Why is 12 * 10-8 is NOT a correct representation of scientific notation?

Answers

Answer 1

Answer:

see below

Step-by-step explanation:

Scientific notation is a * 10 ^b

a must be a number between 1 ( including 1 ) and less than 10

12 is greater than 10 so it is not scientific notation


Related Questions

what is the domain of f(x)=(1/4)^x

Answers

Answer:

B All real numbers

hope you wil understand

Answer:

[tex]\boxed{\sf B. \ All \ real \ numbers}[/tex]

Step-by-step explanation:

The domain is all possible values for x.

[tex]f(x)=(\frac{1}{4} )^x[/tex]

There are no restrictions on the value of x.

The domain is all real numbers.

2. An economist reports that 576 out of a sample of 1,200 middle-income American households participate in the stock market. A confidence interval of [0.468, 0.492] was calculated. What confidence level was used in this calculation

Answers

Answer:

Confidence level  = 59.46%

Step-by-step explanation:

Given that:

An economist reports that 576 out of a sample of 1,200 middle-income American households participate in the stock market.

sample mean = 576

sample size = 1200

The sample proportion [tex]\hat p[/tex] = x/n

The sample proportion [tex]\hat p[/tex] = 576/1200 = 0.48

A confidence interval of [0.468, 0.492] was calculated. What confidence level was used in this calculation?

The  confidence interval level can be determined by using the formula:

[tex]M.E =Z_{critical} \times \sqrt{\dfrac{\hat p (1- \hat p)}{n}}[/tex]

If the calculated confidence interval was [0.468, 0.492]

Then,

[tex]\hat p[/tex]  - M.E = 0.468

0.48 -M.E = 0.468

0.48 - 0.468 = M.E

0.012 = M.E

M.E = 0.012

NOW;

[tex]0. 012 =Z_{critical} \times \sqrt{\dfrac{0.48 (1- 0.48)}{1200}}[/tex]

[tex]0. 012 =Z_{critical} \times \sqrt{\dfrac{0.48 (0.52)}{1200}}[/tex]

[tex]0. 012 =Z_{critical} \times \sqrt{\dfrac{0.2496}{1200}}[/tex]

[tex]0. 012 =Z_{critical} \times \sqrt{2.08\times10^{-4}}[/tex]

[tex]0. 012 =Z_{critical} \times 0.01442[/tex]

[tex]\dfrac{0. 012}{0.01442} =Z_{critical}[/tex]

[tex]Z_{critical} =0.8322[/tex]

From the standard normal tables,

the p - value at [tex]Z_{critical} =0.8322[/tex] =  0.7973

Since the test is two tailed

[tex]1 - \alpha/2= 0.7973[/tex]

[tex]\alpha/2= 1-0.7973[/tex]

[tex]\alpha/2= 0.2027[/tex]

[tex]\alpha= 0.2027 \times 2[/tex]

[tex]\alpha= 0.4054[/tex]

the level of significance = 0.4054

Confidence level = 1 - level of significance

Confidence level = 1 - 0.4054

Confidence level = 0.5946

Confidence level  = 59.46%

Independent random samples taken on two university campuses revealed the following information concerning the average amount of money spent on textbooks during the fall semester.

University A University B
Sample Size 50 40
Average Purchase $260 $250
Standard Deviation (s) $20 $23

We want to determine if, on the average, students at University A spent more on textbooks then the students at University B.

a. Compute the test statistic.
b. Compute the p-value.
c. What is your conclusion? Let α = 0.05.

Answers

Answer:

The calculated Z= 10/4.61 = 2.169

The P value is 0.975 .

Since the calculated value of z= 2.169 falls in the rejection region we therefore reject the null hypothesis at 5 % significance level . On the basis of this we conclude that the students at University A do not spend more on textbooks then the students at University B.

Step-by-step explanation:

We set up our hypotheses as

H0 : x 1= x2 and Ha: x1 ≠ x2

We specify significance level ∝= 0.05

The test statistic if H0: x1= x2 is true is

Z =  [tex]\frac{x_1-x_2}\sqrt\frac{s_1^2}{n_1}+ \frac{s_2^2}{n_2}[/tex]

Z = 260-250/ √400/50 + 529/40

Z= 10 / √8+ 13.225

Z= 10/4.61 = 2.169

The critical value for two tailed test at alpha=0.05 is ± 1.96

The P value is 0.975 .

It is calculated by dividing alpha by 2 for a two sided test and subtracting from 1. When we subtract   0.025 ( 0.05/2)from 1 we get 0.975

Since the calculated value of z= 2.169 falls in the rejection region we therefore reject the null hypothesis at 5 % significance level . On the basis of this we conclude that the students at University A do not spend more on textbooks then the students at University B.

