Help give me an Explanation

Possible number of death rays (D) and claws (C) that Archimedes can build are given by the following system of inequalities: 350D + 200C ≤ 3000. D, C ≥ 0

The first inequality represents the fact that the total amount of gold used to build the machines cannot exceed the 3000 lbs of gold given by the city. The second inequality ensures that the number of death rays and claws cannot be negative.

To explain this system, let us assume that Archimedes builds x death rays and y claws. The amount of gold required to build x death rays and y claws is given by 350x + 200y. The first inequality ensures that this value cannot exceed 3000 lbs of gold. The second inequality ensures that the number of death rays and claws cannot be negative.

Therefore, the solution to this system of inequalities gives us all the possible combinations of death rays and claws that Archimedes can build with the given amount of gold.

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The planes intersect at the point (-19/21, -11/14, 1).

To determine how, if at all, the planes intersect, we need to solve the system of equations given by the three planes:

[2T/3A] 2x + 2y + z - 10 = 0

5x + 4y - 4z = 13

3x – 2z + 5y - 6 = 0

We can use elimination to solve this system. First, we can eliminate z from the second and third equations by multiplying the second equation by 2 and adding it to the third equation:

5x + 4y - 4z = 13

6x - 4z + 10y - 12 = 0

11x + 14y - 12 = 0

Next, we can eliminate z from the first and second equations by multiplying the first equation by 2 and subtracting the second equation from it:

4x + 4y + 2z - 20 = 0

-5x - 4y + 4z = -13

9x - y - 6z - 20 = 0

Now we have two equations in three variables. To eliminate y, we can multiply the second equation by 14 and subtract it from the first equation:

11x + 14y - 12 = 0

-70x - 56y + 56z = -182

-59x - 42z - 12 = 0

Finally, we can substitute this expression for x into one of the previous equations to find z:

3(59/42)z - 12/42 - 2y - 10 = 0

177z - 60 - 84y - 420 = 0

177z - 84y - 480 = 0

Now we have two equations in two variables, z and y. We can solve for y in terms of z from the second equation:

y = (177/84)z - (480/84)

Substituting this expression for y into the third equation, we can solve for z:

177z - 84[(177/84)z - (480/84)] - 480 = 0

177z - 177z + 480 - 480 = 0

This equation simplifies to 0=0, which means that z can be any value. Substituting z=1 into the expression for y, we get:

y = (177/84)(1) - (480/84) = -11/14

Substituting z=1 and y=-11/14 into the expression for x, we get:

x = (59/42)(1) - (12/42) + 2(-11/14) + 10 = -19/21

Therefore, the planes intersect at the point (-19/21, -11/14, 1).

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Using given function 9(x) = ln(3x+11), gl (x) B) 3(-3x + 11)^-1.

We are given 9(x) = ln(3x+11) and we need to find gl(x).

First, we can use the chain rule to differentiate ln(3x+11):

d/dx [ln(3x+11)] = 1/(3x+11) * d/dx [3x+11] = 3/(3x+11)

Now, we can use the given equation 9(x) = ln(3x+11) to find d/dx [9(x)]:

d/dx [9(x)] = d/dx [ln(3x+11)] = 3/(3x+11)

Therefore, gl(x) = d/dx [9(x)] / 3(x) = 3/(3x+11) * 1/3 = (3(-3x+11))^-1.

Therefore, the correct answer is B) 3(-3x+11)^-1.

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The set of ordered pairs that does NOT represent a function is option B) (0, 1), (2, 2), (4, 8), (2, 7), (5, 8), (7, 9)

  A function is a term that do not have two different outputs that is (y) for the same input(x).

Note that

For option A: The x values are {0,2,4,-2,5} Here none of the x values are repeated, So, it represents a function.

For option B: The input value 2 is linked with two different output values (2 and 7), so this set of ordered pairs does not stand a function.

Based on the explanation of the function given above, if x=2, the 'y' can not be equal to 2, 7. So, this is not a function.

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The product of (8-6b)(5-3b), using the distributive property of multiplication is [tex]18b^2 - 54b + 40[/tex].

This problem is actually an algebraic expression involving variables and constants. To find the product of (8-6b)(5-3b), we need to use the distributive property of multiplication.

We can start by multiplying 8 by 5, which gives us 40. Next, we multiply 8 by -3b, which gives us -24b. Then, we multiply -6b by 5, which gives us -30b. Finally, we multiply -6b by -3b, which gives us[tex]18b^2[/tex].

Putting all of these terms together, we get:
(8-6b)(5-3b) = [tex]40 - 24b - 30b + 18b^2[/tex]

Simplifying this expression, we can combine the like terms -24b and -30b to get -54b. So the final answer is:
(8-6b)(5-3b) = [tex]18b^2 - 54b + 40[/tex]

Therefore, the product of (8-6b)(5-3b) is  [tex]18b^2 - 54b + 40[/tex].

