Decompose -6x/(x-6)(x+3)
into partial fractions.

The cost of each stone sphere is $ 36.81

Solution:

Given that,

An architect plans to buy 5 stone spheres and 3 stone cylinders

For the same​ amount, she can buy 2 stone spheres and 6 stone cylinders

Let "x" be that same amount

Let "a" be the cost of each stone sphere

Cost of each stone cylinder = $ 36.81

Therefore,

x = 5 stone spheres and 3 stone cylinders

x = 5a + 3(36.81)

Similarly,

x = 2 stone spheres and 6 stone cylinders

x = 2a + 6(36.81)

Equate both,

[tex]5a + 3(36.81) = 2a + 6(36.81)\\\\5a + 110.43 = 2a + 220.86\\\\5a - 2a = 220.86 - 110.43\\\\3a = 110.43\\\\a = 36.81[/tex]

Thus cost of each stone sphere is $ 36.81

By setting up and solving equations based on the given scenarios, we find that the cost of each stone sphere is also $36.81, the same as one stone cylinder.

Let's denote the cost of one stone sphere as S and the cost of one stone cylinder as C. We know one stone cylinder costs $36.81. We are given two scenarios involving the purchase of stone spheres and cylinders with equal total cost. In the first scenario, 5 stone spheres and 3 stone cylinders are bought, and in the second, 2 stone spheres and 6 stone cylinders are purchased.

Formulating these scenarios into equations gives us:

Substituting the value of C into the equation simplifies it to:

5S + 3(36.81) = 2S + 6(36.81)

Simplifying further:

5S + 110.43 = 2S + 220.86

Subtracting 2S and 110.43 from both sides results in:

3S = 110.43

Dividing both sides by 3 gives:

S = $36.81

Therefore, each stone sphere also costs $36.81.

Final answer:

It will take 400 minutes for the temperature to decrease by 5°F, calculated by dividing 5°F by the rate of ¾°F/h, resulting in 6.6667 hours, which is then converted into minutes (6.6667 hours × 60 minutes/hour = 400 minutes).

Explanation:

To calculate the time it will take for the temperature to decrease by 5°F, knowing that the temperature decreases at a rate of ¾ of a degree per hour, we can use the following steps:

Divide the total desired temperature decrease (5°F) by the rate of temperature decrease per hour (¾°F/h).

Calculate the total hours required for a 5°F decrease.

Since we want to find the time in minutes, we multiply the hours by 60 (as there are 60 minutes in an hour).

Step 1: 5°F divided by ¾°F/h = 5 ÷ 0.75 = 6.6667 hours
Step 2: 6.6667 hours are needed.

Step 3: 6.6667 hours × 60 minutes/hour = 400 minutes

Therefore, it will take 400 minutes for the temperature to decrease by 5°F.

Answer:

x= 3.73737373737......

Step-by-step explanation:

To find x, divide -9.9 to both sides.

-37 = x-9.9

-9.9     -9.9

3.73737373737...= x

This is a repeating decimal.

Answer: n since that number depends on s, the songs he already had.

Final answer:

The dependent variable in Steve's music player scenario is the total number of songs after buying new songs, represented by T = s + n. As the number of new songs purchased (n) changes, the total (T) changes, demonstrating a linear relationship between n and T.

Explanation:

In the scenario where Steve had s songs on his music player and he bought n new songs, the dependent variable is the total number of songs Steve has after the purchase. To define this in terms of a mathematical relationship, we can express the total number of songs (T) as an equation:

T = s + n

Here, s is the original number of songs, and n is the number of new songs purchased. The dependent variable is T (the total number of songs), which depends on the value of n (the number of songs bought). If we change the independent variable n, the total number of songs T will also change.

To illustrate this with a dataset, suppose Steve originally had 5 songs (s = 5). If he buys:

The relationship between the number of new songs purchased (n) and the total number of songs (T) is linear, as T increases by the same amount that n does.

Answer:

3

Step-by-step explanation:

If we already know that 10% of 60 is 6 and that 5% is one half of 10%, we can just divide the 10% value by two to find the 5% of 60.

