-by-step explanation:

Do the long division and you'll find the answer.

The venue will break even after selling 1000 tickets.

The value will be $70,000 at that point.

Step-by-step explanation:

Given,

Charges of band;

C(x) = 55x+15000

Per ticket charges by Venue = $70

V(x) = 70x

For breaking even,

C(x) = V(x)

[tex]55x+15000=70x\\15000=70x-55x\\15000=15x\\15x=15000[/tex]

Dividing both sides by 15

[tex]\frac{15x}{15}=\frac{15000}{15}\\x=1000[/tex]

The venue will break even after selling 1000 tickets.

Putting 1000 in V(x)

[tex]V(1000)=70(1000)\\V(1000)=70000[/tex]

The value will be $70,000 at that point.

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Answer:

x^2 - 6x + x - 6

= x^2 - 5x - 6

Answer:

The plot of land was sold for rs 129,600

Step-by-step explanation:

The cost of the plot of land                  =  rs 108,000

Profit calculation ( 20% x 108,000)      =  rs 21,600

Price land was sold                               = (108,000 + 21,600) = 129,600

Answer:

y = 3x - 1

The rate of change of the function is 3.

Step-by-step explanation:

The equation of the linear function through the points (- 2, - 7)and (1,2) will be given by  

[tex]\frac{y - ( - 7)}{- 7 - 2} = \frac{x - (- 2)}{- 2 - 1}[/tex]

⇒ (y + 7) = 3(x + 2)

⇒ y = 3x - 1 (Answer)

Now, the rate of change of the function is the slope of the line i.e. 3. (Answer)

We know the equation of a straight line which passes through the points [tex](x_{1}, y_{1})[/tex] and [tex](x_{2}, y_{2})[/tex] is given by  

[tex]\frac{y - y_{1}}{y_{1} - y_{2}} = \frac{x - x_{1}}{x_{1} - x_{2}}[/tex]

Answer:

[tex]\large\boxed{\dfrac{1}{x-4}\ \text{for}\ x\neq5\ \wedge\ x\neq4}[/tex]

Step-by-step explanation:

[tex]\dfrac{x-5}{(x-5)(x-4)}\\\\\text{Domain:}\ (x-5\neq0\ \wedge\ x-4\neq0)\Rightarrow(x\neq5\ \wedge\ x\neq4)\\\\\dfrac{x-5}{(x-5)(x-4)}\qquad\text{cancel}\ (x-5)\\\\=\dfrac{1}{x-4}[/tex]

|12x + 1| = 10

Remove the absolute value term and make two equations:

12x +1 = 10

12x +1 = -10

Now solve for both x values:

12x +1 = 10

Subtract 1 from both sides:

12x = 9

Divide both sides by 12:

x = 9/12

12x +1 = -10

Subtract 1 from both sides:

12x = -11

Divide both sides by 12:

x = -11/12

The answer would be (-11/12, 912)

Question:

A computer can sort x objects in t seconds, as modeled by the function below:

t=0.005x^2+0.002x

How many objects are required to keep the computer busy for exactly 9 seconds?

Round to the nearest whole object.

Answer:

42 objects are required to keep the computer busy for exactly 9 seconds

Solution:

Given function is:

Computer can sort x objects in t seconds, as modeled by the function below:

[tex]t = 0.005x^2+0.002x[/tex]

We have to find number of objects required to keep the computer busy for exactly 9 seconds

Therefore t = 9

Substitute t = 9 in given function

[tex]9 = 0.005x^2+0.002x\\\\0.005x^2+0.002x - 9 = 0[/tex]

Let us solve the above equation by quadratic formula,

[tex]\text {For a quadratic equation } a x^{2}+b x+c=0, \text { where } a \neq 0\\\\x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}[/tex]

Using the above formula,

[tex]\text{ for } 0.005x^2+0.002x - 9 = 0 , \text{ we have } a = 0.005 ; b = 0.002 ; c = -9[/tex]

