evaluate the integral. check your results by differentiation. (remember the constant of integration.) (3x3 1)69x2 dx

Answer:

-24

Step-by-step explanation:

-3[-4(3-10)-12]/-2(-1)

-3[-4(-7)-12]/2

-3[28-12]/2

-3[16]/2

-48/2

-24

Hopes this helps

We can change the base formula by:

                           logₐ b = logₓ a /logₓ b

                                         OR

                           logₓ a, logₓ b = logₓ a

The change of base formula is used to change the base of a logarithm.  We know that "log" stands for a logarithm of base 10 and "ln" stands for a logarithm of base e. But there is no option to calculate the logarithm of a number with any other bases other than 10 and e.

The change of base formula solves this issue of changing base from e to 10, and from base 10 to e. Also, it is used in solving several logarithms problems.

Base Formula: -

The change of base formula is used to write a logarithm of a number with a given base as the ratio of two logarithms each with the same base that is different from the base of the original logarithm. This is a property of logarithms. You can see the change of base formula here.

Change of base formula

                            logₐ b = logₓ a /logₓ b

                                         OR

                            logₓ a, logₓ b = logₓ a

Change of Base Formula

Change of base formula can be represented as follows. Here the base of the given logarithm is also changed to a logarithm with a new base. The basic logarithm with a base is transformed to two logarithms with a new and same base. The change of base formula is:

                          logₐ  b = [logₐ x] / [logₐ y]

In this formula,

Change of Base Formula Derivation

Change of base formula derivation can be understood from the following steps. Let us assume that

log ₐ b = p, logₙ a = q, and logₙ b = r.

By converting each of these into exponential form, we get,

a = [tex]b^{p}[/tex], a = [tex]x^{q}[/tex], and b = [tex]n^{r}[/tex]

From the first two equations,

[tex]x^{q}[/tex] = [tex]n^{r}[/tex]

Substituting b =  [tex]n^{r}[/tex] (which is from third equation) here,

([tex]n^{r}[/tex])[tex]^{p}[/tex] = [tex]x^{q}[/tex]

[tex]n^{r}[/tex])[tex]^{p}[/tex] = [tex]x^{q}[/tex](Using a property of exponents)

p r = q

p = q / r

Substituting the values, we get,

logₐ  b = [logₐ x] / [logₐ y]
For Example:

Evaluate the value of log₆₄ 8 using the change of base formula.

Solution:

We will apply the change of base formula (by changing the base to 10). Note that log₁₀ is same as log.

log₆₄ 8 = [log 8] / [log 64]

            = [log 8] / [log 82]

            = [log 8] / [2 log 8]              [∵ log am = m log a]

            = 1 / 2

Answer: log₆₄ 8 = 1 / 2

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Answer:

C

Step-by-step explanation:

The rectangular form of Θ = 3π/4 will be; y + x = 0.

Complex value z is written in a rectangular form as z = x+iy where (x, y) is the rectangular coordinates.

On converting the rectangluar to polar form of the complex number;

x = rcosθ and y = rsinθ

z = r(cosθ+isinθ)

r is the modulus of the complex number and θ is the argument

r =√x²+y² and θ = tan⁻¹y/x

Given the Θ = 3π/4

Therefore, θ = tan⁻¹y/x  = 3π/4

y/x  = tan (3π/4)

Now multiply both side by x

yx/x  = tan (3π/4) x

y =  tan (3π/4) x

Thus, y = -x

in rectangular form y + x = 0

Hence, the rectangular form of Θ = 3π/4 will be; y + x = 0.

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The numbers below to put them in order from least to greatest will be 6.42, 6.453, 6.486, 6.497, 6.53, and 6.66.

A number is a mathematical entity that can be used to count, measure, or name things. For example, 1, 2, 56, etc. are the numbers.

It is given that the numbers are,6.42,6.53,6.497,6.66,6.453,6.486.

Ascending order refers to a list of numbers that are arranged from least to greatest. The numbers are arranged in ascending order in this form. The first figure must be less than the second figure. It is referred to in descending order when the numbers are listed from greatest to least.

Thus, the numbers below to put them in order from least to greatest will be 6.42, 6.453, 6.486, 6.497, 6.53, and 6.66.

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Answer:

Step-by-step explanation:

Define Variables:

Let p= the price of each peach

Let m=the price of each mango

One peach costs $p, so 5 peaches cost 5p. One mango costs $m, so 6 mangos cost 6m. The total cost 5p+6m equals $17.75:

5p+6m=17.75

Likewise, if 2 peaches and 3 mangos cost $8, then:

2p+3m=8

Write System of Equations:

5p+6m=17.75

2p+3m=8

System of equations that could be used to determine the price of each peach and the price of each mango is 5p + 6m = 17.75 and 2p + 3m = 8.