The circumference of the circle is 23.89 ft

The circumference of a circle is the distance around it and it can be calculated by the formula:

Circumference = π x diameter

The circumference is therefore:

= 22/7 x 7.6

= 167.2 / 7

= 23.89 feet

The option given that is closest to 23.89 ft is the right answer.

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Answer:

7/20

Step-by-step explanation:

7/10 * 1/2

By multiplying

7/20

0.35

Answer: 7/20 or .35

Step-by-step explanation: .7 times .5 is .35

Answer:

The denominator should be 3.  Hope this helps.

15/k

15/3

5

Hope this helps! ;)

Answer:

5

Step-by-step explanation:

Since k=3 you would put 3 under 15. Now the equation would look like this: 15/3 instead of 15/k. Then you would divide 15 by 3 which is 5.

Answer:1. 3 inches

Step-by-step explanation:1 inch =2ft 2x3=6

To find the length of the garden in Micah's model, we need to set up a proportion using the given scale. The scale for the figure on the left to the figure on the right can be determined by comparing corresponding lengths.

1.) To find the length of the garden in Micah's model, we need to use the scale 1 inch: 2 feet. Since the garden has a length of 6 feet, we can set up a proportion:

1 inch / 2 feet = x inches / 6 feet

Cross-multiplying, we get 2 feet * x inches = 1 inch * 6 feet

Simplifying, we have 2x inches = 6 inches, so x = 3 inches.

Therefore, the length of the garden in Micah's model is 3 inches.

2.) The scale of the figure on the left to the figure on the right can be determined by comparing the corresponding lengths. If we measure the lengths of a certain side of the left figure and the right figure, we can find the scale. For example, if the length of a side of the left figure is 2 cm, and the corresponding side on the right figure is 4 cm, the scale is 1 cm: 2 cm.

Therefore, the scale is 1 cm: 2 cm.

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Answer:

[tex]x^3y^6z^3[/tex]

Step-by-step explanation:

we have

[tex](xy^2z)^3[/tex]

Applying the property of exponents

[tex](x^m)^n=x^{m*n}[/tex]

[tex](xy^2z)^3=(x^3)(y^2)^3(z^3)=x^3y^{2*3}z^3=x^3y^6z^3[/tex]

The correct simplification of [tex]\((xy^2z)^3\) is \(x^3y^6z^3\).[/tex]

To simplify the expression [tex]\((xy^2z)^3\),[/tex] you need to raise each component within the parentheses to the power of 3.

[tex]\(x^3\)[/tex] represents \(x\) raised to the power of 3, [tex]\(y^{2 \times 3}\)[/tex] represents [tex]\(y^6\)[/tex] (since you multiply the exponents when raising a power to a power), and [tex]\(z^3\)[/tex]represents \(z\) raised to the power of 3.

So, [tex]\((xy^2z)^3\) simplifies to \(x^3y^6z^3\).[/tex]

The given options are:

[tex]a) \(x^3y^3z^3\)\\b) \(x^3y^6z^3\)\\c) \(x^3y^8z^3\)\\d) \(x^4y^5z^4\)[/tex]

Comparing the simplified expression [tex]\(x^3y^6z^3\)[/tex] with the options, the correct answer is option (b) [tex]\(x^3y^6z^3\).[/tex]

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Answer:

  see attached

Step-by-step explanation:

The given function is a vertical expansion of the sine function by a factor of 3 and a translation downward by 5 units.

It will oscillate between -8 and -2, with a midline at y = -5. The multiplier 12 indicates the period of the function will be 2π/12 = π/6 ≈ 0.5233... for the given value of π.

A graphing calculator is a useful tool for creating the graph.

Answer:

Step-by-step explanation:

took the test

Answer:

3.6

Step-by-step explanation:

30%×12= 3.6

30%=0.3

For a system to have no solutions the lines must be parallel (they cannot intersect).

So the answer is the graph on the top right.

Answer: top right and bottom right

Step-by-step explanation:

For there to be a solution the lines have to cross at one time so when they are not touching there is no solution

Step-by-step explanation:

We have,

[tex]\log _{a}x[/tex]

To find, the value of [tex]\log _{a}x[/tex] = ?