Substituting the values of a = 0.005 ; b = 0.002 ; c = -9 in above quadratic formula we get,

[tex]\begin{aligned}&x=\frac{-0.002 \pm \sqrt{0.180004}}{0.01}\\\\&x=\frac{-0.002 \pm 0.4242}{0.01}\\\\&x=\frac{-0.002+0.4242}{0.01} \text { or } x=\frac{-0.002-0.4242}{0.01}\\\\&x=42.22 \text { or } x=-42.62\end{aligned}[/tex]

Ignoring negative value,

x = 42.22 ≈ 42

Thus 42 objects are required to keep the computer busy for exactly 9 seconds

Alejandro read 4 pages in 6 minutes, then it would take him 9 minutes to read 6 pages

Solution:

Given that Alejandro read 4 pages in a book in 6 minutes

Therefore,

4 pages = 6 minutes

Let us find the time taken to read 1 page

time taken to read 1 page = time taken to read 4 pages divided by 4

[tex]\text{ time taken to read 1 page } = \frac{6 minutes}{4} = \frac{3}{2} minutes[/tex]

Thus to read 6 pages at this rate, multiply time taken to read 1 page by 6

[tex]\rightarrow \frac{3}{2} \times 6 = 9[/tex]

Therefore it would take 9 minutes to read 6 pages

Answer:

150

Step-by-step explanation:

60 minutes in 1 hour

3x10= 30 in one hour

30x5=150 30 trucks in 5 hours

Tarzan can load 150 trucks in 5 hours.

To calculate how many trucks Tarzan can load in 5 hours based on the rate that he loaded 3 trucks in 6 minutes. First, we convert 5 hours into minutes to match the units of the given rate. There are 60 minutes in 1 hour, so 5 hours is 300 minutes (5 hours x 60 minutes/hour). Next, we need to find out how many 6-minute intervals there are in 300 minutes, which is 300 minutes / 6 minutes/intervals = 50 intervals. Since Tarzan loads 3 trucks per interval, we multiply the number of intervals by the number of trucks per interval to get the total number of trucks loaded in 5 hours: 50 intervals x 3 trucks/interval = 150 trucks.

Answer:

1) [tex]y> \frac{x}{3}+2[/tex] 2) (0,2) 3) The first and second Points must have x coordinate <-6, or y-coordinate  y >2 e.g. (-7,2), (-6,3)

Step-by-step explanation:

1) To Rewrite it as Slope-intercept form, is to isolate the y on the left side and on the right side the rest of the inequality.

[tex]-x+3y>6\Rightarrow 3y>x+6 \Rightarrow y> \frac{x+6}{3}\Rightarrow y> \frac{x}{3}+2[/tex]

2) Since this is a linear inequality the y intercept is given by "b" parameter.[tex]y> mx+b \Rightarrow y> \frac{x}{3}+2 \Rightarrow b=2[/tex]

So the y-intercept is y > 2, coordinate point (0,2). In the graph, we have a dashed line over 2.

3) Since there no choices, the points that satisfy this inequality lie within the green area. We know that the points for this inequality must satisfy  x < -6 or y> 2:

Testing for (-7,2)  for x<-6 ⇒-7 <-6

[tex]-x+3*y>6\\-(-7)+3*2>6\\7+6>6\\13>6\:\\True\\[/tex]

Testing for (-6,3)  for y>2 ⇒3>2

[tex]-x+3*y>6\\-(-6)+3*3>6\\6+9>6\\15>6\:True\\[/tex]

Answer:

Domain : {10, 15, 19, 32}

Range: {-1, 5, 9}

Step-by-step explanation:

As the table of the relation is given as follows:

X                Y

10               5

15               9

19               -1

32             5  

From this table, we can define the relation by combining the ordered pairs from the table. Each and every order pair consists of x-coordinate and corresponding y-coordinate of any corresponding point.

So, the relation from the table can be made as follows:

                            relation : {(10, 5), (15, 9), (19, -1), (32, 5)}

Domain of a relation consists of all the x-coordinates (first elements) of order pairs.