Factor.
x2 - 7x + 10

(x - 10)(x + 1)
(x + 1)(x - 10)
(x - 5)(x - 2)
(x + 5)(x + 2)

Answers

Answer:

The answer is option C

Step-by-step explanation:

x² - 7x + 10

To factor the expression rewrite - 7x as a difference

That's

x² - 5x - 2x + 10

Factor out x from the expression

x( x - 5) - 2x + 10

Factor - 2 from the expression

x(x - 5) - 2( x - 5)

Factor out x - 5 from the expression

The final answer is

( x - 2)(x - 5)

Hope this helps you

20 points!
Please help.

Answers

Man this is a hard one!

Last question of the day!!

Answers

Answer:

Correct options are 2, 5 and 7.

Step-by-step explanation:

Consider the given vertices of triangle are A(-3,-3), B(-3,2) and C(1,2).

Distance formula:

[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]

Using distance formula, we get

[tex]AB=\sqrt{(-3-(-3))^2+(2-(-3))^2}[/tex]

[tex]AB=\sqrt{(0)^2+(5)^2}[/tex]

[tex]AB=\sqrt{25}[/tex]

[tex]AB=5[/tex]

Similarly,

[tex]BC=\sqrt{(1-(-3))^2+(2-2)^2}=4[/tex]

[tex]AC=\sqrt{(1-(-3))^2+(2-(-3))^2}=\sqrt{16+25}=\sqrt{41}[/tex]

From the above calculation it is clear that AC>AB and AC>BC.

According to Pythagoras theorem, in a right angle triangle, the square of largest side is equal to the sum of squares of two small sides.

[tex]hypotenuse^2=base^2+perpendicular^2[/tex]

[tex]AC^2=(\sqrt{41})^2=41[/tex]

[tex]AB^2+BC^2=(5)^2+4^2=24+16=41=AC^2[/tex]

So, given triangle is a right angle triangle and AC is its hypotenuse.

Therefore, the correct options are 2, 5 and 7.

The graph below shows the quadratic function f, and the table below shows the quadratic function g.



x -1 0 1 2 3 4 5
g(x) 13 8 5 4 5 8 13

Which statement is true?

A.
The functions f and g have the same axis of symmetry and the same y-intercept.
B.
The functions f and g have different axes of symmetry and different y-intercepts.
C.
The functions f and g have the same axis of symmetry, and the y-intercept of f is greater than the y-intercept of g.
D.
The functions f and g have the same axis of symmetry, and the y-intercept of f is less than the y-intercept of g.

Answers

Answer:

D

Step-by-step explanation:

The true statement is:

The functions f and g have the same axis of symmetry, and the y-intercept of f is greater than the y-intercept of g.

What is Function?

A mathematical phrase, rule, or law that establishes the link between an independent variable and a dependent variable (the dependent variable). In mathematics, functions exist everywhere, and they are crucial for constructing physical links in the sciences.

As, per the graph and table is:

From the graph of f(x):

Axis of symmetry will be at x = 2

The maximum value of f(x) = 10

From the table of g(x):

Axis of symmetry will be at x = 2

The minimum value of g(x) = 4

thus, The functions f and g have the same axis of symmetry, and the y-intercept of f is greater than the y-intercept of g.

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Find the area of the shape shown below.
3.5
2
2

Answers

Answer:

26.75 units²

Step-by-step explanation:

Cube Area: A = l²

Triangle Area: A = 1/2bh

Step 1: Find area of biggest triangle

A = 1/2(3.5)(2 + 2 + 5)

A = 1.75(9)

A = 15.75

Step 2: Find area of 2nd biggest triangle

A = 1/2(5)(2)

A = 1/2(10)

A = 5

Step 3: Find area of smallest triangle

A = 1/2(2)(2)

A = 1/2(4)

A = 2

Step 4: Find area of cube

A = 2²

A = 4

Step 5: Add all the values together

A = 15.75 + 5 + 2 + 4

A = 20.75 + 2 + 4

A = 22.75 + 4

A = 26.75

A website developer wished to analyze the clicks per day on their newly updated website. Let the mean number of clicks per day be μ. If the website developer wants to know if the number of clicks per day is different than 200 clicks a day, on average, what are the null and alternative hypotheses?

Answers

Answer:

Null Hypothesis, [tex]H_0[/tex] : [tex]\mu[/tex] = 200 clicks a day

Alternate Hypothesis, [tex]H_A[/tex] : [tex]\mu\neq[/tex] 200 clicks a day

Step-by-step explanation:

We are given that a website developer wished to analyze the clicks per day on their newly updated website.