Let p= the price of each peach

Let m=the price of each mango

One peach costs $p, so 5 peaches cost 5p. One mango costs $m, so 6 mangoes cost 6m. The total cost 5p+6m equals $17.75:

5p+6m=17.75

Likewise, if 2 peaches and 3 mangoes cost $8, then:

2p+3m=8

System of Equations:

5p+6m=17.75

2p+3m=8

Therefore System of equations that could be used to determine the price of each peach and the price of each mango is 5p + 6m = 17.75 and 2p + 3m = 8.

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The dimensions of the rectangle which form the dining room is 7 by 5 meters.

The perimeter of the rectangle is 24 meters.

The total length of a shape's boundaries is its perimeter. The total lengths of all other edges and sides are added to find a shape's perimeter.

Let the length be a and the width be b.

Hence 2(a + b) = 24

or, a+b = 12

Now the area of the rectangle is given as 35 square meters.

Therefore ab = 35

now we will use these two equations to find the dimension of the rectangle

a+b = 12

or, a = 12 - b

again:

ab = 35

or, (12 - b)b = 35

or, 12 b - b² = 35

or,  - b² +12b -35 = 0

or,   b² - 12b + 35 = 0

or,  (b-7) (b-5) = 0

either b = 7 or b = 5 .

Therefore the value of a from the equations is 5 or 7.

Therefore the dimensions of the dining room are 7 by 5 meters.

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As per the given graph, the correct statement is option (A) The median of the density curve is 3.

Median:

Basically, median refers the middle value in a set of data.

And it can be obtained by organize and order the data from smallest to largest. And to find the midpoint value, divide the number of observations by two. And then if there are an odd number of observations, round that number up, and the value in that position is the median.

Given,

Here we have the graph.

Now, we have to identify in with of the option is correct according to the given graph.

While we looking into the given graph,

We have identified that it form the triangle and as the middle point as 3.

So, based on this fact me have identified that the median density of the curve is 3.

Therefore, the option (A) The median of the density curve is 3.

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The nearby tower that casts a shadow that is 39.5 m long is 85 m tall.

Ratio is the relationship between two quantities, normally expressed as the quotient of one divided by the other.

The ratio of the shadow of a pole to its length is 1.4 m : 3 m.

At the same time, the ratio of the shadow of a tower to its length should  also be equal to the ratio of the shadow of a pole to its length, which is 1.4 : 3.

Let x = length of the tower

Using proportion, determine the length of the tower by solving for x.

1.4 : 3 = 39.5 m : x

x = 39.5(3)/1.4

x = 84.64

Round off to the nearest meter.

length of the tower = 85 meters

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Answer:

7x10x13=910=910 kilograms

Step-by-step explanation:

The probability of both the dice showing prime numbers is 1/4.

Prime numbers:

The numbers which have only two factors,  one and itself are known as prime numbers.

A dice has 6 faces numbering from 1 to 6.

Prime numbers from 1 to 6 are 2, 3, and 5.

Probability of one dice showing a prime number

                 = (number of favorable outcomes) / (total number of outcomes)

                 = 3/6

                 = 1/2

Probability of two dice showing a prime number = (1/2) x (1/2)

                                                                                 = 1/4

Therefore, the probability of both the dice showing a prime number is 1/4.

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Answer:

4 answers

Step-by-step explanation:

y = |x² - 4|

3 = |x² - 4|

3 = x² - 4     and -3 = x² - 4

3+4=x² - 4 + 4          and -3+4=x² - 4 + 4

7=x² and 1=x²

x=[tex]\sqrt{7}[/tex] and x=-[tex]\sqrt{7}[/tex] since x²=7

x=-1 and x=1 since x²=1

x= -[tex]\sqrt{7}[/tex], [tex]\sqrt{7}[/tex], -1, 1 ==> 4 answers

Value of x is 1.411

Given Eq.

