The points Q, R, S and T all lie on the same line segment, in that order, such that the
ratio of QR:RS: ST is equal to 1:4:5. If QT = 40, find RS.

Answer:

height = 5 cm

Step-by-step explanation:

a cube has congruent sides (s)

the volume (V) of a cube is calculated as

V = s³

given V = 125 , then

s³ = 125 ( take cube root of both sides )

\(\sqrt[3]{s^3}\) = \(\sqrt[3]{125}\) = \(\sqrt[3]{5^3}\)

s = 5

then height = 5 cm

Mack should make 9 necklaces and 3.6 (rounded to 4) wristbands to maximize his profit.

We have,

The profit function is given by:

P(x,y) = 3x + 2y

The constraints are:

40x + 25y ≤ 360 (time constraint)

x + y ≤ 12 (item constraint)

x, y ≥ 0 (non-negative constraint)

To find the vertices, we need to solve the system of equations for each pair of intersecting lines. The vertices are the points where the lines intersect.

40x + 25y = 360

x + y = 12

Solving this system of equations, we get:

x = 6, y = 6 (vertex 1)

x = 9, y = 3.6 (vertex 2)

x = 0, y = 12 (vertex 3)

Now, we evaluate each vertex in the profit function:

P(6,6) = 3(6) + 2(6) = 24

P(9,3.6) = 3(9) + 2(3.6) = 30.6

P(0,12) = 3(0) + 2(12) = 24

Now,

Vertex 2 yields a maximum profit of $30.60.

Therefore,

Mack should make 9 necklaces and 3.6 (rounded to 4) wristbands to maximize his profit.

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The expressions that are completely factored are:

The expressions are considered correct because they have been completely factored, meaning they have been written in the form of "a common factor times another expression". This means that the expression on the right side of the equal sign can be simplified into its simplest form.

For example, in the expression 16a5−20a3=4a3(4a2−5), the 4a3 factor is common to both 16a5 and 20a3, so it can be factored out. The expression then becomes 4a3 times (4a2−5), which is the completely factored form.

Similarly, in the expression 12a3+8a=4(3a3+2a), the factor of 4 is common to both the expressions on the right side, so it can be factored out. This results in the completely factored form of 4 times (3a3+2a).

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Answer:

16a^5−20a^3=4a^3(4a^2−5)

and

24a^4+18=6(4a^4+3)

Step-by-step explanation:

Answer:

15x−9

Step-by-step explanation:

Answer:

42 rings

Explanation:

If the ratio of the bracelet to ring is 5:2, we can write the following equation

Where x is the number of rings that she has. So, we have the equation:

If we cross multiply, we get

Finally, divide by 5

Therefore, she has 42 rings

The volume of the cylinder is the amount of space on it

The volume of the oblique cylinder is 174.19 cubic units

The given parameters are:

AB = 16 units

Radius (r) = 2 units

\(\theta = 30^o\)

Start  by calculating the height (h) using:

\(h = 16 * \cos(30^o)\)

This gives

\(h = 13.86\)

The volume of the cylinder is then calculated as:

\(V = \pi r^2 h\)

This gives

\(V = 3.142 * 2^2 * 13.86\)

\(V = 174.19248\)

Approximate

\(V = 174.19\)

Hence, the volume of the oblique cylinder is 174.19 cubic units

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Let us call the side length of the square a, then the area of one white triangle is

There are two of these white triangles, and therefore, the area encompassed by them is

The area A of the yellow region is equal to the area of the square minus the area of the white triangles.

The area of the square is a^2; therefore,

Meaning, the area of the yellow region is 1/2 the area of the square.

Answer: -32

Step-by-step explanation:

I'm just going to plug in and break down the equation to make it more understandable and easier to comprehend

-(3^2) - (4*3) -11

-(9)- (12) -11

-9 - 12 - 11

-21 -11

-32

Answer:

a. C(1,000) = 420,491.11

b. c(1,000) = 420.49

c. dC/dx(1,000) = 429.72

d. x = 900

e. c(900) = 420

Step-by-step explanation:

We have a cost function for x units written as:

\(C(x) = 54,000 + 240x + 4x^{3/2}\)

a. The total cost for x=1000 units is:

\(C(1,000) = 54,000 + 240(1,000) + 4(1,000)^{3/2}\\\\C(1,000)=54,000+240,000+4\cdot 31,622.78\\\\C(1,000)=54,000+240,000+ 126,491.11 \\\\C(1,000)= 420,491.11\)

b. The average cost c(x) can be calculated dividing the total cost by the amount of units:

\(c(1,000)=\dfrac{C(1,000)}{1,000}=\dfrac{ 420,491.11 }{1,000}= 420.49\)

c. The marginal cost can be calculated as the first derivative of the cost function:

\(\dfrac{dC}{dx}=240(1)+4(3/2)x^{3/2-1}=240+6x^{1/2}\\\\\\\dfrac{dC}{dx}(1,000)=240+6(1,000)^{1/2}=240+6\cdot 31.62=429.72\)

d. This value for x, that minimizes the average cost, happens when the first derivative of the average cost is equal to 0.