The website developer wants to know if the number of clicks per day is different than 200 clicks a day, on average.

Let [tex]\mu[/tex] = mean number of clicks per day.

So, Null Hypothesis, [tex]H_0[/tex] : [tex]\mu[/tex] = 200 clicks a day

Alternate Hypothesis, [tex]H_A[/tex] : [tex]\mu\neq[/tex] 200 clicks a day

Here, the null hypothesis states that the mean number of clicks per day is 200 clicks a day.

On the other hand, the alternate hypothesis states that the mean number of clicks per day is different than 200 clicks a day.

Hence, this is the correct null and alternative hypotheses.

Answer: Null Hypothesis [tex]H_0:\mu=200[/tex]

Alternate Hypothesis[tex]H_a:\mu\neq200[/tex]

Step-by-step explanation:

Let [tex]\mu[/tex] be the mean number of clicks per day.

Given, a website developer wished to analyze the clicks per day on their newly updated website.

The website developer wants to know if the number of clicks per day is different than 200 clicks a day, on average.

i.e. he wants to check either [tex]\mu=200\text{ or }\mu\neq 200[/tex]

Since a null hypothesis is a hypothesis believes that there is no difference between the two variables whereas an alternative hypothesis believes that there is a statistically significant difference between two variables.

So,  Null Hypothesis [tex]H_0:\mu=200[/tex]

Alternate Hypothesis[tex]H_a:\mu\neq200[/tex]

Hence, the required null and alternative hypotheses.

Null Hypothesis [tex]H_0:\mu=200[/tex]

Alternate Hypothesis[tex]H_a:\mu\neq200[/tex]

Fill in the blanks and explain the pattern

0,1,1,2,3,5,__,__,21,34,55

Answers

Answer:

8,13

Step-by-step explanation:

Look at the pattern :

0,1,1,2,3,5,...,...,21,34,55.

As you see the number in the pattern was made by the sum of 2 numbers behind it. Then, the blanks must be filled by :

3 + 5 = 88 + 5 = 13

So, the blanks must be filled by 8 and 13

Answer:

In the two blanks would be 8, 13.

The pattern is practically the Fibonacci Code.

Step-by-step explanation:

The Fibonacci Code is a mathematical sequencing in which you start with two numbers and add them together to make the third number, then you add the third number and the second number together.  Practically you keep adding each new sum and the number before it in the sequence to find the next new sum.

After 55 in this pattern, the pattern would go 89, 144, 233, 377, 610, 987,...

Simplify to create an equivalent expression.
-k-(-8k+7)
a=7k−7
b=-7k-7
c=7k+7
d=-7k+7
choose one

Answers

Answer:

a. 7k - 7

Step-by-step explanation:

Step 1: Write out expression

-k - (-8k + 7)

Step 2: Distribute negative

-k + 8k - 7

Step 3: Combine like terms

7k - 7

And we have our answer!

the fourth term of an AP is 5 while the sum of the first 6 terms is 10. Find the sum of the first 19 terms​

Answers

Answer: S₁₉ = 855

Step-by-step explanation:

T₄ = a + ( n - 1 )d  = 5 , from the statement above , but n = 4

       a + 3d  = 5 -------------------------1

S₆ = ⁿ/₂[(2a + ( n - 1 )d]  =  10, where n = 6

    = ⁶/₂( 2a + 5d )         = 10

    = 3( 2a + 5d ) = 10

    = 6a + 15d      = 10 -----------------2

Now solve the two equation together simultaneously to get the values of a and d

   a + 3d     = 5

   6a + 15d = 10

from 1,

a = 5 - 3d -------------------------------3

Now put (3) in equation 2 and open the brackets

6( 5 - 3d )  + 15d = 10

30 - 18d + 15d      = 10

30 - 3d                 = 10

            3d            = 30 - 10

             3d           = 20

                         d = ²⁰/₃.

Now substitute for d to get a in equation 3

           a = 5 - 3( ²⁰/₃)

           a = 5 - 3 ₓ ²⁰/₃

              = 5 - 20

          a  = -15.

Now to find the sum of the first 19 terms,

we use the formula

S₁₉ = ⁿ/₂( 2a + ( n - 1 )d )

     = ¹⁹/₂( 2 x -15 + 18 x ²⁰/₃ )

     = ¹⁹/₂( -30 + 6 x 20 )

     = ¹⁹/₂( -30 + 120 )

     = ¹⁹/₂( 90 )

     = ¹⁹/₂ x 90

     = 19 x 45

     = 855

Therefore,

S₁₉ = 855

 

5 STARS IF CORRECT! In general, Can you translate a phrase or sentence into symbols? Explain the answer.

Answers

Answer:

Step-by-step explanation:

I answered this already a few minutes ago.

