Darien ordered soda for $2.75, a sandwich for $8.50, and a dessert for $3.85. Sales tax was $1.15. Use mental math to find the total amount of the bill. Explain

Answers

Answer 1

What I would do is start with the price of the dessert and the sales tax $3.85+ $1.15= $5. I started with this because when you add the cents they equal a dollar which makes it easier to continue.

Then I would add the 2.75. This would be 7.75.

Then add 7.75 + 8.50. To simplify this, you can add 7 + 8, then .75+ .50

Which would equal 15 + 1.25 + 16.25

Answer 2

This question is based on the mental math. Therefore, the total amount of bill is $16.25.

Given:

Darien ordered soda for $2.75, a sandwich for $8.50, and a dessert for $3.85. Sales tax was $1.15.

We need to determined the total amount of the bill.

Let start with the price of the dessert and the sales tax $3.85+ $1.15= $5.

Now, added 2.75 to $5 .

We get,

$2.75 + $5 = $ 7.75

Then, add 7.75 + 8.50.

We get,

= $16.25

Therefore, the total amount of bill is $16.25.

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Related Questions

Two less than twice a number is the same as four times the number

Answers

x is the number
-2+2x=4x
minus 2x both sides
-2=2x
divide by 2
-1=x
the number is -1

Final answer:

The algebraic expression representing 'two less than twice a number is the same as four times the number' is solved, resulting in the number being -1.

Explanation:

The student's question involves solving a simple algebraic equation. We are given that two less than twice a number is the same as four times the number. To represent this algebraically, let's let the unknown number be n. The phrase 'twice a number' can be written as 2n. 'Two less than' this expression would be 2n - 2. The statement implies this is equal to four times the number, which is 4n. Therefore, the equation we need to solve is 2n - 2 = 4n. To solve this equation, we need to isolate the variable n on one side of the equals sign.

Subtract 2n from both sides: -2 = 2n.

Divide both sides by 2 to find the value of n: n = -1.

In conclusion, the number that satisfies the condition given is -1.

please help on b (left of c) and c !!!

rewrite each of the following expressions so that your answer has no negative or fractional exponents

Answers

alrighty

remember
[tex](ab)^c=(a^c)(b^c)[/tex]
and
[tex]x^\frac{m}{n}=\sqrt[n]{x^m}[/tex]
and
[tex](x^m)^n=x^{mn}
and
[tex]x^0=1[/tex] for all real numbers x
and
[tex]x^{-m}=\frac{1}{x^m}[/tex]

b.
[tex](x^5y^4)^\frac{1}{2}=((x^5)^\frac{1}{2})((y^4)^\frac{1}{2})[/tex]=
[tex](x^\frac{5}{2})(y^\frac{4}{2})=(\sqrt{x^5})(\sqrt{y^4})=x^2y^2\sqrt{x}[/tex]

c.
x^0=1
so
that (x^-3y)^0=1
because exponents first in pemdas
so we are left with
x^2y^-1
[tex]x^2y^{-1}=(x^2)(y^{-1})=(x^2)(\frac{1}{y^1})=\frac{x^2}{y}[/tex]

[tex](x^5y^4)^{ \frac{1}{2} }= \sqrt{x^5y^4} =x^2y^2 \sqrt{x} \\ \\ \\ (x^2y^{-1})(x^{-3}y)^0= (x^2y^{-1})*1= \frac{x^2}{y} [/tex]

The area of one circle is 4 times as large as a smaller circle with a radius of 3 inches. the radius of the larger circle is

Answers

The area of small circle= π(3^2)=9π
As the area of other circle is four times greater than small one
So the area of larger circle will be= 4(9π)=36π
So the radius will be
area=π r^2
36π=π(r^2)
r^2=36
√r^2=√36
r=6 inchs
ANSWER IS THAT THE RADIUSOF LARGER CIRCLE WILL BE
6 INCHES

How do I solve this? (Geometry)

Answers

Draw a perpendicular from the center of the

circle to the midpoint of a side of the triangle.