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The area of the given region under the curve of a function f(x) = ax + b on the interval [0, 4] is 16 square units. So, all the options satisfy the value of a and b except (7, -9).

The area of the given region under the curve of a function f(x) = ax + b on the interval [0, 4] is 16 square units.

f(x) is a linear function, the area under the curve on the interval [0,4] is a trapezoid with a height of 4 and bases of lengths f(0) and f(4).

The area of a trapezoid is the height times the average of the bases.

a. f(x)=-2x+8  f(0)=8, f(4)=0;

  area = 4(8/2) = 16

b. f(x)=x+2; f(0)=2, f(4)=6;

  area = 4(8/2) = 16

c. f(x)=3x-2; f(0)=-2, f(4)=10;

   area = 4(8/2) = 16

d. f(x)=5x-6; f(0)=-6, f(4)=14;

   area = 4(8/2) = 16

e. f(x)=7x-9; f(0)=-9, f(4)=19;

    area = 4(10/2) = 20

Thus, The area is NOT 16 for choice e.

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The value of k that is less than or equal to 2.6

To solve the inequality 5.7 ≥ k + 3.1, you should subtract 3.1 from each side of the inequality.

To isolate the variable k, we need to perform the same operation on both sides of the inequality. In this case, we need to subtract 3.1 from each side:

5.7 - 3.1 ≥ k + 3.1 - 3.1

This simplifies to:

2.6 ≥ k

Therefore, the correct answer is:

k ≤ 2.6

We subtracted 3.1 from each side to isolate the variable k, resulting in the inequality k ≤ 2.6. This means that any value of k that is less than or equal to 2.6 will satisfy the original inequality 5.7 ≥ k + 3.1.

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Cif a = 2.74 mi, b = 3.18 mi and ZC = 41.9° and c^2 = a^2 + b^2 - 2ab*cos(C)where C is the angle opposite to side c, c comes to be ≈ 4.26 mi.

The Law of Cosines is a numerical formula that relates the side lengths and points of any triangle. It expresses that the square of any side of a triangle is equivalent to the number of squares of the other different sides short two times the result of those sides and the cosine of the point between them. To get side c, we can use the law of cosines, which states that c² = a² + b² - 2ab cos(C).
Plugging in the given values, we get:
c² = (2.74)² + (3.18)² - 2(2.74)(3.18)cos(41.9°)
c² ≈ 18.126
Taking the square root of both sides, we get:
c ≈ 4.26 mi
Rounding to 2 decimal places, c ≈ 4.26 mi.
Therefore, the answer is: c ≈ 4.26 mi.

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a. The tube dilution in tubes 1, 3, and 5:
- Tube 1: 1:5 (1 ml serum + 4 ml saline)
- Tube 3: 1:125 (1:5 dilution from Tube 1 x 1:5 dilution from Tube 2 x 1:5 dilution from Tube 3)
- Tube 5: 1:3125 (1:125 dilution from Tube 3 x 1:5 dilution from Tube 4 x 1:5 dilution from Tube 5)

b. The solution dilution in tubes 1, 2, and 7:
- Tube 1: 1:5
- Tube 2: 1:25 (1:5 dilution from Tube 1 x 1:5 dilution from Tube 2)
- Tube 7: 1:78125 (1:3125 dilution from Tube 5 x 1:5 dilutions for Tubes 6 and 7)

c. The total volume and solution dilution in tube 10 before transfer:
- Total volume: 3 ml (2 ml saline + 1 ml transferred from Tube 9)
- Solution dilution: 1:1953125 (1:78125 dilution from Tube 7 x 1:5 dilutions for Tubes 8, 9, and 10)

d. The amount or volume of serum in tube 6 before transfer and after transfer:
- Before transfer: 0.00064 ml (2 ml x 1:3125 dilution from Tube 5)
- After transfer: 0.00032 ml (1 ml x 1:3125 dilution from Tube 5, as half the volume was transferred to Tube 7)

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Answer:

(-2,-8)

Step-by-step explanation:

First, we have to make these linear equations into standard form:

-3x+y=-2

and

9x-y=-10

Now we tell my using elimination method, we can cross out the y variables because when added(y+(-y)) is just 0, so we just cross them out

Add liked terms

6x=-12

Solve for X:

X=-2

Plug 2 for X in any equation (lets do -3x+y=-2)

Plug in -2 for X:

-3(-2)+y=-2

Thus we get 6+y=-2

Solve for Y:

y=-8

Now that we have both our variables, we know that the answer is (-2,-8)

The mean number of points Kathleen made is 38.