Answer:

yes you can

Step-by-step explanation:

you can write algebraic expressions and use variables for the unknown

The areas of two similar octagons are 4 m² and 9 m². What is the scale factor of their side lengths? PLZ PLZ HELP PLZ

Answers

Answer:

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

Step-by-step explanation:

Area of Octagon A = 4 m²

Side length of Octagon A = a

Area of Octagon B = 9 m²

Side length of Octagon B = b

The scale factor of their side lengths = [tex] \frac{a}{b} [/tex]

According to the area of similar polygons theorem, [tex] \frac{4}{9} = (\frac{a}{b})^2 [/tex]

Thus,

[tex] \sqrt{\frac{4}{9}} = \frac{a}{b} [/tex]

[tex] \frac{\sqrt{4}}{\sqrt{9}} = \frac{a}{b} [/tex]

[tex] \frac{2}{3} = \frac{a}{b} [/tex]

Scale factor of their sides = [tex] \frac{2}{3} [/tex]

Answer:

3:5

Step-by-step explanation:

square root of 9 is 3.

square root if 25 is 5.

therefore, 3:5.

A right triangle has the following vertices Find the area of the triangle
(7,-3) (4,-3) (4,9)
20 pnts

Answers

Answer:

Area = 18 square units

Step-by-step explanation:

To find the area of the triangle, let's go through the following steps:

(i) Let the vertices be;

A = (7, -3)

B = (4, -3)

C = (4, 9)

(ii) The sides of the triangle are therefore,

AB, BC and CA

(iii) Using the distance formula, calculate the lengths of AB, BC and CA

[tex]AB = \sqrt{(7-4)^2 + ( -3 - (-3))^2}[/tex]

[tex]AB = \sqrt{3^2 + (0)^2}\\[/tex]

[tex]AB = \sqrt{9}[/tex]

[tex]AB = 3[/tex]

[tex]BC = \sqrt{(4-4)^2 + ( -3 - 9)^2}[/tex]

[tex]BC = \sqrt{0^2 + (-12)^2}[/tex]

[tex]BC = \sqrt{144}[/tex]

[tex]BC = 12[/tex]

[tex]CA = \sqrt{(4-7)^2 + ( 9 - (-3))^2}[/tex]

[tex]CA = \sqrt{(-3)^2 + (12)^2}[/tex]

[tex]CA = \sqrt{9 + 144}[/tex]

[tex]CA = \sqrt{153}[/tex]

[tex]CA = 12.4[/tex]

(iv) Now that we have all the sides, let's calculate the area of the triangle using the Heron's formula.

Area = [tex]\sqrt{p(p-a)(p-b)(p-c)}[/tex]

Where;

p = [tex]\frac{a + b + c}{2}[/tex]

a, b and c are the sides of the triangle.

In our case,

let

a = AB = 3

b = BC = 12

c = CA = 12.4

∴ p = [tex]\frac{3 + 12 + 12.4}{2}[/tex]

p = 13.7

Area = [tex]\sqrt{p(p-a)(p-b)(p-c)}[/tex]

Area = [tex]\sqrt{13.7(13.7-3)(13.7-12)(13.7-12.4)}[/tex]

Area =  [tex]\sqrt{13.7(10.7)(1.7)(1.3)}[/tex]

Area =  [tex]\sqrt{323.9639}[/tex]

Area  = 17.999

Area = 18 square units

OR

To get the area of the triangle, we can use a much simpler approach.

Since the triangle is a right triangle,

(i) the hypotenuse, which is the longest side is CA = 12.4

(ii) the other two sides are AB and BC. These two sides will form the right angle.

Therefore, we can use the relation:

Area = [tex]\frac{1}{2}[/tex] x base x height

Where;

the base or height can either be AB or BC

Area = [tex]\frac{1}{2}[/tex] x 3 x 12

Area = 18 square units

PS: In a right triangle, the other two sides apart from the hypotenuse form the right angle.

Can somebody please solve this problem for me!