Now draw a line from the center to the vertex

of that side. You now have a right triangle with

a base angle of 30º, side adjacent to that angle

of 3, and hypotenuse = r, the radius of circle. So,

cos 30º = 3/r, r = 3.46, the diameter

is twice that, 6.92 round to 6.9

A company can buy packages of 500 sheets of paper for $4. At that rate, how much paper can be bought for $2000?

Answers

250,000 sheets of paper will be bought with $2,000 dollars

Using the unitary method, we can estimate that $2000 will be spent on the purchase of 250,000 sheets of paper.

What is unitary method?The unitary method is a technique in mathematics for solving a problem by finding the value of a single unit, I.e.,1,(by dividing) and then finding the necessary value by multiplying the single unit value.A method of solving a problem that involves first determining the value of a single unit, And then multiplying that value to determine the required value.

According to the question,

A company can buy packages of 500 sheets of paper for $4. At that rate, how much paper can be bought for $2000?let the number of paper be "X".$4=500$2000= 250,000

Hence we can say that Using the unitary method, we can estimate that $2000 will be spent on the purchase of 250,000 sheets of paper.

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A given line has the equation 10x + 2y = −2.

What is the equation, in slope-intercept form, of the line that is parallel to the given line and passes through the point (0, 12)?

Answers

y = -5x + 12 since the slope of that line is -5x and in order to pass through that point the y intercept must be 12

Step 1

Find the slope of the given line

we have

[tex]10x+2y=-2[/tex]

Isolate the variable y

Subtract [tex]10x[/tex] both sides

[tex]2y=-10x-2[/tex]

Divide by [tex]2[/tex] both sides

[tex]y=-5x-1[/tex]

The slope of the given line is

[tex]m=-5[/tex]

Step 2

Find the equation of the line that is parallel to the given line and passes through the point [tex](0, 12)[/tex]

we know that

If two lines are parallel. then their slope are equal

In this problem we have

[tex]m=-5[/tex]

[tex](0, 12)[/tex]

The equation of the line into slope-intercept form is equal to

[tex]y=mx+b[/tex]

substitute the values

[tex]12=-5*0+b[/tex]

[tex]b=12[/tex]

the equation of the line is

[tex]y=-5x+12[/tex]

therefore

the answer is

[tex]y=-5x+12[/tex]

One student can paint a wall in 10 minutes. another student can paint the same wall in 15 minutes. working together, how long will it take for them to paint the wall?

Answers

the answer:
One student can paint a wall in 10 minutes
another student can paint the same wall in 15 minutes.
If they paint together the wall, the result time will be

15mn - 10mn = 5 mn

proof:
let / = minute

student 1:    / /     /     /    /    /    /    /    /    /

student2:     / /     /     /    /    /    /    /    /    /    /    /    /    /    /
                   x----------------------------------------x
                                already done
the remain time is / / / / / =   five minutes

Answer:

6 days

Step-by-step explanation:

Given that one student can paint a wall in 10 minute and another student in 15 minutes.

Since if number of persons increase, painting time decreases, this is a question of inverse proportion

Hence if they work together they can paint in one day

[tex]\frac{1}{10} +\frac{1}{15}[/tex] part of the work

i.e. work completed in 1 day when they work together

=[tex]\frac{1}{10} +\frac{1}{15} \\=\frac{9+6}{90} \\=\frac{1}{6}[/tex]

Hence in 6 days they can together complete the full work

A square garden plot has an area of 75 ft^2. Find the length of each side in simplest radical form. Calculate the length of each side to the nearest tenth of a foot..

Answers

Hello.

The area of a square is calculated by the formula:

A = l²

As A = 75 ft², we have:

75 = l²

l = √75

Now, note that: 75 = 3 . 25 = 3 . 5²

So:

l = √(3 . 5²) = √3 . √(5²)

l = 5√3 ft

Now, we can assume √3 = 1.73

l ≈ 5 * 1.732

l ≈ 8,7 ft (Note that I have already put it in the nearest tenth)

OK :)

The lengths of the sides of a triangle are 6, 8, 10. Can the triangle be a right triangle? Yes or no?