∴ [tex]\log _{a}x[/tex] , where a, b  and x are positive

a ≠ 1 and b ≠ 1

We know that,

The logarithm identity,

[tex]\log_{p}m=\dfrac{\log_{y}m}{\log_{y}p}[/tex]

∴ [tex]\log _{a}x[/tex] = [tex]\dfrac{\log_{b}x}{\log_{b}a}[/tex]

Where, b is the common base of logarithm

∴ The value of [tex]\log _{a}x[/tex] = [tex]\dfrac{\log_{b}x}{\log_{b}a}[/tex]

Thus, the required option A) [tex]\dfrac{\log_{b}x}{\log_{b}a}[/tex] is correct.

Answer:

There are  10 pigs and 30 chickens

Step-by-step explanation:

Let the number of pigs be x

let the number of chickens be y

Then there are 40 heads . So the total number of pigs and chickens is 40

x + y = 40-----------------------(1)

We know that pigs have 4 legs and chickens have 2 legs

4x + 2y = 100----------------------(2)

Solving (1) and (2)

multiply  (1) by 2

2x + 2y = 80-------------------------(3)

subtract (3) from (2)

4x + 2y = 100

2x + 2y = 80

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

2x = 20

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

[tex]x = \frac{20}{2}[/tex]

x = 10

Now Substituting x in (1), we get

10 + y = 40

y = 40 - 10

y = 30

Answer:

It's the third one

Guaranteed!!

Hence, the graph second is correct.

What is the simplification?

Simplification is reducing the expression/fraction/problem in a simpler form. It makes the problem easy with calculations and solving.

Here,

[tex]f(x)=-x\\g(x)=2x[/tex]

So,

[tex]g(x)-f(x)=2x-(-x)\\\\(g-f)(x)=2x+x\\\\(g-f)(x)=3x[/tex]

Draw the graph:-

Hence, the graph second is correct.

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Answer:

The annual rate of depreciation of the car is 8.05% or 72.41/9

Step-by-step explanation:

1. Let's review the information given to us to answer the question correctly:

Price of the purchase of the car = $ 39,150

Price after 9 years = $ 10,800

2. Determine the rate at which the van depreciates in value.

Let's calculate what is the percentage of depreciation after 9 years, this way:

Percentage of depreciation after 9 years = [1 - (Price after 9 years/Price of the purchase of the car)] * 100

Replacing with the values we know:

Percentage of depreciation after 9 years = [1 - (10,800/39,150)] * 100

Percentage of depreciation after 9 years = [1 - 0.2759] * 100

Percentage of depreciation after 9 years = 72.41%

Annual rate of depreciation = 72.41/9 = 8.05%

Answer:

[tex]\frac{\sqrt{3} }{2}[/tex]

Step-by-step explanation:

Remember SOHCAHTOA

The cosine of an angle in a right triangle is the adjacent sides length divided by the hypotenuses length

In this case, for the angle 30 degrees, the adjacent length is [tex]\sqrt{3}[/tex] and the hypotenuse's length is 2

So, the cosine of the 30 degree angle is the first option:

[tex]\frac{\sqrt{3} }{2}[/tex]

Answer:

  80%

Step-by-step explanation:

The percentage change between two numbers can be calculated as ...

  percentage change = ((new value)/(old value) -1) × 100%

  = (2700/1500 -1) × 100% = (1.80 -1) × 100% = 80%

Traffic volume increased 80%.

Answer:

6

Step-by-step explanation:

11x + 7 = 73

     - 7    - 7

     11x = 66

divide 66 by 11 and you get 6

Answer:

x = 6

Step-by-step explanation:

Step 1:  Subtract 7 from both sides

11x + 7 - 7 = 73 - 7

11x = 66

Step 2:  Divide both sides by 11

11x = 66

11x / 11 = 66 / 11

x = 6

Answer:  x = 6

Answer:

  (x, y, z) = (1, 12, 15)

Step-by-step explanation:

As with any set of linear equations, there are many possible routes to a solution. We might simplify the notation a bit by writing the coefficients in an augmented matrix. The columns, left to right, represent the coefficients of x, y, and z, in order, and the constant term.