\(c(x)=\dfrac{C(x)}{x}=\dfrac{54,000+240x+4x^{3/2}}{x}=54,000x^{-1}+240+4x^{1/2}\\\\\\ \dfrac{dc}{dx}=54,000(-1)x^{-2}+0+4(1/2)x^{-1/2}=0\\\\\\\dfrac{dc}{dx}=-54,000x^{-2}+2x^{-1/2}=0\\\\\\2x^{-1/2}=54,000x^{-2}\\\\\\x^{-1/2+2}=54,000/2=27,000\\\\\\x^{3/2}=27,000\\\\\\x=27,000^{2/3}=900\)

e. The minimum average cost is:

\(c(900)=54,000(900)^{-1}+240+4(900)^{1/2}\\\\c(900)=60+240+120\\\\c(900)=420\)

Step-by-step explanation:

a) k(x) = (x - 2)² - 4 → x = (y - 2)² - 4 → (y - 2)² = x + 4 → y - 2 = ±√( x + 4) → y = 2 ±√( x + 4) → k-¹(x) = 2 ±√( x + 4)

b) f(x) = ½(x + 2)² - 3 → x = ½(y + 2)² - 3 → ½(y + 2)² = x + 3 → (y + 2)² = 2x + 6 → y + 2 = ±√(2x + 6) → y = -2 ±√(2x + 6) → f-¹(x) = - 2 ±√(2x + 6)

Answer:

this list

Step-by-step explanation:

1×12000=12000

2×6000=12000

3×4000=12000

4×3000=12000

5×2400=12000

6×2000=12000

8×1500=12000

10 ×1200=12000

12 ×1000=12000

Answer:

The intensity of the 2011 earthquake was about 6 times the intensity of the 2003 earthquake.

Step-by-step explanation:

To compare the intensities, we first need to convert the magnitudes to intensities using the log formula. Then we will set up a ratio to compare the intensities.

Convert the magnitudes to intensities and write them in exponential form.

R=logI

2011 earthquake:

9.1I=logI=109.1

2003 earthquake:

8.3I=logI=108.3

Form a ratio of the intensities.

intensity for 2011intensity for 2003

Substitute in the values and divide by subtracting the exponents to find

109.1108.3100.8≈6.

The intensity of the 2011 earthquake was about 6 times the intensity of the 2003 earthquake.

Your answer:

The intensity of the 2011 earthquake was about 15 times the intensity of the 2003 earthquake.

The intensities of the two earthquakes will be 6.3095.

Algebra is the study of abstract symbols, while logic is the manipulation of all those ideas.

In 2011, Japan encountered a serious quake with an extent of 9.1 on the Richter scale. In 2003, Japan encountered another extraordinary seismic tremor that deliberate 8.3 on the Richter scale.

The intensity of the earthquake is given as,

㏒ (I₁ / I₂) = M₁ - M₂

㏒ (I₁ / I₂) = 9.1 - 8.3

Simplify the equation, then we have

㏒ (I₁ / I₂) = 9.1 - 8.3

㏒ (I₁ / I₂) = 0.8

Take anti log, then we have

(I₁ / I₂) = 6.3095

The intensities of the two earthquakes will be 6.3095.

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Answer:

v = 260

Step-by-step explanation:

You have to multiply all the values given together (13 x 3 x 20) and then divide that by 3.

The other three equations do not have a constant rate of change, and therefore do not represent a proportional relationship between the variables.

A proportional relationship between two variables means that as one variable increases or decreases, the other variable changes in direct proportion. In other words, the ratio of the two variables remains constant.

A linear equation that shows a proportional relationship must have a constant slope or rate of change. This means that the equation must be in the form of y = mx, where m is a constant.

Out of the four options given, the equation that shows a proportional relationship is:

y = 4x

This equation is in the form of y = mx, where m = 4, which means that for every increase of 1 in x, y increases by 4. The ratio of y to x is always 4, which means that the variables are directly proportional to each other.

