Write a point-slope equation for the line that has slope 5 and passes through the point (6, 22). Do not use parenthesis on the y side.

The total volume of the flask will be 50.06 [tex]\rm inches ^3[/tex] and if both the sphere and the cylinder are dilated by a scale factor of 2, the resulting volume would be '8' times the original volume.

Given :

Flask can be modeled as a combination of a sphere and a cylinder.

The volume of Sphere is given by the:

[tex]V_s = \dfrac{4}{3}\pi r^3[/tex]

Given - diameter of sphere = 4.5 inches. Therefore, radius is 2.25 inches.

Now, the volume of sphere of radius 2.25 inches will be:

[tex]V_s = \dfrac{4}{3}\times \pi\times (2.25)^3[/tex]

[tex]\rm V_s = 47.71\; inches^3[/tex]

The volume of Cylinder is given by the:

[tex]V_c = \pi r^2h[/tex]

Given - diameter of cylinder = 1 inches then radius is 0.5 inches and height is 3 inches.

Now, the volume of cylinder of radius 0.5 inches and height 3 inches will be:

[tex]V_c = \pi\times (0.5)^2 \times 3[/tex]

[tex]\rm V_c = 2.35\; inches^3[/tex]

Therefore the total volume of the flask will be = 47.71 + 2.35 = 50.06 [tex]\rm inches ^3[/tex].

Now, if both the sphere and the cylinder are dilated by a scale factor of 2 than:

Radius of sphere = [tex]2.25\times 2[/tex] = 4.5 inches

Radius of cylinder = [tex]0.5\times 2[/tex] = 1 inch

Height of cylinder = [tex]3\times 2[/tex] = 6 inches

Now, the volume of sphere when radius is 4.5 inches will be:

[tex]V_s' = \dfrac{4}{3}\times \pi \times (4.5)^3[/tex]

[tex]\rm V_s' = 381.70\; inches ^3[/tex]

And the volume of cylinder when radius is 1 inch and height is 6 inches will be:

[tex]V_c' = \pi \times (1)^2\times 6[/tex]

[tex]\rm V_c'=18.85\;inches^3[/tex]

Therefore the total volume of the flask after dilation by a scale factor of 2 will be = 381.70 + 18.85 = 400.55 [tex]\rm inches ^3[/tex].

Now, divide volume with dilation by theorginal volume of the flask.

[tex]\dfrac{400.55}{50.06}=8[/tex]

Therefore, if both the sphere and the cylinder are dilated by a scale factor of 2, the resulting volume would be '8' times the original volume.

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Answer:

-6 = -11

Distributive Property

Step-by-step explanation:

A.

4x + 2(x – 3) = 4x + 2x – 11

4x + 2x - 6 = 4x + 2x - 11

6x - 6 = 6x - 11

-6 = -11

since -6 is not equal to -11, so there's no solution

B. Distributive property

The sum of first five terms for the geometric series is 775. The correct answer is (B).

In a geometric sequence, the ratio of two consecutive terms are always equal.

The general expression for the nth term is given as aₙ = a₁ × rⁿ⁻¹ .

The sum of n terms for this sequence is given as Sₙ = (rⁿ - 1)/(r - 1)

The given series is  400 + 200 + 100 + …

It is a geometric series.

Its first term is a₁ = 400.

And, common ratio is r = 200/400 = 0.5.

Now, the expression for the sum of n terms is given as below,

Sₙ = a(1 - rⁿ)/(1 - r)

Then, substitute the value of a₁, a₅ and n = 5 to obtain,

S₅ = 400(1 - 0.5⁵)/(1 - 0.5)

⇒ S₅ = 775.

Hence, the sum of first five terms of the given series is 775.

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Answer:

97.

For finding f(g(x)) we will plugin value of g(x) in place of x in f(x).

99.

Step-by-step explanation:

The solution is, 10√3cm  is the length of a diagonal of a cube with a side length of 10 cm.

Pythagorean theorem, the well-known geometric theorem that the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse (the side opposite the right angle)—or, in familiar algebraic notation, a2 + b2 = c2.

here, we have,

given that,

the side length is 10cm ,

then the length of diagonal of the base of the cube(x) is

X² =(10)²+(10)²

    =200cm  

so x = 10√2cm

so the length of diagonal of the cube(y) is

y² = (10)² + (10√2)²

   =300  

so y =10√3cm

Hence, The solution is, 10√3cm  is the length of a diagonal of a cube with a side length of 10 cm.

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Answer:

Perpendicular lines.

Step-by-step explanation:

We have been given that line BC has an equation of a line [tex]y=2x+3[/tex] and line EF has an equation of a line [tex]y=-\frac{1}{2}x+4[/tex]. We are asked to determine what these both equations represent.

We know that slope of two perpendicular lines is negative reciprocal of each other. This means the product of slope of both lines is equal to [tex]-1[/tex].

Let us find the product of slopes of both lines.

[tex]2\times \frac{-1}{2}[/tex]

Upon cancelling 2 with 2 we will get,

[tex]=-1[/tex]

Therefore, the given two equations represent equations of two perpendicular lines.

Answer:

[tex]\frac{1}{12}[/tex].

Step-by-step explanation:

Given : Two 6-sided dice are rolled.

To find : what is the probability the sum of the two numbers on the die will be 4.

Solution : We have given

Two 6-sided dice are rolled.

Dice have number { 1,2,3,4,5,6}  { 1,2,3,4,5,6} .

[tex]Probability =\frac{outcome\ happn}{total\ outcome}[/tex].

sum of the two numbers on the die will be 4.

Case (1) : first dice rolled 3 and second dice rolled 1.

{3,1}

3 +1 = 4 .

Case (2) : first dice rolled 1 and second dice rolled 3 .

{1,3}

1 + 3 = 4 .

