#6. To raise money a club sells slices of pizza after school. Pepperoni slices cost $3 and cheese slices
cost $2. 12. 7 slices are sold and $326 are raised. How many of each slice were sold?

The final answer is: ∫(2x-1)÷[(x-3)(x+2)]dx = ln|x-3| + ln|x+2| + C

What is Integration ?

In calculus, integration is the inverse operation of differentiation. It is a mathematical technique used to find the integral of a function. The integral of a function f(x) is another function F(x), which gives the area under the curve of f(x) from a certain point to another.

To perform the integration of the given function:

∫(2x-1)÷([tex]x^{2}[/tex]-x-6)dx

First, we need to factor the denominator:

[tex]x^{2}[/tex]- x - 6 = (x-3)(x+2)

So we can rewrite the integral as:

∫(2x-1)÷[(x-3)(x+2)]dx

Next, we need to decompose the fraction into partial fractions:

(2x-1)÷[(x-3)(x+2)] = A÷(x-3) + B÷(x+2)

Multiplying both sides by (x-3)(x+2), we get:

2x-1 = A(x+2) + B(x-3)

Substituting x=3, we get:

5A = 5

A = 1

Substituting x=-2, we get:

-5B = -5

B = 1

So we have:

(2x-1)÷[(x-3)(x+2)] = 1÷(x-3) + 1÷(x+2)

Substituting this back into the integral, we get:

∫(2x-1)÷[(x-3)(x+2)]dx = ∫[1÷(x-3) + 1÷(x+2)]dx

Using the first rule of integration, we get:

∫[1÷(x-3) + 1÷(x+2)]dx = ln|x-3| + ln|x+2| + C

where C is the constant of integration.

Therefore, the final answer is: ∫(2x-1)/[(x-3)(x+2)]dx = ln|x-3| + ln|x+2| + C

To learn more about Integration from given link.

https://brainly.com/question/30018203

#SPJ1

[tex] \displaystyle \int \rm \: \dfrac{2x - 1}{ {x}^{2} - x - 6 } dx[/tex]

Answer:

[tex] \underline{\boxed{\rm = ln |x + 2| + ln |x - 3| + C}}[/tex]

Step-by-step explanation:

[tex] = \displaystyle \int \rm \: \dfrac{2x - 1}{ {x}^{2} - x - 6 } dx[/tex]

[tex] = \displaystyle \int \rm \: \dfrac{2x - 1}{ {x}^{2} - 3x + 2x - 6 } dx[/tex]

[tex] = \displaystyle \int \rm \: \dfrac{2x - 1}{ x(x - 3) + 2(x - 3) } dx[/tex]

[tex] = \displaystyle \int \rm \: \dfrac{2x - 1}{ (x + 2) (x - 3)} dx[/tex]

[tex] \rm \: Let : \displaystyle \rm \: \dfrac{2x - 1}{ (x + 2) (x - 3) } = \dfrac{A }{x + 2} + \dfrac{B}{x - 3} [/tex]

[tex]\rm\implies\displaystyle \rm \: \dfrac{2x - 1}{ (x + 2) (x - 3) } = \dfrac{A(x - 3) + B(x + 2) }{(x + 2)(x - 3)} \\ [/tex]

[tex] \rm \implies\displaystyle \rm \: {2x - 1}{ } = {A(x - 3) + B(x + 2) } \\ [/tex]

Put x = 3 , we get

[tex] \rm \implies\displaystyle \rm \: {6 - 1}{ } = {A(3- 3) + B(3 + 2) } \\ [/tex]

[tex] \rm \implies\displaystyle \rm \: {5}{ } = 5 B \\ [/tex]

[tex] \implies \rm \: B = 1[/tex]

Again

put put x = -2

[tex] \rm \implies\displaystyle \rm \: { - 4- 1}{ } = {A( - 2- 3) + B( - 2 + 2) } \\ [/tex]

[tex] \rm \implies\displaystyle \rm \: { - 5}{ } = {A( - 5) } \\ [/tex]

[tex] \rm \implies\displaystyle \rm A = 1 \\ [/tex]

