Three balls are packaged in a cylindrical container as shown below. The balls just touch the top, bottom, and sides of the cylinder. The diameter of each ball is 13 cm. a. What is the volume of the cylinder? Explain your method for finding the volume.
b. What is the total volume of the three balls? Explain your method for finding the total volume.
c. What percent of the volume of the container is occupied by the three balls? Explain how you would find the percent.

Answers

Answer 1

Answer: We know the following:
Cylinder volume: V₁ = π r² h
Ball (sphere) volume:V₂ = $(4)/(3)$ π r³
where:
V - volume
r - radius of base of cylinder and diameter of ball
h - height of cylinder.

R = 13 cm ⇒ r = 13 ÷ 2 = 6.5
π = 3.14

a) Since balls touch all sides of cylinder (as shown in image), it can be concluded that height of cylinder is equal to sum of diameters of 3 balls and that radius of base of cylinder is equal to radius of ball:
h = 3 × r = 3 × 13 cm = 39 cm
r = 6.5 cm
So,
V₁ = π r² h
V₁ = 3.14 × (6.5 cm)² × 39 cm
V₁ = 5,173.9 cm³

b. The total volume of three balls is the sum of volumes of each ball:
Vₐ = 3 × V₂
Vₐ = 3 × $(4)/(3)$ π r³
Vₐ = 3 × $(4)/(3)$ 3.14 (6.5 cm)³
Vₐ = 3,449.3 cm³

c. Percentage of the volume of the container occupied by three balls ould be expressed as ratio of volume of three balls and volume of cylinder:
V = $( V_(1))/( V_(a) )$ ×100
V = $(3,449.3)/(5,173.9)$ ×100
V = 0.6666 ×100
V = 66.66%

Answer 2

Answer: A. The volume of the cylinder can be computed because the statement "The balls just touch the top, bottom, and sides of the cylinder" gives a clue of the height and radius of the cylinder. The height is equal to 3 times the diameter of the balls, and its radius is the same as the balls' radius.

Vcylinder = πr^2h = π(13/2)^2 (3*13) = 5176.56 cm^3

B. The volume of the three balls is computed using the equation of V for spheres:

Vball = 3*4*π*r^2/3 = 3451.05 cm^3

C. The answer is the equal to the total volume of the balls divided by the volume of the cylinder

3451.05(100%)/5176.56 = 66.67%

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Use Synthetic Division to Factor the following polynomials completely by given factors or roots and Find all the zeros; When you reach quadratic equation, performance regular factoring or Quadratic Formula. $x^(5) -9x^(3) -x^(2) +9;(x-3),(x-1)$ , as factors and -3 as a root

Answers

Answer:

The factors are (x - (1 - i√3)/2) , (x - (1 + i√3)/2) , (x - 3) , (x + 3) , (x - 1)

The zeroes are 1 , 3 , -3 , (1 - i√3)/2 , (1 + i√3)/2

Step-by-step explanation:

$(x^(5)+0-9x^(3) -x^(2)+0+9)$ ÷ (x - 3) =

$(x^(4)-9x^(3)-x^(2)+0+9)$ ÷ (x - 3) =

$x^(4)+3x^(3)+(-x^(2)+0+9)$ ÷ (x - 3) =

$x^(4)+3x^(3)-x+(-3x+9)$ ÷ (x - 3) =

$(x^(4)+3x^(3)-x-3)$

$(x^(4)+3x^(3)+0-x-3)$ ÷ (x - 1) =

x³ + (4x³ + 0 - x - 3) ÷ (x - 1) =

x³ + 4x² + (4x² - x - 3) ÷ (x - 1) =

x³ + 4x² + 4x + (3x - 3) ÷ (x - 1) =

x³ + 4x² + 4x + 3

∵ -3 is a root ⇒ (x + 3) is a factor

(x³ + 4x² + 4x + 3) ÷ (x + 3) =

x² + (x² + 4x + 3) ÷ (x + 3) =

x² + x + (x + 3) ÷ (x + 3) =

x² + x + 1 ⇒ use the formula to find the factors of this quadratic

∵ a = 1 , b = 1 and c = 1

∴ $x=\frac{-1+\sqrt{(1)^(2)-4(1)(1)} }{2(1)}=(-1+√(-3))/(2)=(-1+i√(3))/(2)$

∴ $x=(-1-i√(3) )/(2)$

∴ The factors are (x - (1 - i√3)/2) , (x - (1 + i√3)/2) , (x - 3) , (x + 3) , (x - 1)

The zeroes:

x - 3 = 0 ⇒ x = 3

x + 3 = 0 ⇒ x = -3

x - 1 = 0 ⇒ x = 1

x - (1 - i√3)/2 = 0 ⇒ x = (1 - i√3)/2

x - (1 + i√3)/2 = 0 ⇒ x = (1 + i√3)/2

The zeroes are 1 , 3 , -3 , (1 - i√3)/2 , (1 + i√3)/2

Find the slope through the points (2,4) and (6,12)

Answers

Answer:

The slope is 2

Step-by-step explanation:

Slope = $(rise(changeiny))/(run(changeinx)) =(y2-y1)/(x2-x1) =(12-4)/(6-2) =(8)/(4)$

8/4 reduces to 2/1 which makes the slope 2

What is the factored form of 2x2 − 7x + 6?

Answers

split -7 up so that one is a multiplule of 2 and the other is a factor of 6

2x^2-4x-3x+6
(2x^2-4x)+(-3x+6)
undistribute
(2x)(x-2)+(-3)(x-2)
undistribute
(2x-3)(x-2)

the factored form of 2x² - 7x + 6 is (2x - 3)(x - 2).

To find the factored form of the quadratic expression 2x² - 7x + 6, we can look for two binomials that, when multiplied, give us the original expression.

First, we need to determine two numbers that multiply to give 12 (the product of the leading coefficient, 2, and the constant term, 6) and add up to -7 (the coefficient of the linear term, -7).

The numbers -4 and -3 fit these conditions.

Now, we can express the quadratic expression in factored form:

2x² - 7x + 6 = (2x - 3)(x - 2)

Therefore, the factored form of 2x² - 7x + 6 is (2x - 3)(x - 2).

Learn more about factor here

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