The row operations we'll use are multiplying a row by a value and adding that result to another row, replacing the other row by the sum.

We can make things a little simpler by writing the second equation first. Then the augmented matrix we're starting with is ...

  [tex]\left[\begin{array}{ccc|c}4&-1&1&7\\1&1&-1&-2\\1&-3&2&-5\end{array}\right][/tex]

Adding the second row to the first, we get ...

  [tex]\left[\begin{array}{ccc|c}5&0&0&5\\1&1&-1&-2\\1&-3&2&-5\end{array}\right][/tex]

Dividing the first row by 5 gives ...

  [tex]\left[\begin{array}{ccc|c}1&0&0&1\\1&1&-1&-2\\1&-3&2&-5\end{array}\right][/tex]

Subtracting this from the second row, and again from the third row, we are left with ...

  [tex]\left[\begin{array}{ccc|c}1&0&0&1\\0&1&-1&-3\\0&-3&2&-6\end{array}\right][/tex]

Multiplying the second row by 3 and adding that to the third row, we get ...

  [tex]\left[\begin{array}{ccc|c}1&0&0&1\\0&1&-1&-3\\0&0&-1&-15\end{array}\right][/tex]

Subtracting the third row from the second gives ...

  [tex]\left[\begin{array}{ccc|c}1&0&0&1\\0&1&0&12\\0&0&-1&-15\end{array}\right][/tex]

Finally, multiplying the last row by -1, we have the solution:

  [tex]\left[\begin{array}{ccc|c}1&0&0&1\\0&1&0&12\\0&0&1&15\end{array}\right][/tex]

This matrix corresponds to the equations ...

_____

The purpose of our choice of row operations is to make the diagonal elements 1 and the off-diagonal elements 0. That is how we end up with the final equations shown.

As we said, there are many ways to go about this. In general, one can ...

Mitchel can transport a 1:1 ratio of goats to calves in his livestock trailer, equating to 11 goats and 11 calves per trip, given their respective average weights of 90 and 360 pounds and the trailer's weight limit of 5000 pounds.

Mitchel has a livestock trailer with a weight limit of 5,000 pounds. Given that each goat weights about 90 pounds and each calf about 360 pounds, we need to calculate how many of each animal the trailer can accommodate within its weight limit, while taking into account their average weights.

Let's look at one possible solution, using goats and calves in a 1:1 ratio. This would be 90 + 360 = 450 pounds per set of goat and calf. To fit within the trailer's weight limit, we then divide the total weight capacity by the weight of one set of animals: 5,000 / 450 = 11.11, which we round down to 11. This tells us that Mitchel can transport 11 goats and 11 calves in his trailer for each trip.

It's important to mention that this distribution can be adjusted according to need, as long as the total weight of the animals is kept below the trailer's weight limit.

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Answer: y= -2/3+8

Step-by-step explanation:

Answer:

h=18m, b=2m

Step-by-step explanation:

Let the base be b and height be h

Condition 1:

b=2h ----(1)

Condition 2:

Area=b*h

b*h=648 -----(2)

Putting (1) into (2)

2h*h=648

2h^2=648

Dividing both sides by 2

h^2=324

Taking sq root on both sides

h=18 m

Now

Putting value of h in eq (1)

b=2(18)

b=36 m

Answer:

Explanation:

Please, find attached an image with the table that accompanies this question.

1. Pattern

The table is:

Years:               2        4       5       8       9

Height (in.):      34      50   58     82    90

The most simple pattern is a linear pattern. A linear pattern has a constante rate of change.

The rate of change between two points is:

Find the rate of change for the data:

Hence, the heigth and the years since the tree was transplantated show a linear relationship: every year the tree grew 8 inches.

2. Intial height:

You can find the initial height of the tree by using the rate of change of the height.

3. Relationship

You can describe the relationship in terms of the initial height and the numbers of years since the tree was transplantated.

Then, the height of the tree is 18 inches plus 8 inches for each year since the tree was transplanted.

You can even write an equation (function):