Answers

Yes because Pythagorean theorem. Comment if u don't know what that is

HELP if f(x)=-14x-2, then f^-1(x)=?

Answers

Replace f(x) with y. Then swap x and y. Once the swap has been done, solve for y to get the inverse.

[tex]f(x) = -14x - 2 [/tex]

[tex]y = -14x - 2 [/tex]

[tex]x = -14y - 2 [/tex]

[tex]x+2 = -14y [/tex]

[tex]-14y = x+2 [/tex]

[tex]y = -\frac{x+2}{14} [/tex]

[tex]f^{-1}(x) = -\frac{x+2}{14} [/tex]

Use Gauss-Jordan elimination to solve the following system of equations. 3x + 5y = 7 6x − y = −8 A. x = 2, y = 1 B. x = 5, y = 6 C. x = 3, y = −1 D. x = −1, y = 2

Answers

hello :
(-1, 2) verifies the 2 equations Answer D
because :
3(-1)+5(2) = 7...right
6(-1)-(2) = -8...... right

What is the solution to 2x-8<12 ?
A. X<2
B. X<8
C. X<10
D. X<40

Answers

2x - 8 < 12
2x < 12 + 8
2x < 20
x < 20/2
x < 10

Answer:

x < 10

Step-by-step explanation:

i need help find the volume of this hexagon based pyramid!
please write steps and its ok to have more than one person answering

Answers

In solid geometry, the volume of any pointed solid is equal to one-third of the product of the area of its base and the vertical height. In equation, it is written as:

V = (1/3)*Bh
where
B is area of the base
h is height

We know the height to be 27 in. The missing information is the area of the hexagonal base. So, let's focus on this shape as drawn in the attached picture. The area of any given polygon is one-half the product of its apothem and perimeter:

B = (1/2)*aP

The apothem is a line drawn from the center of the polygon and projected down to the center of its base. So, an apothem is a perpendicular bisector denoted by the red line. Each interior angle of a hexagon is equal to 60° because one revolution divided by 6 sides is: 360/6 = 60°. When an apothem is drawn, it makes a right angle with an angle of 30° and a base of half of 23 inches. Using the pythagorean theorem, the apothem is equal to:

tan 30° = (23/2) ÷ a
a = 23*sqrt(3)/2 in

The perimeter is the sum of all sides of the polygon. Assuming the hexagon is regular, all its sides are equal measuring 23 inches. So, the perimeter is equal to:

P = 23(6) = 138 in

So, the area of the base is equal to:
B = (1/2)(23*sqrt(3)/2 in)(138 in)
B = 1,374.3823 in²

So, we can finally solve for V:
V = (1/3)(1,374.3823 in²)(27 in)
V = 12,369.44 in³

Find the missing length.

Answers

Find this using the pythagoream theorem for right triangles.

a^2+b^2=c^2
12^2+9^2=c^2

144+81= c^2
225=c^2
15=c

Final answer: c=15

What is the circumference of this circle, in millimeters? use 22/7 for pi

r = 49

Answers

Circumference=[tex]2\pi r=2\times \dfrac{22}{7} \times 49 = \boxed{308 \text{ mm}}[/tex]

Answer: circumference = 308mm

Step-by-step explanation: the formula for the circumference of a circle is given by

C=2πr

Given that r=49mm

Pi=22/4

C=2*22/7*49

C=44*7=308mm

In geometry, the circumference of a circle is the distance around it. That is, the circumference would be the length of the circle if it were opened up and straightened out to a line segment. Since a circle is the edge of a disk, circumference is a special case of perimeter.

Subtract: (2x2 − 7x + 5) − (−6x2 − 4x − 2)

Answers

2x^2 - 7x + 5 - (-6x^2 - 4x - 2) =
2x^2 - 7x + 5 + 6x^2 + 4x + 2 =
8x^2 - 3x + 7 <==

Answer:

8x^2 - 3x + 7

Step-by-step explanation:

I took the test.