Thus ,

[tex] \displaystyle \int \rm \: \dfrac{2x - 1}{ (x + 2) (x - 3)} dx = \int\dfrac{1}{x + 2} dx + \int \dfrac{1}{x - 3} dx[/tex]

[tex] \rm = ln |x + 2| + ln |x - 3| + C[/tex]

[tex] \tt\int \dfrac{dx}{ {x}^{2} + {a}^{2} } = \frac{1}{a} { \tan}^{ - 1} \frac{x}{a} + c \\ [/tex]

[tex] \tt\int \dfrac{dx}{ {x}^{2} - {a}^{2} } = \frac{1}{2a} log \frac{x - a}{x + a} + c \\ [/tex]

[tex] \tt\int \dfrac{dx}{ {a}^{2} - {x}^{2} } = \frac{1}{2a} log \frac{a + x}{a - x} + c \\ [/tex]

[tex] \tt\int \: \dfrac{dx}{ \sqrt{ {x}^{2} + {a}^{2} } } = log|x + \sqrt{ {a}^{2} + {x}^{2} } | + c \\ [/tex]

[tex] \tt\int \: \dfrac{dx}{ \sqrt{ {x}^{2} - {a}^{2} } } = log|x + \sqrt{ {x}^{2} - {a}^{2} } | + c \\ [/tex]

[tex] \tt \int \: \dfrac{dx}{ {a}^{2} - {x}^{2} } = { \sin }^{ - 1} \bigg(\dfrac{x}{a} \bigg) + c \\ [/tex]

[tex] \tt \int \: \sqrt{ {x}^{2} + {a}^{2} } dx \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\= \tt \dfrac{x}{2} \sqrt{ {a}^{2} + {x}^{2} } + \dfrac{ {a}^{2} }{2} log |x + \sqrt{ {x}^{2} + {a}^{2} }| + c[/tex]

Answer:

3

Step-by-step explanation:

HCF(a, b) x LCM(a, b) = a x b

We are given that LCM(18, 21) = 126. Let's use this to find the HCF:

HCF(18, 21) x 126 = 18 x 21

HCF(18, 21) = (18 x 21) / 126

HCF(18, 21) = 3

Therefore, the HCF of 18 and 21 is 3.

Answer:

To show how x² + 6x + 5 is equivalent to (x + 5)(x + 1), we can use the FOIL method, which stands for First, Outer, Inner, and Last.

First, we multiply the first term of each factor: x and x, which gives x².

Next, we multiply the outer terms of each factor: x and 1, which gives x.

Then, we multiply the inner terms of each factor: 5 and x, which gives 5x.

Finally, we multiply the last term of each factor: 5 and 1, which gives 5.

Adding up these terms, we get:

x² + x + 5x + 5

Simplifying by combining like terms, we get:

x² + 6x + 5

This is the same as the original expression. Therefore, we have shown that:

x² + 6x + 5 = (x + 5)(x + 1)

Step-by-step explanation:

Answer:

6)180-99=81

81+43=124

180-124=56

b=56

The sum οf [tex](x - 5x^2 - 12)[/tex] and [tex](4 + 11x - 3x^2)[/tex] is [tex]12x - 8x^2 - 8.[/tex]

The term "sum" can alsο be used tο describe a specific sum οf mοney. Yοu cοuld spend a cοnsiderable amοunt οf mοney οn a new car. Hοwever, if yοu tοtal up οr add up all οf its advantages, yοu might be able tο justifiably justify spending sο much.

When yοu add up the cοsts οf everything yοu οrdered at the restaurant, yοu can determine the final tοtal. It's nοt necessary fοr sum tο οnly refer tο numerical values. A summary οr general statement abοut sοmething is what yοu are giving when yοu sum sοmething up.

Tο find the sum οf [tex](x - 5x^2 - 12)[/tex] and [tex](4 + 11x - 3x^2)[/tex], we need tο add the like terms. Like terms are thοse terms that have the same variable raised tο the same pοwer.