Range of a relation consists of all the y-coordinates (second elements) of ordered pairs.

So, domain  and range of relation will be as follows:

                                  Domain : {10, 15, 19, 32}

                                  Range: {-1, 5, 9}

Note: If there is any duplicate element in any x or y-coordinate of any ordered pair, it will be written only once when we determine domain and range. Here, in this example, 5 is duplicate, so, it will be mentioned only one time when we determine the range of this relation.

Keywords:  domain, relation, range

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The domain and range of the relation given in the table are the sets of x-values and y-values respectively. In this case, the domain is {-1, 5, 9} and the range is {10, 15, 19, 32}, which corresponds to Option A.

To determine the domain and range of the relation shown in a table, we must look at the set of ordered pairs provided. The domain consists of all the first elements in these pairs, which represent the x-values, while the range consists of all the second elements, which are the y-values. Looking at the table, we can list the x and y values separately to clearly identify the domain and range.

If the table of the relation presents the ordered pairs as (X, Y), interpreting the question that the domain should be {-1, 5, 9} and the range should be {10, 15, 19, 32}, then the correct option would be:

Option A.

Domain: \{-1, 5, 9\} and Range: \{10, 15, 19, 32\}.

It is essential to note that a function is a special relation where each element of the domain maps to a unique element in the range. In this case, as each x-value maps to a distinct y-value, it suggests the relation is indeed a function.

Answer:

[tex]\frac{9}{4}[/tex] or [tex]2\frac{1}{4}[/tex] or [tex]2.25[/tex]

Step-by-step explanation:

1 cm = 25000cm = 1/4 km. 9*1/4=9/4=2 1/4= 2.25

Answer:

degree = 6

Step-by-step explanation:

Given [tex]f(x)=3x^2[/tex], and [tex]g(x)=4^3+1[/tex], we can find the composition of functions: [tex]fog(x)[/tex] by applying the definition of composition and performing the needed algebra.

Recall that the composition of functions is defined as: [tex]fog(x)=f(g(x))[/tex], where we use as input for the function f(x) the actual expression in terms of "x" of the function g(x):

[tex]f(g(x))=f(4x^3+1)\\f(g(x))=3(4x^3+1)^2\\f(g(x))=3\,(4x^3+1)\,(4x^3+1)\\f(g(x))=3\,[16x^6+4x^3+4x^3+1]\\f(g(x))=3\,[16x^6+8x^3+1]\\f(g(x))=48x^6+24x^3+3[/tex]

Therefore, the degree of this expression is "6" (the highest power at which the variable "x" appears)

Answer:

the answer is 6

Step-by-step explanation:

don't have one

You would use long division to determine that 8/12 = 0.667 approximately. The steps would go something like this

-------------

Step 1) Write 8 under a long division bar and place 12 to the left and outside the bar

Step 2) Change the 8 into 8.0000 (you can have as many trailing zeros as you want)

Step 3) While 12 doesn't go into 8, 12 does go into 80. So you can think of 8.0 as 80. Ask yourself "how many times does 12 go into 80?". The answer is 6 times because 6*12 = 72 (note how 7*12 = 84 is too big). So we write a "6" above the 8.0000 such that it is above the first "0". So far we have 0.6 as the quotient.

Step 4) Multiply that 6 by 12 to get 72. We write the 72 under the 80 and then subtract to get 80 - 72 = 8. Pull down the second 0 from "8.0000" to change that 8 into 80.

Step 5) We effectively repeat steps 3 and 4 again. So thats why we get another 6 as the next part of the quotient, and why the 6's repeat forever. The only reason why there's a 7 at the end is because of rounding.

-----------------------

The long division is shown in the image attachment below. The gridlines are there to show how things line up though when you write out the long division steps (to show to your teacher) you wont use any grid lines.

side note: the fraction 8/12 reduces to 2/3. You divide both top and bottom by the GCF 4. So 8/4 = 2 and 12/4 = 3.