Find the area of the circle with the given radius or diameter. Use = 3.14.

r = 6

A =

37.68 sq. units
113.04 sq. units
226.08 sq. units

Answers

Radius [ r ] = 6 units

Area of a circle = [tex] \pi r^{2} [/tex] = [tex]3.14 * 6 * 6 = 3.14 * 36 = 113.04 [/tex] sq. units.

Hence, the answer is B.

Answer: 113.04 sq. units

A kayaker spends a morning paddling on a river. She travels 9 miles upstream and 9 miles downstream in a total of 6 hours. In still water, she can travel at an average speed of 4 miles per hour. What is the average speed of the river's current in miles per hour?
A) 1 mi/h
B) 2 mi/h
C) 3 mi/h
D) 1.5 mi/h

Answers

To answer this item, we have 4 as the speed of the kayaker in still water and the speed of current be y.

When the karayaker moves upstream or against the current, his speed would be 4 - y. Further, if he moves downstream or with the current, the total speed would be 4 + y. The time utilized for the travel is equal to the ratio of the distance and the speed.

Total time = 9/(4 - y) + 9/(4 + y) = 6

We multiply the equation by (4-y)(4+y)
9(4-y) + 9(4 + y) = 6(4-y)(4+y)

Simplifying,
72 = 96 - 6y²
Transposing all the constants to only one side of the equation and rearranging,
6y² = 96 - 72
y² = 4
y = 2

Hence, the speed of the river's current is 2 miles/hr. The answer is letter B.) 2 miles/hour.

Average speed is the ratio of the total distance traveled to the total time taken. The average speed of the river's current is 2 mi/h.

What is Average speed?

Average speed is the ratio of the total distance traveled to the total time taken.

As we know that the total time taken by the boat to travel upstream and downstream is 6 hours. And the distance traveled by Kayaker is 9 miles, each time while going upstream and downstream.

We know that when the boat is traveling upstream the water current will try to resist the boat, therefore, the speed of the boat while going upstream is (4-x), where x is the speed of the boat. Similarly, the speed of the boat when going downstream will be (4+x), as the water stream will try to provide a push to the boat. Therefore, the total time taken by the Kayaker can be written as,

Total Time

= Time taken while going upstream + Time taken while going downstream

[tex]\rm 6 = \dfrac{Distance\ upstream}{Speed\ upstream} + \dfrac{Distance\ Downstream}{Speed\ Downstream}[/tex]

[tex]\rm 6 = \dfrac{9}{(4+x)} + \dfrac{9}{(4-x)}[/tex]

Taking the LCM,

[tex]6 = \dfrac{9(4+x)+9(4-x)}{(4+x)(4-x)}\\\\6\times (4+x)(4-x) = 9(4+x)+9(4-x)\\\\6(16-x^2) = 36+9x+36-9x\\\\96 - 6x^2 = 72\\\\-6x^2 = 72-96\\\\6x^2 = 24\\\\x^2 = 4\\\\x =2[/tex]

Hence, the average speed of the river's current in miles per hour is 2 mi/h.

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I need to know the solution

Answers

For the first one, you have to get x by itself and make it positive. So divide both sides by -11. Then, you get x=-7. The answer is x=-7.

For the second one, get x by itself as a whole number. Multiply each side by 8. You get x=16. The answer is x=16. Hope this helps! ;)

a. x=7
and
c. x=16
you can also get an app that is called script calculator that helps with that

A club decides to sell T-Shirts for 15$ as a fund-raiser. It cost $20 plus $9 per T-Shirt to make them. How many T-Shirts need to be made to make a profit of at least $150?