The other three equations do not have a constant rate of change, and therefore do not represent a proportional relationship between the variables.

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Answer:

  y = 0.5(3^x)

Step-by-step explanation:

When matching a function to a table of values, the functions we usually try are linear, quadratic, and exponential. We can determine which of these might be appropriate by looking at differences in table values.

__

Subtracting each height (y) value from the next, we get successive first differences of ...

  1, 3, 9

These are not constant, and they do not have a constant difference of their own. However, we do notice they are related by a multiplicative factor of 3.

First differences are constant at a level equal to the degree of polynomial required to match the table values. If first differences are constant, then the table can be represented by a first-degree (linear) polynomial. Similarly, if second differences are constant, a second-degree (quadratic) polynomial will model the table.

If an exponential model is appropriate, differences will have the same constant ratio at every level. Here, first differences have ratios of 3/1 = 9/3 = 3. The second differences of 2 and 6 likewise have a ratio of 6/2 = 3. That ratio is the base of the exponential function.

We have determined that the base of the exponential function is 3. The multiplier is the y-intercept (the value when x=0). The table tells us that is 0.5. Then we have ...

  y = a·b^x . . . . . . 'a' = y-intercept; b = common ratio

  y = 0.5·(3^x)

_____

Additional comment

The above discussion of differences applies to tables where the x-differences are constant. If they are other than 1, then the resulting function will need to be horizontally scaled. If the x-differences are not constant, other methods of regression analysis and interpolation are appropriate.

Answer:

10

Step-by-step explanation:

the sequence follows the rule of adding the number and then the number -1 depending on the number before it. for example, you can see in order to get from 2 to 2, it adds 0 or subtracts by 0. After that to get from 2 to 4, you add 2. After that, you subtract 1 from 2 and you subtract that number which is 1 from 4 to get 3 and so on. Following that rule, you get 10 as the next number in the sequence

Answer:

4 times 1 over 10

Step-by-step explanation:

i think it is this hope it is right

Answer:

Step-by-step explanation:

it would be 4 times 1 over 10

Answer: B. 473.176

Step-by-step explanation: 3.78541 Liters are in one gallon, so if you multiply 3.78541 Liters by 125 gallons then you will end up with 473.176 liters. If you have to add tax then it would be A. 475.250

To solve systems of equations word problems, we can delegate certain variables to subjects mentioned in the word problem. From there, we can form equations based on the given values.

Let a be the older sibling's age and b be the younger sibling's age.

We're given:

All we've done so far is translate the given sentences into equations. Now, we have to solve for b, which is the age of the youngest sibling. Solve using substitution:

\(a+b=55\\(b+9)+b=55\\b+9+b=55\\2b+9=55\\2b=46\\b=23\)

Therefore, the younger sibling is 23 years old.

23 years old.

Answer:

1)  The functions are not inverses.

\(\textsf{2)} \quad g^{-1}(n)=&\dfrac{3}{8}n-\dfrac{7}{8}\)

\(\textsf{3)} \quad g^{-1}(x)&=\sqrt[3]{\dfrac{1}{2}-\dfrac{1}{2}x}\)

Step-by-step explanation:

The inverse composition rule states that if two functions are inverses of each other, then their compositions result in the identity function.

Given functions:

\(g(x) = 4 + \dfrac{7}{2}x \qquad \qquad f(x) = 5 - \dfrac{4}{5}x\)

Find g(f(x)) and f(g(x)):

\(\begin{aligned} g(f(x))&=4+\dfrac{7}{2}f(x)\\\\&=4+\dfrac{7}{2}\left(5 - \dfrac{4}{5}x\right)\\\\&=4+\dfrac{35}{2}-\dfrac{14}{5}x\\\\&=\dfrac{43}{2}-\dfrac{14}{5}x\\\\\end{aligned}\)                 \(\begin{aligned} f(g(x))&=5 - \dfrac{4}{5}g(x)\\\\&=5 - \dfrac{4}{5}\left(4 + \dfrac{7}{2}x \right)\\\\&=5-\dfrac{16}{5}-\dfrac{14}{5}x\\\\&=\dfrac{9}{5}-\dfrac{14}{5}x\end{aligned}\)

As g(f(x)) or f(g(x)) is not equal to x, then f and g cannot be inverses.

\(\hrulefill\)

To find the inverse of a function, swap the dependent and independent variables, and solve for the new dependent variable.