[tex]9^{x+2} = 2^{5x-4} \\\\[/tex]

Applying exponent rules

[tex](x+2)ln(9) = (5x-4)ln(2)\\[/tex]

Solving for x

[tex]x = \frac{4ln(6)}{5ln(2)-2ln(3)}[/tex]

[tex]x = \frac{1.791}{5(0.693)-2(1.098)}[/tex]

[tex]x = \frac{1.791}{3.465-2.196} \\[/tex]

[tex]x = \frac{1.791}{1.269}[/tex]

[tex]x = 1.411[/tex]

Value of x is 1.411

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In the first function have vertical compression by 1/2, in second function have horizontal compresion by 1/4, in third compression have Vertical reflection and in fourth function have Horizontal reflection and horizontal compression by 1/2.

In the given question we have to find the transformations that map the function y = [tex]8^x[/tex] onto each function.

Since we have to find the transformation, so we know that in a transformation;

The transformation from the first equation to the second one can be found by finding a, h and k for each equation.

[tex]y=ab^{x-h}+k[/tex]

The horizontal is define by the value of h. The horizontal shift is defined as:

g(x)=f(x+h) - The graph is shifted to the left h units.

g(x)=f(x−h) - The graph is shifted to the right h units.

Horizontal Shift: None

The vertical shift define on the value of k. The vertical shift is defined as:

g(x)=f(x)+k - The graph is shifted up k units.

g(x)=f(x)−k - The graph is shifted down k units.

Vertical Shift: None

The value of a describes the vertical stretch or compression of the graph.

a>1 is a vertical stretch

0<a<1 is a vertical compression.

(a) The given first function is [tex](\frac{1}{2})8^{x}[/tex].

y = [tex]8^x[/tex]

Combine [tex]8^x[/tex] and 1/2.

y = [tex]8^x[/tex]/2

Assume that

y = [tex]8^x[/tex]  is f(x)=[tex]8^x[/tex] and y= [tex](\frac{1}{2})8^{x}[/tex] is g(x)= [tex](\frac{1}{2})8^{x}[/tex].

On comparing the equation [tex]y=ab^{x-h}+k[/tex]

The value of a, h, and k for f(x)=[tex]8^x[/tex].

a=1, h=0, k=0

The value of a, h and k for g(x)=[tex](\frac{1}{2})8^{x}[/tex].

a=1/2, h=0, k=0

Since the value of h and k so tnere is no horizontal and vertical shift.

SInce the value of a changes from 1 to 1/2 and 1/2 is greater that 0 but less than 1 so there is vertical compression by 1/2.

(b) The given second function is [tex]8^{4x}[/tex].

y= [tex]8^{4x}[/tex] is g(x)= [tex]8^{4x}[/tex].

On comparing the equation [tex]y=ab^{x-h}+k[/tex]

The value of a, h and k for g(x)=[tex]8^{4x}[/tex].

a=1/4, h=0, k=0

Parent Function: f(x)=8^x

So, Horizontal Shift: None, Vertical Shift: None, Reflection about the x-axis: None, Vertical Compression or Stretch: None, horizontal compresion by 1/4.

(c) The given third function is [tex]-8^{x}[/tex].

y= [tex]-8^{x}[/tex] is g(x)= [tex]-8^{x}[/tex].

The value of a, h and k for g(x)=[tex]-8^{x}[/tex].

a=-1, h=0, k=0

Parent Function: f(x)=8^x

So, Horizontal Shift: None, Vertical Shift: None, Reflection about the x-axis: Reflected, Vertical Compression or Stretch: None.

Vertical reflection

(d) The given fourth function is [tex]8^{-2x}[/tex].

y= [tex]8^{-2x}[/tex] is g(x)= [tex]8^{-2x}[/tex].

The value of a, h and k for g(x)=[tex]8^{-2x}[/tex].

a=-1/2, h=0, k=0

Parent Function: f(x)=8^x

So, Horizontal Shift: None, Vertical Shift: None, Reflection about the x-axis: Reflected, Vertical Compression or Stretch: None.

Horizontal reflection and horizontal compression by 1/2.

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Check the picture below.

A) Chi-square test can be used if one of the samples has a value of less than five is the correct option.

Chi-square distribution and Chi-square test for independence:

The assumptions for the chi-square test are:

Hence, options (b), (c), and (d) are the assumptions that are true for chi-square test. Therefore, the options (b), (c), and (d) are considered incorrect.

Hence, option (a), that is, “a Chi-square test can be used if one of the samples has a value of less than five” is the correct option.

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The number of students that are in both the Categories A and B are 16 students.

In the question ,

it is given that ,

the total number of students in category A and category B is 32 students .

which is represented as A union B ,

that is n(AUB) = 32 ,

the number of students in category A = n(A) = 24

the number of students in category B = n(B) = 24

we need to find the number of students that are in both category ,

that is n(A∩B) .

we know that ,

n(AUB) = n(A) + n(B) - n(A∩B)

Substituting the values , we get

n(A∩B) = 24 + 24 - 32

= 48 - 32

= 16

Therefore , The number of students that are in both the Categories are 16 students.