Sο, the sum οf  [tex](x - 5x^2 - 12)[/tex] and [tex](4 + 11x - 3x^2)[/tex] is:

[tex](x + 11x) + (-5x^2 - 3x^2) + (-12 + 4)[/tex]

Simplifying, we get:

[tex]12x - 8x^2 - 8[/tex]

Therefοre, the sum οf [tex](x - 5x^2 - 12)[/tex] and [tex](4 + 11x - 3x^2)[/tex]  is [tex]12x - 8x^2 - 8.[/tex]

learn more about sum

https://brainly.com/question/13013054

#SPJ1

I disagree with Brian's assertion that there is less fluctuation in the number of mystery novels simply because the mean number of mystery books is fewer than the mean number of adventure books.

The mean is merely one measure of central tendency and provides no information about the data's dispersion or variability. We would need to look at a measure of dispersion such as the range, variance, or standard deviation to see whether there is less variety in the quantity of mystery novels. If the range or standard deviation of the number of mystery books is less than that of adventure novels, we may claim that the number of mystery books is less variable. If, however, the If the range or standard deviation of the number of mystery novels is greater than that of adventure books, Brian's statement is false. As a result, we cannot establish whether or not there is less fluctuation in the quantity of mystery novels without knowing the dispersion of the data.

learn more about  books here:

https://brainly.com/question/28339193

#SPJ4

Answer:

One possible rational expression that simplifies to 2x/(x+1) and meets the given restrictions is:

(4x^2 - 2x - 3) / [(x + 1)(x - 3)(x - 2)]

To see why this expression simplifies to 2x/(x+1), we can simplify the numerator and denominator separately:

Numerator:

2x(2x-1) = 4x^2 - 2x

Denominator:

(x+1)(x-3)(x-2)

Multiplying the numerator and denominator by -1 gives:

(-2x)(2x-1) / [(3-x)(2-x)(1+x)]

Then, we can rearrange the factors in the denominator to get:

(-2x)(2x-1) / [(x+1)(x-2)(x-3)]

Now we have the desired rational expression that simplifies to 2x/(x+1) and has the given restrictions on x.

Step-by-step explanation:

The approximate volume of rubber used to make the ball is 636.24 cubic centimeters.

To find the volume of the concave space inside the ball, we need to abate the volume of the ball with a  lower compass. The compass of the inside  face of the ball can be  set up by abating the consistence of the rubber from the compass of the outside  face   = r_outside-

consistence  =  6- 2  =  4 centimeters  

The volume of the concave space inside the ball can be  set up using the same formula as below but with the  lower compass   = (4/3)πr_i

nside3  = (4/3) π( 4) 3  

= (4/3) π( 64)

≈268.08 boxy centimeters  

Eventually, the volume of rubber used to make the ball is the difference between the volume of the entire ball and the volume of the concave space inside   =  V-V_hollow  ≈

=904.32-268.08  

=636.24 cubic centimeters

Learn more volumes at

https://brainly.com/question/14846560

#SPJ4

By answering the presented question, we may conclude that As a result, the probability of picking a clown performer from the circus is 1/3, or 0.33. (rounded to two decimal places).

Probabilistic theory is a branch of mathematics that calculates the chance that an event or statement will occur or be true. A risk is a number between 0 and 1, where 1 denotes certainty and 0 indicates how probable an event is to occur. Probability is a mathematical term for the likelihood of a certain event occurring. Probabilities can also be expressed as integers between 0 and 1 or as percentages ranging from 0% to 100%. In relation to all other outcomes, the ratio of occurrences among equally likely alternatives that result in a certain event.

The likelihood of picking a clown performer from the circus is proportional to the number of clowns to the total number of circus artists.

The probability of picking a clown performer is the number of clowns divided by the total number of performers.

So,

The likelihood of picking a clown performer is 5/15.

By dividing the fraction's numerator and denominator by 5, we get:

The likelihood of picking a clown performer is 1/3.

As a result, the likelihood of picking a clown performer from the circus is 1/3, or 0.33. (rounded to two decimal places).

To know more about probability visit:

https://brainly.com/question/11234923

#SPJ1

Answer:

1

Step-by-step explanation:

i think

From the given data, the effectiveness of epoetin alfa, which is a medication used to treat anemia, can be determined through a blood test called a complete blood count (CBC).