Answers

Answer:

x = 200.674

Step-by-step explanation:

tan∅ = opposite/adjacent

Step 1: Find length of z

tan70° = 119/z

ztan70° = 119

z = 119/tan70°

z = 43.3125

Step 2: Find length z + x (denoted as y)

tan26° = 119/y

ytan26° = 119

y = 119/tan26°

y = 243.986

Step 3: Find x

y - z = x

243.986 - 43.3125 = x

x = 200.674

What is the value of this expression when x = -6 and y = — 1/2? 4(x^2+3) -2y A. -131 B. -35 C. 57 1/2 D. 157

Answers

Answer:

D

Step-by-step explanation:

[tex]4(x^2+3)-2y\\\\=4((-6)^2+3)-2(\frac{-1}{2} )\\\\=4(36+3)+1\\\\=4(39)+1\\\\=156+1\\\\=157[/tex]

The value of the expression 4(x² + 3) - 2y is 157, when x = -6 and y = -1/2.

What is an algebraic expression?

An algebraic expression is consists of variables, numbers with various mathematical operations,

The given expression is,

4(x² + 3) - 2y

Substitute x = -6 and y = -1/2 to find the value of expression,

= 4 ((-6)² + 3) - 2(-1/2)

= 4 (36 + 3) + 1

= 4 x 39 + 1

= 156 + 1

= 157

The required value of the expression is 157.

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A poker hand consisting of 7 cards is dealt from a standard deck of 52 cards. Find the probability that the hand contains exactly 3 face cards. Leave your answer as a reduced fraction.

Answers

Answer:

The probability is 2,010,580/13,378,456

Step-by-step explanation:

Here is a combination problem.

We want to 7 cards from a total of 52.

The number of ways to do this is 52C7 ways.

Also, we know there are 12 face cards in a standard deck of cards.

So we are selecting 3 face cards from this total of 12.

So also the number of cards which are not face cards are 52-12 = 40 cards

Out of all these 40, we shall be selecting exactly 4. The number of ways to do this 40C4

Thus, the required probability will be;

(40C4 * 12C3)/52C7 = (91,390 * 220)/133,784,560

= 20,105,800/133,784,560 = 2,010,580/13,378,456

Identify each x-value at which the slope of the tangent line to the function f(x) = 0.2x^2 + 5x − 12 belongs to the interval (-1, 1).

Answers

Answer:

Step-by-step explanation:

Hello, the slope of the tangent is the value of the derivative.

f'(x) = 2*0.2x + 5 = 0.4x + 5

So we are looking for

[tex]-1\leq f'(x) \leq 1 \\ \\<=> -1\leq 0.4x+5 \leq 1 \\ \\<=> -1-5=-6\leq 0.4x \leq 1-5=-4 \\ \\<=> \dfrac{-6}{0.4}\leq 0.4x \leq \dfrac{-4}{0.4} \\\\<=> \boxed{-15 \leq x\leq -10}[/tex]

Hope this helps.

Do not hesitate if you need further explanation.

Thank you

Using derivatives, it is found that the x-values in which the slope belong to the interval (-1,1) are in the following interval is (-15,-10).

What is the slope of the tangent line to a function f(x) at point x = x_0?

It is given by the derivative at x = x_0, that is:

m = f'(x_0)

In this problem, the function is:

f(x) = 0.2x^2 + 5x − 12

Hence the derivative is:

f'(x) = 0.4x + 5

For a slope of -1, we have that,

0.4x + 5 = -1

0.4x = -6

x = -15.

For a slope of 1, we have that,

0.4x + 5 = 1.

0.4x = -4

x = -10

Hence it is found that the x-values in which the slope belong to the interval (-1,1) are in the following interval is (-15,-10).

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Emily made a pot of cream of pumpkin soup for thanksgiving dinner. She put 5
cups of cream in the soup. She poured the soup into 24 small soup bowls. How
much cream (measured in oz.) is used for each small bowl of soup?

Answers

Answer:

1 2/3 ounces in each bowl

Step-by-step explanation:

We need to convert 5 cups to ounces

1 cup = 8 ounces

5 cups = 5*8 = 40 ounces

We divide the 40 ounces into 24 bowls

40 ounces / 24 bowl

5/3 ounces per bowl

1 2/3 ounces in each bowl

Answer:

each bowl can contain 5/3 oz. of soup.

Step-by-step explanation:

1 cup = 8 oz.

                   8 oz.

5 cups x --------------  =  40 oz.

                    1 cup

to get the measurement of each bowl,

40 oz. divided into 24 bowls.

therefore, each bowl can contain 5/3 oz. of soup.

The graphs below have the same shape. Complete the equation of the blue
graph. Enter exponents using the caret (; for example, enter x2 as x^2.

Answers

Answer: (x+4)^2-3

This is equivalent to x^2+8x+13

======================================================

Explanation:

The vertex of f is at (0,0). The vertex of g is at (-4, -3). The vertex has moved 4 units to the left and 3 units down.