Calculate the inverse of g(n):

\(\begin{aligned}y &= \dfrac{8}{3}n + \dfrac{7}{3}\\\\n &= \dfrac{8}{3}y + \dfrac{7}{3}\\\\3n &= 8y + 7\\\\3n-7 &= 8y\\\\y&=\dfrac{3}{8}n-\dfrac{7}{8}\\\\g^{-1}(n)&=\dfrac{3}{8}n-\dfrac{7}{8}\end{aligned}\)

Calculate the inverse of g(x):

\(\begin{aligned}y &= 1-2x^3\\\\x &= 1-2y^3\\\\x -1&=-2y^3\\\\2y^3&=1-x\\\\y^3&=\dfrac{1}{2}-\dfrac{1}{2}x\\\\y&=\sqrt[3]{\dfrac{1}{2}-\dfrac{1}{2}x}\\\\g^{-1}(x)&=\sqrt[3]{\dfrac{1}{2}-\dfrac{1}{2}x}\\\\\end{aligned}\)

Answer:

1.

If the composition of two functions is the identity function, then the two functions are inverses. In other words, if f(g(x)) = x and g(f(x)) = x, then f and g are inverses.

For\(\bold{g(x) = 4 + \frac{7}{2}x\: and \:f(x) = 5 -\frac{4}{5}x}\), we have:

\(f(g(x)) = 5 - \frac{4}{5}(4 + \frac{7}{2}x)\\ =5 - \frac{4}{5}(\frac{8+7x}{2})\\=5 - \frac{2}{5}(8+7x)\\=\frac{25-16-14x}{5}\\=\frac{9-14x}{5}\)

\(g(f(x)) = 4 + (\frac{7}{5})(5 - \frac{4}{5}x) \\=4 + (\frac{7}{5})(\frac{25-4x}{5})\\=4+ \frac{175-28x}{25}\\=\frac{100+175-28x}{25}\\=\frac{175-28x}{25}\)

As you can see, f(g(x)) does not equal x, and g(f(x)) does not equal x. Therefore, g(x) and f(x) are not inverses.

Sure, here are the inverses of the functions you provided:

2. g(n) = (8/3)n + 7/3

we can swap the roles of x and y and solve for y to find the inverse of g(n). In other words, we can write the equation as y = (8/3)n + 7/3 and solve for n.

y = (8/3)n + 7/3

n =3/8*( y-7/3)

Therefore, the inverse of g(n) is:

\(g^{-1}(n) = \frac{3}{8}(n - \frac{7}{3})=\frac{3}{8}*\frac{3n-7}{3}=\boxed{\frac{3n-7}{8}}\)

3. g(x) = 1 - 2x^3

We can use the method of substitution to find the inverse of g(x). We can substitute y for g(x) and solve for x.

\(y = 1 - 2x^3\\2x^3 = 1 - y\\x = \sqrt[3]{\frac{1 - y}{2}}\)

Therefore, the inverse of g(x) is:

\(g^{-1}(x) =\boxed{ \sqrt[3]{\frac{1 - x}{2}}}\)

9514 1404 393

Answer:

Step-by-step explanation:

In 1954, the value of the bonds was said to be ...

  (54 bonds)×(2500 oz/bond)×($350/oz) = $47.25 million

Then the value for any year after 1954 is said to be ...

  value = $0.04725×1.00019^(365t) . . . . billions

where t = years after 1954.

In 2012, t = 2012 -1954 = 58, so the value is about ...

  value = $0.04725×1.00019^21170 ≈ $2.64 billion

In 2024, t = 2024 -1954 = 70, so the value is about ...

  value = $0.04725×1.00019^25550 ≈ $6.06 billion

_____

Additional comment

If the bonds cannot be redeemed, their value is zero, regardless of any formula.

In 2010 a lawsuit was brought in US court to force the Federal Republic of Germany to make good on the bonds owned by US investors. Germany claims the debt was settled by a 1954 treaty agreement, and that the deadline for any claims was 1958. Hence the bonds are very likely worthless, though investors continue to be hopeful.

The equation which describes the relationship between the side length and shaded area of the quilt is y=0.5x²

Side length, x = 1,3,4,5,8

Shaded Area, y = 0.5, 4.5, 8, 12.5, 32

The relationship can be modeled as a quadratic function. Using a graphing calculator for the quadratic function written in the form y = ax² + bx + c

a = 0.5 ; b = 0 ; c = 0

Therefore, the quadratic function can be written as y = 0.5x²

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The cost of the carpet will be $2,335.5.

What is Multiplication?

To multiply means to add a number to itself a particular number of times. Multiplication can be viewed as a process of repeated addition.