The CBC measures the number of red blood cells, white blood cells, and platelets in the blood, as well as the levels of hemoglobin and hematocrit.

In patients receiving epoetin alfa, the goal is to increase the hemoglobin level and hematocrit to a target range that is appropriate for their condition. Therefore, monitoring these levels through regular CBCs can help determine whether the medication is effective and whether the dosage needs to be adjusted.

Learn more about hemoglobin here: brainly.com/question/11211560

#SPJ1

Check the picture below.

Make sure your calculator is in Degree mode.

[tex]\tan(43^o )=\cfrac{\stackrel{opposite}{4.1}}{\underset{adjacent}{x}}\implies x=\cfrac{4.1}{\tan(43^o )}\implies x\approx 4.4[/tex]

Using a trigonometric relation we can see that x = 4.4 units.

Here we can see a right triangle, where we can see that 4.1 is one of the legs, and x is the other leg.

We also can see that the angle adjacent to x is 43°.

Then we can use the trigonometric relation to find the value of x:

tan(43°) = 4.1/x

Solving for x:

x = 4.1/tan(43°)

x = 4.4

Learn more about right triangles at:

https://brainly.com/question/2217700

#SPJ2

The probability of observing an 84°F temperature or higher in LA during a randomly chosen day in June is 0.1151 and the temperature of the coldest 10% of the days (days with lowest average high temperature) during June in LA is approximately 71.6°F.

Let X be the random variable that represents the average daily high temperature in LA in June. Then X ~ N(μ = 78, σ = 5). The probability of observing an 84°F temperature or higher in LA during a randomly chosen day in June is given by: P(X > 84) = P(Z > (84 - 78) / 5) = P(Z > 1.2) = 0.1151 (rounded to 4 decimal places)

To find the temperature of the coldest 10% of the days (days with lowest average high temperature) during June in LA, we need to find the 10th percentile of the distribution. Using a z-score table, we can find the z-score corresponding to the 10th percentile: z = -1.28. Thus, the temperature of the coldest 10% of the days during June in LA is given by: x = μ + zσ= 78 + (-1.28)(5)≈ 71.6°F (rounded to 1 decimal place)

To know more about probability, refer here:

https://brainly.com/question/30144287#

#SPJ11

False: It is the same to reflect the graph of y = cos x across the y-axis as it is across the x-axis.

A 180° rotation about the origin is a transformation that would have the same outcome as a reflection over the x-axis followed by a reflection over the y-axis. The x-coordinate of each point must be negated while reflecting across the Y axis, but the -value must remain unchanged.

By graphing y=-f(x), we may reflect the graph of any function f about the x-axis, and by graphing y=f, we can reflect the graph about the y-axis (-x). By graphing y=-f, we can even reflect it about both axes (-x).

To know more about reflect visit:-

https://brainly.com/question/1527462

#SPJ1

Answer: a. The equation for the total expenses E for n bikers is:

E = 30n + 125n + 700

b. The equation for the total income I for n bikers is:

I = 350n

c. The equation for the profit P for n bikers is:

P = I - E = 350n - (30n + 125n + 700) = 195n - 700

The numbers 8 and 7, which have fixed values that never shift, are the constants.

A coefficient in mathematics is an integer or symbol that multiplies a variable or a variable product. In algebraic formulas, equations, and polynomials, coefficients are used. For instance, the coefficient of the variable x in the equation 3x + 5 is 3, and the coefficient of the constant term 5 is 1. The factors of x and y in the equation 2x + 3y = 7 are 2 and 3, respectively. Coefficients can be whole integers, fractions, or decimals and can have a positive, negative, or zero sign. They aid in the simplification of mathematical expressions and equations and serve to illustrate the relationship between variables.

given

3 coefficients

x is a variable.

8 and 7 are constants.

The coefficient in the equation 3x +  8 = 7 multiplies the variable x by the number 3.

We are looking for an unknown number, represented by the variable x. The numbers 8 and 7, which have fixed values that never shift, are the constants.

To know more about coefficient visit:

https://brainly.com/question/28975079

#SPJ1

Answer:

34

Step-by-step explanation:

In all right triangles, the two shortest sides squared equals the longest side squared, given by the equation [tex]a^2+b^2=c^2[/tex].