To shift to the left, we replace x with x+4. So we have f(x) = x^2 turn into f(x+4) = (x+4)^2 so far.

Then we subtract off 3 to move everything 3 units down. We have

g(x) = (x+4)^2-3

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

Optionally we can expand things out like so

g(x) = (x+4)^2-3

g(x) = (x+4)(x+4)-3

g(x) = x(x+4)+4(x+4)-3

g(x) = x^2+4x+4x+16-3

g(x) = x^2+8x+13

Showing that (x+4)^2-3 is equivalent to x^2+8x+13

Question 2 Rewrite in simplest radical form 1 x −3 6 . Show each step of your process.

Answers

Answer:

√(x)

Step-by-step explanation:

(1)/(x^-(1/2)) that's 3 goes into -3 leaving 1 and goes into 6 leaving 2

1/2 is same as 2^-1

so therefore we can simplify the above as

x^-(-1/2)

x^(1/2)

and 4^(1/2)

is same as √(4)

so we conclude as

√(x)

Suppose the radius of a circle is 5 units. What is its circumference?​

Answers

Answer:

C≈31.42

Step-by-step explanation:

C=2πr

C=2xπx5

C≈31.42

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Tom is afraid of heights above 9 feet. He is asked to repair a side of a high deck. The bottom of the ladder must be placed 6 feet from a deck. The ladder is 10 feet long. How far above the ground does the ladder touch the deck? Is Tom afraid of the height?

Answers

Answer:

8 ftno

Step-by-step explanation:

The height on the side of the deck (h) can be found using the Pythagorean theorem. It tells you ...

  6^2 + h^2 = 10^2

  h = √(10^2 -6^2) = √64 = 8

The ladder touches the deck 8 feet above the ground. Tom is not afraid of that height.

What word phrase can you use to represent the algebraic expression 7x?

A. 7 more than a number x
B. the product of 7 and a number x
C. the quotient of 7 and a number x
D. 7 less than a number x

Answers

Answer:

B. the product of 7 and a number x

Step-by-step explanation:

7x is 7 multiplied by x.

Answer:

b is the product

Step-by-step explanation:

Use spherical coordinates. Evaluate e x2 + y2 + z2 dV, E where E is enclosed by the sphere x2 + y2 + z2 = 25 in the first octant.

Answers

Answer:

[tex]\displaystyle \iiint_E {e^{\sqrt{x^2 + y^2 + z^2}}} \, dV = \frac{\pi (17e^5 - 2)}{2}[/tex]

General Formulas and Concepts:
Calculus

Integration

Integrals

Integration Rule [Reverse Power Rule]:
[tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]

Integration Rule [Fundamental Theorem of Calculus 1]:
[tex]\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)[/tex]

Integration Property [Multiplied Constant]:
[tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]

Integration Property [Addition/Subtraction]:
[tex]\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx[/tex]

Integration Method [Integration by Parts]:
[tex]\displaystyle \int {u} \, dv = uv - \int {v} \, du[/tex]

[IBP] LIPET: Logs, Inverses, Polynomials, Exponentials, Trig

Multivariable Calculus

Triple Integrals

Cylindrical Coordinate Conversions:

[tex]\displaystyle x = r \cos \theta[/tex][tex]\displaystyle y = r \sin \theta[/tex][tex]\displaystyle z = z[/tex][tex]\displaystyle r^2 = x^2 + y^2[/tex][tex]\displaystyle \tan \theta = \frac{y}{x}[/tex]

Spherical Coordinate Conversions:

[tex]\displaystyle r = \rho \sin \phi[/tex][tex]\displaystyle x = \rho \sin \phi \cos \theta[/tex][tex]\displaystyle z = \rho \cos \phi[/tex][tex]\displaystyle y = \rho \sin \phi \sin \theta[/tex][tex]\displaystyle \rho = \sqrt{x^2 + y^2 + z^2}[/tex]

Integral Conversion [Spherical Coordinates]:
[tex]\displaystyle \iiint_T {f( \rho, \phi, \theta )} \, dV = \iiint_T {\rho^2 \sin \phi} \, d\rho \, d\phi \, d\theta[/tex]

Step-by-step explanation:

*Note:

Recall that φ is bounded by 0 ≤ φ ≤ 0.5π from the z-axis to the x-axis.

I will not show/explain any intermediate calculus steps as there isn't enough space.

Step 1: Define

Identify given.