Answers

The expression for T-shirt production is 20+9T
The expression for total price of selling the T-shirts is 15T

Profit = Total cost of selling - Total cost of buying
Profit = 15T - (20+9T)
Profit = 15T - 20 - 9T
Profit = 6T - 20

To make profit ≥150

6T - 20 ≥ 150
6T ≥ 170
T ≥ 170/6
T ≥ 28.3

The minimum number of T-shirts needed is 29 T-shirts

Calculate the upper and lower limit for a 95% confidence interval about this mean. a family needs a new car, but isn't sure they can fit the payment into their budget. a sample of 36 months of grocery bills yields a mean of $94 with a standard deviation of $10. if the upper limit of a 95% confidence level is below $100, the family can afford to buy the car. standard error = (standard deviation)/(square root of sample size) upper limit (dollars and cents) lower limit (dollars and cents)

Answers

To find the upper and lower limits of a 95% confidence interval for the given data, calculate the standard error, use the multiplier of 2, and apply the formula Sample mean ± Multiplier * Standard error.

To calculate the upper and lower limit for a 95% confidence interval, we use the formula: Confidence interval = Sample mean ± Multiplier * Standard error. For this case, the sample mean is $94 and the standard deviation is $10. The standard error is calculated as $10 / √36 = $10 / 6 ≈ $1.67.

With a 95% confidence level, the multiplier is approximately 2. Therefore, the upper limit would be $94 + 2($1.67) = $94 + $3.34 ≈ $97.34, and the lower limit would be $94 - $3.34≈ $90.66.

In the diagram, PN¯¯¯¯¯ is the perpendicular bisector of AB¯¯¯¯¯ and is also the angle bisector of ∠CPD. If m∠CPD = x, which quantity is equal to sin ∠DPB?

sin x2

sinx2

cosx2

cos x2

Answers

Since ∠CPD = x and segment PN is the angle bisector of this angle, therefore segment PN equally divides ∠CPD into two angles. Which means that:

∠CPN = ∠NPD = x / 2

Further, segment PN is also the perpendicular bisector of AB which further means that the intersection formed by PN and AB creates a right angle (90°). Therefore:

∠NPD + ∠DPB = 90°

x/2 + ∠DPB = 90°

∠DPB = 90 – x/2

Therefore:

sin∠DPB = sin(90 – x/2) which is not in the choices

However we know that the relationship of sin and cos is:

sin(π/2 - θ) = cos θ

Where,

π/2 = 90

θ = x/2

Therefore:

sin(90 – x/2) = cos(x/2)

Answer:

cos(x/2)

The quantity which is equal to sin ∠DPB is:

cos(x/2)

What is an Angle?

This refers to the figure which is formed by two rays with a common endpoint.

Hence, we know that

∠CPN = ∠NPD = x / 2

If we segment PN which is the bisector of AB, it would crerate angle 90 and this would give us:

∠NPD + ∠DPB = 90°

x/2 + ∠DPB = 90°

∠DPB = 90 – x/2

With this in mind, there is the relation between sin and cos, which would be:

sin(π/2 - θ) = cos θ

We are aware that

π/2 = 90θ = x/2

Hence,

sin(90 – x/2) = cos(x/2)

=cos(x/2)

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Constantine picks two letters at random from the word constantinople with replacement. what is the probability that both letters picked are consonants?

Answers

Constantinople has a total of 14 letters of which 9 are consonants

P(picking a consonant) = 9/14

P(2 letters are consonants with replacement) = 9/14 * 9/14 = 81/196

Mason and Nora decided to swim across the river. Mason began swimming 8 seconds earlier than Nora.

Mason swam at a speed of 5 feet per second.
Nora swam at a speed of 9 feet per second.
For how many seconds had Mason been swimming at the moment when the two swimmers had swam exactly the same distance?

Answers

This is a distance = rate * time problem. The easiest way to solve these is to make a table with the information. Since d= rt, we will set up the table like that:

                  distance              rate               time
Mason            d                       5                  t + 8
Nora               d                       9                    t

Let me explain the values in the table. The problem says "...when the swimmers had swam exactly the same distance"; therefore, we put a d there to indicate that, although we have no idea the distance they swam, both distances were the same. The rates are easy; they are self-explanatory. The time could be a little tricky too though. If Mason began swimming 8 seconds earlier than Nora, Nora's time is t and Mason's time is t + 8, which is Nora's time with 8 seconds added to it. Because d = rt, we set up the equations like that: d = 5(t+8), and d = 9t. Because the 2 d's are the same, we set them equal to each other: 5(t+8) = 9t. Simplifying that gives you 5t + 40 = 9t and 40 = 4t and t = 10. Now put that t value of 10 into Mason's time to solve the question they are asking you: t + 8 with the substitution is 10 + 8 = 18. So Mason had been swimming for 18 seconds when they had both swam the same distance.