We know the two shortest sides, so we can plug in

[tex]16^2+30^2=c^2\\256+900=c^2\\1156=c^2\\c=\sqrt{1156} \\c=34[/tex]

1. 55%

2. 60%

3. 35%

4. 95%

5. 40%

1. The probability that a randomly selected student is male is 0.55 (55%).

2. The probability that a randomly selected female student is registered in the 2-year diploma program is 0.6 (60%).

3. The probability that a randomly selected male student is registered in the 1-year certificate program is 0.35 (35%).

4. The probability that a randomly selected student is female or is registered in a 2-year diploma program is 0.95 (95%).

5. The events “1-year program” and “male” are not independent as the probability of one event affects the probability of the other event. For example, the probability of a randomly selected male student being registered in the 1-year certificate program is 0.35 (35%), which is lower than the overall probability of a randomly selected student being registered in the 1-year certificate program (0.4 or 40%).

Learn more about probability

brainly.com/question/30034780

#SPJ4

You would need to deposit approximately $47.83 now to be able to withdraw $650 at the end of each year for 20 years, assuming an annual interest rate of 11% compounded annually.

It is typically expressed as an annual percentage rate (APR). Interest rates can be fixed, meaning they remain the same for the entire term of the loan, or variable, meaning they can change over time based on market conditions or other factors. The interest rate is an important factor to consider when borrowing money, as it affects the overall cost of the loan. It is also a key factor in investment decisions, as it determines the return on an investment.

To calculate the present value of an investment that will yield a future value of $650 for 20 years at 11% annual interest, we can use the formula for present value of an annuity:

PV = FV / (1 + r)ⁿ

where PV is the present value, FV is the future value, r is the interest rate per period, and n is the number of periods.

In this case, we want to find the present value of a 20-year annuity that pays $650 at the end of each year, with an annual interest rate of 11%. Plugging in the values, we get:

PV = $650 / (1 + 0.11)²⁰

PV = $650 / 13.584

PV = $47.83 (rounded to the nearest cent)

Therefore, you would need to deposit approximately $47.83 now to be able to withdraw $650 at the end of each year for 20 years, assuming an annual interest rate of 11% compounded annually.

To know more about withdraw visit:

https://brainly.com/question/22727019

#SPJ1

Answer:

102

Step-by-step explanation:

(b•h)/2

(12•17)/2

204/2

There will be six possibilities in the sample space if a notebook is selected at random.

the shop sells two types of notebooks and each notebook is offered in red, blue or yellow.

each of the two notebook kinds is available in three different colors.

As a result, there are a total of the following possibilities in the sample space:2 types of notebooks x 3 colors each = 6

Hence, the sample space contains six distinct alternatives.

they are,

Red type 1 notepad

Blue Type 1 notepad

Yellow type 1 notepad

Red type 2 notepad

Blue Type 2 notepad

Yellow type 2 notepad

therefore, the sample space contains six distinct alternatives.

learn more about probabilities

brainly.com/question/30034780

#SPJ4

Answer:

Step-by-step explanation:

708588 hope this helps

Step-by-step explanation:

Let x be the number of suede coats and y be the number of cotton coats. Then, the system of inequalities representing the situation is:

150x + 69y ≥ 6200 (the total cost of the women's coats must be at least $6200)

x ≥ 0, y ≥ 0 (the number of coats cannot be negative)

Note that this system assumes that the store only sells suede and cotton coats for women, and that there are no other costs associated with these coats (such as shipping or storage costs).

The percentage gain is 45%

To calculate the percentage gain on your investment, you need to find the difference between the selling price and the buying price, divide that difference by the buying price, and then multiply by 100 to get the percentage.

The difference between the selling price and the buying price is:

$21.75 - $15 = $ 6.75

So the gain on the investment is $ 6.75 per share.

To find the percentage gain, you divide the gain by the buying price:

$6.75 ÷ $15 = 0.45

Then, multiply by 100 to convert this into a percentage :

0.45 x 100% = 45%

Learn more about percentage gain here

brainly.com/question/24344510

#SPJ4

Answer:

5h

Step-by-step explanation:

300÷60=5

km÷km/h=h

=5h

Using the conversion factor we know that Maya sled down a total of 60 ft of both hills.