Case (3) : first dice rolled 2 and second dice rolled 2.

{2,2}

2 + 2 = 4.

Then there are 3 possible outcomes where the sum of the two dice is equal to 4.

The number of total possible outcomes = 36.

[tex]Probability =\frac{3}{36}[/tex].

[tex]Probability =\frac{1}{12}[/tex].

Probability of getting sum of two dice is [tex]\frac{1}{12}[/tex].

Therefore,  [tex]\frac{1}{12}[/tex].

divide total miles by speed

125/65 = 1.923 hours

there are 60 minutes per hour

multiply 1.923*60 = 115.384 minutes

 rounded off to nearest whole number = 115 minutes

The probability that Jonas will get an orange and Beth will get an apple is 20/121.

To find the probability that Jonas will get an orange and Beth will get an apple, we need to find the probability of Jonas getting an orange and then multiply it by the probability of Beth getting an apple.

The probability of Jonas getting an orange is the number of oranges in the basket (5) divided by the total number of fruits (11):

P(orange for Jonas) = 5/11

The probability of Beth getting an apple is the number of apples in the basket (4) divided by the total number of fruits (11):

P(apple for Beth) = 4/11

To find the combined probability, we multiply these two probabilities together:

P(orange and apple) = P(orange for Jonas) * P(apple for Beth) = (5/11) * (4/11) = 20/121

Answer:

OQ > PQ > OP

Step-by-step explanation:

In this question measure of all angles in a triangle has been given.

m∠ p = (10x - 2)

m∠ q = (3x - 3)

m∠ o = (3x + 9)

Since total of all angles is 180°, so we will equate the total of all angles to 180°

∠p + ∠q + ∠o = 180°

(10x - 2) + (3x - 3) + (3x + 9) = 180°

(10x + 3x + 3x) -2 - 3 + 9 = 180

16x - (2 + 3 - 9) = 180°

16x + 4 = 180

16x = 180 - 4

16x = 176

x = [tex]\frac{176}{16}=11[/tex]

Now we will find the measure of each angle.

∠p = (10x - 2) = 10×11 - 2 = (110 - 2) = 108°

∠q = (3x - 3) = 3×11 - 3 = (33 - 3) = 30°

∠o = (3x + 9) = 3×11 + 9 = (33 + 9) = 42°

As we know in a triangle, angle opposite to the largest side is largest and angle opposite side is the smallest.

Since ∠p > ∠o > ∠q

Therefore, in Δ OPQ opposite sides to these angles will be in the same ratio.

OQ > PQ > OP

The transformation from f to g represents a Vertical stretch.

We are given a parent function f(x) as:

                      [tex]f(x)=\sqrt{x}[/tex]

and the transformed function g(x) as:

        [tex]g(x)=6\sqrt{x}[/tex]

We know that any function transformation of the type:

      f(x) → a f(x)

represents either a vertical stretch stretch or compression depending on the value of a.

If   0<a<1 then the transformation is a vertical compression and if a>1 then the transformation is a vertical stretch.

Here we have a=6>1

Hence, the transformation is a VERTICAL STRETCH.

Answer:

the answer is 2 and 3

Step-by-step explanation:

I took the test

area = L x w

42 = 6 x w

w=42/6 = 7

rectangle is 6 x 7

square would be 6 x 6 = 36 square feet

36/42 = 0.857 = 85.7% ( Round answer if needed)

The value of the fourth term in the geometric sequence is 3.75.

The value of the fourth term in a geometric sequence can be found using the formula:

an = a1 * r(n-1)

Given that a1 = 30 and r = 1/2, we can substitute these values into the formula and solve for a4:

a4 = 30 * (1/2)(4-1) = 30 * (1/2)3 = 30 * (1/8) = 3.75

Therefore, the value of the fourth term in the geometric sequence is 3.75.

Final answer:

The value of the fourth term in the geometric sequence is 3.75.

Explanation:

A geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a constant ratio. In this case, the first term (a1) is 30 and the common ratio (r) is 1/2. To find the fourth term, we can use the formula:

an = a1 * r(n-1)

Substituting the values given, we have:

a4 = 30 * (1/2)(4-1)

Simplifying this expression, we get:

a4 = 30 * (1/8) = 3.75

Therefore, the value of the fourth term in this geometric sequence is 3.75.

The roots of the equation [tex]y=4x^2+2x-30[/tex] are x = -3 and x = 5/2

The given quadratic equation is:

[tex]y=4x^2+2x-30[/tex]

Set y = 0

[tex]4x^2+2x-30=0[/tex]

Factor the trinomial completely

[tex]4x^2-10x+12x-30=0\\\\2x(2x-5)+6(2x-5)=0\\\\(2x-5)(2x+6)=0[/tex]

Set each factor to zero and solve

2x  -  5  =  0

2x  =  5

x  =  5/2

2x  +  6  =  0

2x  =  -6

x  =  -6/2

x  =  -3

The roots of the equation [tex]y=4x^2+2x-30[/tex] are x = -3 and x = 5/2

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To solve the equation, we cross multiply to eliminate the fractions and solve for r. The solution is r = 13. We can check the solution by substituting it back into the original equation.

To solve the equation (r - 3) / 10 = r / 13, we can cross multiply to eliminate the fractions. This gives us 13(r - 3) = 10r. Distributing, we get 13r - 39 = 10r. We can then solve for r by subtracting 10r from both sides and adding 39 to both sides. This gives us 3r = 39. Dividing both sides by 3, we find that r = 13.

To check the solution, we substitute r = 13 back into the original equation. The left side becomes (13 - 3) / 10 = 10 / 10 = 1, and the right side becomes 13 / 13 = 1. Since both sides are equal to 1, we can confirm that the solution r = 13 is correct.

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