A spherical ballon with a radius r inches has volume V(r)=4/3 pir^3. Find a function that represents the amount of air required to inflate the balloon from a radius inches to a radius of r+1 inches.

Answers

Answer 1

Answer:

Answer:

The function that represents the amount of air is $\Delta V =\cfrac 43 \pi (3r^3+3r+1)$

Step-by-step explanation:

The amount of air here represents the difference between V(r) and V(r+1), so we can start working by finding an expression for the volume at r+1, and then subtract the original volume V(r).

Volume of balloon of radius r+1 inches.

We can replace r with r+1 on the formula and we get:

$V(r+1)=\cfrac43 \pi (r+1)^3$

We can expand $(r+1)^3$ since we will use it to simplify it later on.

So we will have first

$(r+1)^2 = (r+1)(r+1)\n(r+1)^2 =r^2+r+r+1\n(r+1)^2 = r^2+2r+1$

We can multiply that result by (r+1) to get $(r+1)^3$

$(r+1)^3= (r+1)^2 (r+1)\n(r+1)^3=(r^2+2r+1)(r+1)\n(r+1)^3= r^3+2r^2+r+r^2+2r+1\n(r+1)^3 =r^3+3r^2+3r+1$

Thus the volume equation at r+1 will be

$V(r+1)=\cfrac 43 \pi (r^3+3r^2+3r+1)$

Finding the amount of air required to inflate from r to r+1

The amount required to inflate is the difference of volumes, so we have

$V(r+1)-V(r)=\cfrac 43 \pi (r^3+3r^2+3r+1) \cfrac 43 \pi r^3$

Combinging both into one term by factor $\cfrac 43 \pi$ give us

$V(r+1)-V(r)=\cfrac 43 \pi (r^3+3r^2+3r+1-r^3)$

Simplifying

$V(r+1)-V(r)=\cfrac 43 \pi (3r^2+3r+1)$

And that function represents the amount required to inflate the balloon from r to r+1 inches.

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Three consecutive even integers have a sum of 18. Find the integers.

Answers

Let
x = first consecutive even
x + 2 = second consecutive even
x + 4 = third consecutive even

Based on the problem, we equate
x + (x + 2) + (x + 4) = 18
Solving for x,
3x + 6 = 18
3x = 18 - 6
3x = 12
x = 12/3
x = 4
and x + 2 = 4 + 2 = 6
and x + 4 = 4 + 4 = 8

Therefore, the three consecutive even integers are 4, 6, 8.

What is the slope of 6x-12y=36

Answers

The slope is −1/2 . Hope that helps:)

What is the slope of a line perpendicular to the line whose equation is 2y=-6x+8 1)-3
2)1/3
3)1/6
4)-6

Answers

2y=-6x+8

First, the slope of any equation, is always the coefficient with the variable "x".

So, the slope for this line, is -6x.

To get rid of the coefficient with y, we need to divide 2, from both sides of the equal sign.

2y=-6x+8

y=-3+4

Now that we have our new equation, we can see that the slope of our line has also changed, from -6x to-3x. Because -6÷2=-3.

And to find the perpendicular slope, you change the reciprocal of the slope so from -3 to 1/-3 and then you change the sign of the slope so from 1/-3 to 1/3. That is your final answer.

The answer is 2) 1/3.

Your supposed to get Y by its self. So do this- (-6x+8)/2You should get y=-3x+4So the slope would be -3 of the original line.The perpendicular slope would be the reciprical and opposite sign of it, so it should be 1/3. hope this helps

what is the algebraic expression represents the phrase “the quotient of negative eight and the sum of a number and three”?

Answers

You set it up like this.

So quotient means that you're dividing, so put a ÷
Next, it says the dividend, or the part being divided is -8, so now you have this: -8 ÷
Lastly, it says the divisor, or the part doing the dividing, is a quantity equal to the sum of "some number" (we'll use x) and 3. Since the divisor is one quantity collectively, you have to put it in parentheses.
This gives you: (x + 3)
When you put it all together, you get -8 ÷ (x + 3).
This can also be written as a fraction by putting -8 over (x+3).

The algebraic expression represents the phrase “The quotient of negative eight and the sum of a number and three” as –8 / (x + 3).

What is Algebra?

Algebra is the study of graphic formulas, while logic is the interpretation among those signs.

The algebraic expression represents the phrase is given below.

“The quotient of negative eight and the sum of a number and three”

Let x be the number.

Then the sum of a number and three. Then the expression will be

⇒ x + 3

Then the quotient of negative eight and the sum of a number and three. Then the expression will be given as,

⇒ –8 / (x + 3)

Thus, the quotient of negative eight and the sum of a number and three will be –8 / (x + 3).

More about the Algebra link is given below.

brainly.com/question/953809

#SPJ6

Type the correct answer in each box. If necessary, round your answers to the nearest hundredth. The vertices of ∆ABC are A(2, 8), B(16, 2), and C(6, 2). The perimeter of ∆ABC is (blank) units, and its area is (blank) square units.

Answers

Answer:

The perimeter of ∆ABC is 32.44 units, and its area is 30 square units....

Step-by-step explanation:

The perimeter is the sum of three distances.

To find the distances BC, CD, DB use the distance formula:

D=√(y2-y1)^2+(x2-x1)^2

We have given A(2, 8), B(16, 2), and C(6, 2).

AB= √(16-2)^2+(2-8)^2

AB= √(14)^2+(-6)^2

AB=√196+36

AB=√232 = 15.23

BC=√(6-16)^2+(2-2)^2

BC=√(-10)^2+0

BC=√100 = 10

CA = √(2-6)^2+(8-2)^2

CA=√(-4)^2+(6)^2

CA=√16+36

CA=√52 = 7.21

Perimeter = AB+BC+CA

Perimeter = 15.23+10+7.21

Perimeter = 32.44 units

For the area BC is the parallel to x-axis

area = (1/2)base * height

=(1/2)10*(ya-yb)

=(1/2)10*(8-2)

=(1/2)10*6

=30 unit²

The perimeter of ∆ABC is 32.44 units, and its area is 30 square units....

Answer:

perimeter of ∆ABC is 32.44 units, and its area is 30 square units

Step-by-step explanation:

plato

What adds to 15 and multiplies to 25

Answers

five 5+5+5=15 5*5=25

Solutions

5 + 5 + 5 = 15

5 x 5 = 25

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