[tex]\displaystyle \iiint_E {e^{\sqrt{x^2 + y^2 + z^2}}} \, dV[/tex]

[tex]\displaystyle \text{Region E:} \ x^2 + y^2 + z^2 = 25 \ \text{bounded by first octant}[/tex]

Step 2: Integrate Pt. 1

Find ρ bounds.

[Sphere] Substitute in Spherical Coordinate Conversions:
[tex]\displaystyle \rho^2 = 25[/tex]Solve:
[tex]\displaystyle \rho = 5[/tex]Define limits:
[tex]\displaystyle 0 \leq \rho \leq 5[/tex]

Find θ bounds.

[Sphere] Substitute in z = 0:
[tex]\displaystyle x^2 + y^2 = 25[/tex][Circle] Graph [See 2nd Attachment][Graph] Identify limits [Unit Circle]:
[tex]\displaystyle 0 \leq \theta \leq \frac{\pi}{2}[/tex]

Find φ bounds.

[Circle] Substitute in Cylindrical Coordinate Conversions:
[tex]\displaystyle r^2 = 25[/tex]Solve:
[tex]\displaystyle r = 5[/tex]Substitute in Spherical Coordinate Conversions:
[tex]\displaystyle \rho \sin \phi = 5[/tex]Solve:
[tex]\displaystyle \phi = \frac{\pi}{2}[/tex]Define limits:
[tex]\displaystyle 0 \leq \phi \leq \frac{\pi}{2}[/tex]

Step 3: Integrate Pt. 2

[Integrals] Convert [Integral Conversion - Spherical Coordinates]:
[tex]\displaystyle \iiint_E {e^{\sqrt{x^2 + y^2 + z^2}}} \, dV = \iiint_E {e^{\sqrt{x^2 + y^2 + z^2}} \rho^2 \sin \phi} \, d\rho \, d\phi \, d\theta[/tex][dρ Integrand] Rewrite [Spherical Coordinate Conversions]:
[tex]\displaystyle \iiint_E {e^{\sqrt{x^2 + y^2 + z^2}}} \, dV = \iiint_E {e^{\rho} \rho^2 \sin \phi} \, d\rho \, d\phi \, d\theta[/tex][Integrals] Substitute in region E:
[tex]\displaystyle \iiint_E {e^{\sqrt{x^2 + y^2 + z^2}}} \, dV = \int\limits^{\frac{\pi}{2}}_0 \int\limits^{\frac{\pi}{2}}_0 \int\limits^5_0 {e^{\rho} \rho^2 \sin \phi} \, d\rho \, d\phi \, d\theta[/tex]

We evaluate this spherical integral by using the integration rules, properties, and methods listed above:

[tex]\displaystyle \begin{aligned} \iiint_E {e^{\sqrt{x^2 + y^2 + z^2}}} \, dV & = \int\limits^{\frac{\pi}{2}}_0 \int\limits^{\frac{\pi}{2}}_0 \int\limits^5_0 {e^{\rho} \rho^2 \sin \phi} \, d\rho \, d\phi \, d\theta \\ & = \int\limits^{\frac{\pi}{2}}_0 \int\limits^{\frac{\pi}{2}}_0 {\bigg[ (\rho^2 - 2 \rho + 2) e^{\rho} \sin \phi \bigg] \bigg| \limits^{\rho = 5}_{\rho = 0}} \, d\phi \, d\theta\end{aligned}[/tex]

[tex]\displaystyle \begin{aligned}\iiint_E {e^{\sqrt{x^2 + y^2 + z^2}}} \, dV & = \int\limits^{\frac{\pi}{2}}_0 \int\limits^{\frac{\pi}{2}}_0 {(17e^5 - 2) \sin \phi} \, d\phi \, d\theta \\& = \int\limits^{\frac{\pi}{2}}_0 {\bigg[ -(17e^5 - 2) \cos \phi \bigg] \bigg| \limits^{\phi = \frac{\pi}{2}}_{\phi = 0}} \, d\theta \\& = \int\limits^{\frac{\pi}{2}}_0 {17e^5 - 2} \, d\theta \\& = (17e^5 - 2) \theta \bigg| \limits^{\theta = \frac{\pi}{2}}_{\theta = 0} \\& = \frac{\pi (17e^5 - 2)}{2}\end{aligned}[/tex]

∴ the given integral equals [tex]\displaystyle \bold{\frac{\pi (17e^5 - 2)}{2}}[/tex].