The perimeter of a square is 96 inches. if the side length is 2x + 4, what is the value of x and the length of each side?

Answers

subtract 4 from 96 and divide by 2

Which value is in the domain of f(x)?
A.) –7
B.) –6
C.) 4
D.) 5

Answers

C.) 4 is in the domain of f(x)

f(x) = -2x + 3 0<x <= 4

hope that helps

Answer:

C) 4

Step-by-step explanation:

The given function is

[tex]f(x)=\left \{ {{2x+5,\:-6\:<\:x\le0} \atop {-2x+3,\:0\:<x\le4}} \right.[/tex]

The function is defined on two intervals.

The first interval is

[tex]-6\:<\:x\le0[/tex] and the second interval is [tex]\:0\:<x\le4[/tex].

[tex]-7[/tex] does not belong to any of these intervals.

[tex]-6[/tex] does not also belong to any of these intervals.

[tex]4[/tex] belongs to the interval [tex]\:0\:<x\le4[/tex].

Hence 4 is in the domain of f(x).

[tex]5[/tex] does not also belong to any of the intervals.

Therefore the correct answer is C.

Melissa exercises for 20 minutes every day. She decides to increase her daily exercise time by 5 minutes each week. However, according to her doctor’s orders, she can spend no more than 45 minutes a day exercising. For how many weeks can Melissa increase her exercise time this way?

Answers

Let's start by finding the difference between Melissa's maximum time limit and her every day exercising time.

45-20 = 25 minutes.

Now we know that each week she increases her time by 5 minutes. Therefore, we have to find how many 5's can fit in the 25 minute difference.

25/5 = 5 weeks

Melissa increases 5 minutes every week for 5 weeks until she reaches her limit of 45 minutes.

I hope that helps!

Answer:

at most 5 weeks

Step-by-step explanation:

There are 7 black marbles and 9 white marbles in a bag. what is the probability of drawing 2 black marbles then a white marble without replacement

Answers

The probability of drawing to marbles then a white one without replacement will be as follows;
total number of marbles=16
total number of black marbles=7
total number of white marbles=9
probability of drawing a black marble on first=7/16
probability of drawing a black marble on second draw=6/15
probability of drawing a white marble on the 3rd draw=9/14
thus the P(B and B and W)
=7/16*6/15*9/14
=9/80
the answer is 9/80

A ball is thrown from a height of 70 feet with an initial downward velocity of 4/fts. The ball's height h (in feet) after t seconds is given by the following.

h=70-4t-16t^2

How long after the ball is thrown does it hit the ground?

Round your answer(s) to the nearest hundredth.
(If there is more than one answer, use the "or" button.)

Answers

In this one, you must make h = 0.

You get : [tex]-16t^2-4t+70[/tex] which can be further simplified to [tex]8t^2+2t-35[/tex] if you divide all of the numbers by -2.

Here you can use the quadratic formula again!
You get the numbers to be : 1.97 and -2.22

I had made a mistake to think that negative numbers can be included but in these questions, you can't have negative numbers as your answer. So the correct answer is 1.97!

Distance is a scalar quantity that refers to "how much ground an object has covered" during its motion. 1.97 is the time taken by the ball to hit the ground.

What is Distance?

Distance is a scalar quantity that refers to "how much ground an object has covered" during its motion.

Given that a ball is thrown from a height of 70 feet with an initial downward velocity of 4/fts. The ball's height h (in feet) after t seconds is given by the following.

h=70-4t-16t²

Now we can take h=0

h=-16t²+70-4t

-16t²+70-4t

Divide by 2

-8t²-2t+35

Now apply quadratic formula

a=-8, b=-2, c=35

t=-b±√b²-4ac/2a

t=2±√-2²-4(-8)(35)/2(-8)

t=2±√4+1120/-16

we get t= 1.97 and t= -2.22

You get the numbers to be : 1.97 and -2.22

We do not consider negative values. So the correct answer is 1.97

Hence 1.97 is the time taken by the ball to hit the ground.