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Answer:

No, Eduardo made a mistake from Step 1 to Step 2. Eduardo should have subtracted 4 from 6 instead of evaluating the exponent. The order of operations says to perform all operations inside parentheses first.

Answer:

For example, a population with a 2% annual growth would have a doubling time of 35 years.Mar 24, 2015

Step-by-step explanation:

Total students who graduated this year from this university is 3000.

This is a normal distribution problem with

Mean , μ = $20,000

Standard deviation, σ = $8,000

If 189 of the recent graduates have salaries of at least $32240.

Then, We first find the percentage of LU graduates with salaries more than $32240.

Required probability = P(x ≥ 32240)

We first normalize or standardize $32,240, then as we know,

z = (x - μ)/σ

= (32240 - 20000)/8000 = 1.53

The required probability:

P(x ≥ 32240) = P(z ≥ 1.53)

We'll use data from the normal probability table for these probabilities

P(x ≥ 32240) = P(z ≥ 1.53) = 1 - P(z < 1.53)

= 1 - 0.93699

= 0.06301

= 6.301%

So, 6.301% of the graduates this year = 189

Total Number of graduates this year = (189/0.06301) = 2999.5 = 3000 graduates this year.

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Answer: 30 weeks

Step-by-step explanation:

let w = number of weeks

you have = 60 + 7w

she has = 120 + 5w

60+7w = 120 + 5w

7w-5w = 120-60

2w = 60

w = 30 weeks

Step-by-step explanation:

in that setup they will both explode after traveling the same distance.

A is just going faster and has to launch therefore a little bit later.

so, we need to make an equation based on distance traveled and find in that process the number of seconds needed to travel that distance.

speed is distance/time, so we need to multiply speed by time to get distance.

x = seconds for B

x - 0.25 = seconds for A

A is launched after B but explodes at the same time.

so, A's trip is of shorter time than B's trip.

(x - 0.25)×360 = x×320

360x - 90 = 320x

40x - 90 = 0

40x = 90

4x = 9

x = 9/4 = 2.25 seconds

both fireworks will explode 2.25 seconds after B launches (and 2.25 - 0.25 = 2 seconds after A launches).

the explosion will be at

320×2.25 = 360×2 = 720 feet height.

The difference quotient for the given quadratic equation is:

d = (2h + 4x + 4)

Here we have the quadratic function:

f(x)= 2x² + 4x - 3

And we want to get the difference quotient that is defined as:

d =  ( f(x + h) - f(x))/(h)

Replacing our function there, we get:

d = ( 2(x + h)² + 4*(x + h) - 3 - (2x² + 4x - 3)/h

d = (2x² + 2h² + 4xh + 4x + 4h - 3 - 2x² - 4x + 3)/h

d = (2h² + 4xh + 4h)/h

d = (2h + 4x + 4)

That is the difference quotient.

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The rate when his distance from second base changes and it is two third of the way to first base is - 13.86 feet/s.

let us consider the following values

distance between batter to Second base = d feet

distance between batter to First base

= x feet

distance between first to second base

= 90 feet

Using the Pythagorean theorem:

d² = x² + 90²

Knowing that the velocity of the bat to first base is 25 ft/s,

the distance x can be written as: x

= 90 - 25t

The equation for is,d² = 90²+(90 -25t)²--(1)

we want to find the rate of change of d.

let it be d' and from implicit differentiation:

2d.d' = 2(90-24t) × (-25)

d' = -50(90 - 25t) / (2d)

d' = -25 (90 - 24t) / d ---(*)

We can determine t and d from the fact that the batter is two third to his first base: d = √90^2 + 60^2 = √1 = 108.2 feet

putting the values of d in (1) we get,

=> 90 - 25t = 60 => 25t = 30 => t = 6/5.

Substitute d' in the formula equation (*)

d' = -25 (90 - 24t) / d = -25(90-30)/108.2

d'= - 13.86

d' = - 13.86 feet/s

negative sign means decreasing distance from second base (as it approaches the first

bases, it approaches the second base).

Hence,The required speed is 13.86 feet/s

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Answer:

x = 62

y = 41

Step-by-step explanation:

y + 21 = x

x + y + 77 = 180

2y+77 = 180

2y = 82

y = 41

x = 62