A conversion factor is a number that is used to multiply or divide one set of units into another.

If a conversion is required, it must be done using the correct conversion factor to get an identical value.

For instance, 12 inches equals one foot when converting between inches and feet.

Conversion factors' function in the Medicare fee structure.

Every year, the CF is calculated using the CF from the year before, with the Medical Economic Index, the Update Adjustment Factor, Legislative Change, and Budget Neutrality factors are taken into account.

So, we know that:

1 inch = 0.0833333 ft

Then,

432 inches = 36 ft

Then, total ft Maya sled down:

= 24 ft + 36 ft

= 60 ft

Therefore, using the conversion factor we know that Maya sled down a total of 60 ft of both hills.

Know more about the conversion factor here:

https://brainly.com/question/97386

#SPJ1

The length of the ramp is approximately 16 meters.

Trigonometric ratios are ratios of the sides of a right triangle that relate the angles of the triangle to its sides.

There are 3 basic trigonometric ratios are given by

sin θ = opposite side /hypotenuse.

cos θ = adjacent side /hypotenuse.

tan θ  = opposite side /adjacent.

Here we have

The angle of the incline of the ramp is 32 degrees.

The height of the ramp is 10m.

Represent the following data as a right-angled triangle  

where the height of the ramp will be opposite side to the angle of the incline and the length of the ramp will be adjacent side

Let's assume the length of the ramp is "x".  

From the trigonometric ratios,

tan(32°) = 10/x

Multiply both sides by x:

x × tan(32°) = 10

Divide both sides by tan(32°):

x = 10 / tan(32°)

x = 10/0.62

x = 16.0033 meters

x = 16 meters

Therefore,

The length of the ramp is approximately 16 meters.

Learn more about Trigonometric ratios at

https://brainly.com/question/22926215

#SPJ1

P = 83.57% after rounding to the closest two decimal places.

As a result, roughly 83.57% of these procedures are successful and infection-free.

We must deduct the percentage of operations that fail, are infected, or both from 100% in order to get the proportion of operations that are successful and free of infection.

Let P represent the proportion of procedures that are both successful and infection-free.

We are aware that 14% of surgeries fail due to repair failure, and 3% of operations result in infection. Hence, 3% + 14% - 0.57% = 16.43% of surgeries have either an infection or a failure, or both.

Hence, 100% - 16.43% = 83.57% is the proportion of surgeries that are successful and free of infection.

P = 83.57% after rounding to the closest two decimal places.

As a result, roughly 83.57% of these procedures are successful and infection-free.

learn more about decimal places here:

https://brainly.com/question/50455

#SPJ4

The sequence is 3, 9, 15, 21, 27, and the recursive formula for this sequence is a_1 = 3, a_n = a_{n-1} + 6, and the next term is 33.

The sequence is an arithmetic sequence with a common difference of 6, starting at 3. A recursive formula for this sequence can be written as:

a_1 = 3

a_n = a_{n-1} + 6, for n > 1

This formula means that the first term in the sequence is 3, and every subsequent term is found by adding 6 to the previous term.

To find the next term in the sequence, we can use this formula to compute a_6:

a_6 = a_5 + 6

a_6 = 27 + 6

a_6 = 33

Therefore, the next term in the sequence is 33.

The given sequence is an arithmetic sequence, where each term is 6 more than the previous term, starting at 3.

A recursive formula is a mathematical formula that is used to define a sequence in terms of its previous terms. In this case, we can use the recursive formula a_1 = 3 and a_n = a_{n-1} + 6 to define the given sequence. The formula says that the first term in the sequence is 3, and each subsequent term can be found by adding 6 to the previous term.

Using this recursive formula, we can find the next term in the sequence, which is 33. We can continue to apply the formula to find any term in the sequence.

Learn more about arithmetic sequence here:

https://brainly.com/question/15412619

#SPJ4

Answer:

6 + 5h
h = The number of hours he can work in the yard.