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A volcano fills the volume between the graphs z=0 and z=1/(x^2+y^2)^10 and outside the cylinder x+y=1. find the volume.

Answers

For this case, we use the cylindrical coordinates:
x² + y² = r²
dV = r dz dr dθ

The limits are:
z = 0 to z = 1/(r²)^10 = 1/r^20
r = 1 to ∞
θ = 0 to 2π

Integrating over the limits:
V = ∫ [0 to 2π] ∫ [1 to ∞] ∫ [0 to 1/r^20] r dz dr dθ
V = ∫ [0 to 2π] ∫ [1 to ∞] rz | [z = 0 to 1/r^20] dr dθ
V = ∫ [0 to 2π] ∫ [1 to ∞] 1/r^19 dr dθ
V = ∫ [0 to 2π] −1/(18r^18) |[1 to ∞] dθ
V = ∫ [0 to 2π] 1/18 dθ
V = θ/18 |[0 to 2π]
V = π/9

The volume of the volcano is an illustration of definite integral

The volume of the volcano is: [tex]\mathbf{\frac{1}{9}\pi}[/tex]

The graphs are given as:

[tex]\mathbf{z = 0}[/tex] and [tex]\mathbf{z = \frac{1}{(x^2 + y^2)^{10}}}[/tex]

The cylinder is:

[tex]\mathbf{x + y =1}[/tex]

For cylindrical coordinates, we have:

[tex]\mathbf{r^2 =x^2 + y^2}[/tex]

So, we have:

[tex]\mathbf{z = \frac{1}{(r^2)^{10}}}[/tex]

[tex]\mathbf{z = \frac{1}{r^{20}}}[/tex]

Where:

[tex]\mathbf{r = 1 \to \infty}[/tex]

[tex]\mathbf{\theta = 0 \to 2\pi}[/tex]

So, the integral is:

[tex]\mathbf{V = \int\limits^{2\pi}_0 {\int\limits^{\infty}_1 {\frac{1}{r^{20}}} \, r\ dr } \, d\theta }[/tex]

Cancel out r

[tex]\mathbf{V = \int\limits^{2\pi}_0 {\int\limits^{\infty}_1 {\frac{1}{r^{19}}} \, dr } \, d\theta }[/tex]

Rewrite as:

[tex]\mathbf{V = \int\limits^{2\pi}_0 {\int\limits^{\infty}_1 {r^{-19}} \, dr } \, d\theta }[/tex]

Integrate

[tex]\mathbf{V = \int\limits^{2\pi}_0 {{-\frac{1}{18}r^{-18}}} |\limits^{\infty}_1 \, d\theta }[/tex]

Expand

[tex]\mathbf{V = \int\limits^{2\pi}_0 {{-\frac{1}{18}(\infty^{-18} -1^{-18}) }} , d\theta }[/tex]

[tex]\mathbf{V = \int\limits^{2\pi}_0 {{-\frac{1}{18}(0 -1) }} , d\theta }[/tex]

[tex]\mathbf{V = \int\limits^{2\pi}_0 {{-\frac{1}{18}( -1) }} , d\theta }[/tex]

[tex]\mathbf{V = \int\limits^{2\pi}_0 {{\frac{1}{18} }} , d\theta }[/tex]

Integrate

[tex]\mathbf{V = \frac{1}{18}(\theta)|\limits^{2\pi}_0}[/tex]

Expand

[tex]\mathbf{V = \frac{1}{18}(2\pi - 0)}[/tex]

[tex]\mathbf{V = \frac{1}{18}(2\pi )}[/tex]

Cancel out 2

[tex]\mathbf{V = \frac{1}{9}\pi}[/tex]

Hence, the volume is: [tex]\mathbf{\frac{1}{9}\pi}[/tex]