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Learn more about spherical coordinates: https://brainly.com/question/16415822

Learn more about multivariable calculus: https://brainly.com/question/4746216

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Topic: Multivariable Calculus

Unit: Triple Integrals Applications

please help with this

Answers

Answer:

[tex]\sin \left(\theta \right)-\frac{1}{2}\cos \left(2\theta \rightt)+C[/tex]

Step-by-step explanation:

We are given the graph of r = cos( θ ) + sin( 2θ ) so that we are being asked to determine the integral. Remember that [tex]\:r=cos\left(\theta \right)+sin\left(2\theta \right)[/tex] can also be rewritten as [tex]\int \cos \left(\theta \right)+\sin \left(2\theta \right)d\theta \right[/tex].

Let's apply the functional rule [tex]\int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx[/tex],

[tex]\int \cos \left(\theta \right)+\sin \left(2\theta \right)d\theta \right[/tex] = [tex]\int \cos \left(\theta \right)d\theta \right+\int \sin \left(2\theta \right)d\theta \right[/tex]

At the same time [tex]\int \cos \left(\theta \right)d\theta \right=\sin \left(\theta \right)[/tex] = [tex]sin( \theta \right ))[/tex], and [tex]\int \sin \left(2\theta \right)d\theta \right[/tex] = [tex]-\frac{1}{2}\cos \left(2\theta \right)[/tex]. Let's substitute,

[tex]\int \cos \left(\theta \right)d\theta \right+\int \sin \left(2\theta \right)d\theta \right[/tex] = [tex]\sin \left(\theta \right)-\frac{1}{2}\cos \left(2\theta \right)[/tex]

And adding a constant C, we receive our final solution.

[tex]\sin \left(\theta \right)-\frac{1}{2}\cos \left(2\theta \rightt)+C[/tex] - this is our integral

An apple orchard has an average yield of 32 bushels of apple per acre. For each unit increase in tree density, yield decreases by 2 bushels per tree. How many trees per acre should be planted to maximize yield

Answers

An apple orchard has an average yield of 32 bushels of apples per tree if tree density is 26 trees per acre. For each unit increase in tree density, the yield decreases by 2 bushels per tree. How many trees per acre should be planted to maximize the yield?

Answer:

Step-by-step explanation:

From the given information:

Let assume that  26+x trees per acre  are planted

then the  yield per acre will be (26+x)(32-2x)

However;

As x = 0 (i.e. planting 26 per acre), we have;

= (26+0) (32 - 2 (0))

= 26 × 32

= 832

As x = 1 (i.e planting 19 per acre), we have:

= (26+1) (32-2(1)

= 27 × 30

= 810

As x = 2 (i.e. planting 20 per acre), we have:

= (26 +2 ) ( 32 - 2(2)

= 28 × 28

= 784

The series continues in a downward direction for the yield per acre.

Thus,  for maximum plant 19 per acre, it can achieved by method of calculus given that the differentiation of the maximum point of x = 1

Finally, due to integer solution, it is not advisable to use calculus as such other methods should be applied.

Each student in a school was asked, "What is your favorite color?" The circle graph below shows how they answered

Which color was chosen by approximately one fourth of the students?

Approximately what percentage of the students chose purple or green?

Answers

Answer:

a). BLUE color

b). 20%

Step-by-step explanation:

a). "Which color was chosen by approximately one fourth of the students?"

  Since one fourth of the students will be represented by one fourth area of the circle given.

That means color of choice represented by the quarter of the circle will be the color liked by one fourth students.

In the figure attached, BLUE color is the choice of one fourth students in the class.

b). Area represented by purple, green and other colors is a quarter of the circle.

If we divide this quarter into five equal sections, then the total of purple and green will be  [tex]4\times \frac{1}{5}[/tex] of the the quarter of the circle.

Measure of the angle defined by purple or green sections = [tex]\frac{4}{5}\times 90[/tex]

                                                                                                     = 72°

Percentage of the students who preferred purple or green = [tex]\frac{72}{360}\times 100[/tex]

                                                                                                     = 20%

Answer:

blue

20%

Step-by-step explanation:

A small publishing company is planning to publish a new book. The production costs will include one-time fixed costs (such as editing) and variable costs (such as printing). The one-time fixed costs will total $ 63,042 . The variable costs will be $ 11.25 per book. The publisher will sell the finished product to bookstores at a price of $ 25.50 per book. How many books must the publisher produce and sell so that the production costs will equal the money from sales?

Answers

Answer:

  4424 books

Step-by-step explanation:

After the revenue from each book pays for its own cost, it can contribute to the payment of the fixed costs. That "contribution margin" is ...

  $25.50 -11.25 = $14.25

If each book sold contributes that much to the recovery of fixed costs, then the total number of books that must be sold to break even is ...

  $63,042/($14.25/book) = 4424 books

4424 books must be produced and sold so production costs equal sales.

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