Given that;

The dimensions of the carpet are;

⇒ 15 feet by 18 feet

Now,

The area of the carpet = 15 x 18

                                   = 279 feet²

                                   = 270 / 3 yards²

                                   = 90 yards²

Since, The carpet costs = $25.95 per square yard.

Hence, The carpet cost for 90 yards² = 90 x $25.95

                                                          = $2,335.5

Thus, The cost of the carpet will be $2,335.5.

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Standard Deviation for given set of data is 0.25634797778466.

What is standard deviation?

Given that,

Sample size :12.3,11.9,12.5,12.1,12.6,11.9,12.2,12.1

Count, N: 8

Sum, submission x: 97.6

Mean, x: 12.2

Variance, s2:  0.065714285714286

s^2 = Σ(xi - x)^2/N - 1

= (12.3 - 12.2)2 + ... + (12.1 - 12.2)^2/8 - 1

= 0.46/7

=  0.065714285714286

s =  √0.065714285714286

=  0.25634797778466

Standard Deviation for given set of data is 0.25634797778466.

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Answer:

C. 6kg to 7kg(is the old one)

Solving steps:

Answer:

60 : 72 = 5 : 6

Solution:

Try to reduce the ratio further with the greatest common factor (GCF).

The GCF of 60 and 72 is 12

Divide both terms by the GCF, 12:

60 ÷ 12 = 5

72 ÷ 12 = 6

Try to reduce the ratio further with the greatest common factor (GCF).

The GCF of 120 and 144 is 24

Divide both terms by the GCF, 24:

120 ÷ 24 = 5

144 ÷ 24 = 6

The ratio 120 : 144 can be reduced to lowest terms by dividing both terms by the GCF = 24 :

120 : 144 = 5 : 6

Therefore:

120 : 144 = 5 : 6

A) The ratio 60 : 72 can be reduced to lowest terms by dividing both terms by the GCF = 12 :

60 : 72 = 5 : 6

Therefore:

60 : 72 = 5 : 6

Answer:

30 : 36 = 5 : 6

b) Solution:

Try to reduce the ratio further with the greatest common factor (GCF).

The GCF of 30 and 36 is 6

Divide both terms by the GCF, 6:

30 ÷ 6 = 5

36 ÷ 6 = 6

The ratio 30 : 36 can be reduced to lowest terms by dividing both terms by the GCF = 6 :

30 : 36 = 5 : 6

Therefore:

30 : 36 = 5 : 6

Answer:

6 : 7 = 6 : 7

c) Solution:

Try to reduce the ratio further with the greatest common factor (GCF).

The GCF of 6 and 7 is 1

Divide both terms by the GCF, 1:

6 ÷ 1 = 6

7 ÷ 1 = 7

The ratio 6 : 7 cannot be reduced further by dividing both terms by the GCF = 1 :

This ratio is already in lowest terms.

Therefore:

6 : 7 = 6 : 7

Answer:

15 : 18 = 5 : 6

d) Solution:

Try to reduce the ratio further with the greatest common factor (GCF).

The GCF of 15 and 18 is 3

Divide both terms by the GCF, 3:

15 ÷ 3 = 5

18 ÷ 3 = 6

The ratio 15 : 18 can be reduced to lowest terms by dividing both terms by the GCF = 3 :

15 : 18 = 5 : 6

Therefore:

15 : 18 = 5 : 6

Answer:

5 : 6 = 5 : 6

e) Solution:

Try to reduce the ratio further with the greatest common factor (GCF).

The GCF of 5 and 6 is 1

Divide both terms by the GCF, 1:

5 ÷ 1 = 5

6 ÷ 1 = 6

The ratio 5 : 6 cannot be reduced further by dividing both terms by the GCF = 1 :

This ratio is already in lowest terms.

Therefore:

5 : 6 = 5 : 6

The layout of the fast-food restaurant of dimensions 6 by 8 5 feet squares is presented in the attached table created with Sheets.

The dimensions of the facility are;

Horizontal = 6 squares

Vertical = 8 squares

The side length of each square = 5 feet

Therefore;

Area of each square = 5² ft.² = 25 ft.²

Number of squares, n, required by each dependent are therefore;

CB department, n = 300 ÷ 25 = 12 squares

CF department, n = 200 ÷ 25 = 8 squares

PS department, n = 8 squares

DD department, n = 8 squares

CS department, n = 12 squares

A layout for the fast-food restaurant is therefore;

Please see the attached table layout created using Sheets.

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