To calculate the mean number of points Kathleen made, we will use the following terms: mean, sum, and total number of assignments.

The mean is the average value of a set of numbers. To find the mean, we need to sum all the given values and then divide the sum by the total number of values in the set.

Kathleen's assignment scores are 29, 38, 45, 42, and 36 points. To find the sum, we add these numbers together: 29 + 38 + 45 + 42 + 36 = 190 points.

Now, we need to determine the total number of assignments. Kathleen has completed five assignments. So, we will divide the sum of her points (190) by the total number of assignments (5) to find the mean.

Mean = Sum / Total number of assignments
Mean = 190 / 5
Mean = 38

The mean number of points Kathleen made on her assignments is 38 points. This indicates that on average, she scored 38 points per assignment. Calculating the mean gives us a general idea of her performance across all assignments, allowing us to gauge her overall progress.

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Answer:

See explanation.

Step-by-step explanation:

The arc BX, central angle BCX, and inscribed angle BNX are connected through the following relationships:

1. The central angle BCX subtends the arc BX. This means that the central angle is formed by two radii connecting the center of the circle to the endpoints of the arc BX.

2. The inscribed angle BNX subtends the same arc BX. This means that the inscribed angle is formed by two chords connecting a point on the circumference of the circle to the endpoints of the arc BX.

3. The relationship between the central angle BCX and the inscribed angle BNX is given by the Inscribed Angle Theorem, which states that the measure of an inscribed angle is half the measure of the central angle subtending the same arc. In other words, if θ is the measure of the central angle BCX and α is the measure of the inscribed angle BNX, then:

[tex]\alpha =\frac{1}{2}[/tex] θ

She requires 55 consecutive successful throws to get a success rate of 60%

The number of successful throws she made is 8 out of 50

Now to calculate the percentage we will use the formula

[tex]\frac{8}{50} X 100 = 16[/tex]

According to the problem, she needs to make her success percentage 60% and for that, we need to estimate the number of consecutive successful throws. Hence the new equation will be

[tex]\frac{8+x}{50+x} X 100 = 60[/tex]

[tex]or, \frac{8+x}{50+x} = 0.6[/tex]

This is the equation concerned.

Now,to solve this, we will take the denominator on the other side will give us

[tex]or, 8+x = 0.6(50+x)[/tex]

[tex]or, 8+x = 30+ 0.6x[/tex]

[tex]or, x - 0.6x = 30 -8[/tex]

[tex]or, 0.4x= 22[/tex]

or, x = 55

Hence we see that she requires 55 consecutive successful throws to get a success rate of 60%

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Complete Question

A basketball player made 8 out of 50 free throws she attempted which is 16%. She wants to know how many consecutive free throws more in a row she would have to make to raise her overall successful baskets divided by attempts or percent of successful free throws to 60%.

(a) Write an equation to represent this situation.

(b) Solve the equation. How many consecutive free throws would she have to make to raise her percent to 60%?

The maximum population density evaluated is 1200 citizens per square mile, under the condition that the city is 30 miles long and two-thirds as wide, and 555,000 citizens currently live there.

Now to evaluate the maximum population density that is considered in the mayor's calculation is
Let us first calculate the area of the city which is (2/3) × (30 miles)
= 20 miles.
So, now we can calculate the current population density which is
555,000 / (20 × 20)
= 1387.5 citizens per square mile.
Hence the mayor evaluates that at least 75,000 people must transfer out of the city for adequate services to be exercised, we can find the new population as
555,000 - 75,000
= 480,000 citizens.
Therefore, the new population density would be 480,000 / (20 × 20)
= 1200 citizens per square mile

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4.20 cm² is the area of the triangle STU to the nearest 10th of a square centimeter.

Given, t = 3.4 cm, u = 6.9 cm and ∠S = 21°

We know that the formula for the area of the triangle = 1/2 * t*u *sin(S)

Substituting the values

Area = 1/2 × 3.4 × 6.9 × sin(21°)

Area = 11.73 × sin(21°)

Area = 11.73 ×  0.3583

Area = 4.2016

Rounding to the nearest 10th of a square centimeter .

Area = 4.20 cm²

Hence, 4.20 cm² is the area of the triangle STU to the nearest 10th of a square centimeter.

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The volume of the solid is given by V = 9r/2 - 3.

To calculate the volume of this solid, we will use a triple integral, which involves integrating a function of three variables over a three-dimensional region. The triple integral is denoted by ∭f(x, y, z) dV, where f(x, y, z) is the function we are integrating, and dV is the volume element.

In our problem, the function f(x, y, z) is equal to 1, which means we are integrating a constant function. Therefore, we can simplify the triple integral to V = ∭dV, where V represents the volume of the solid.

To evaluate the triple integral, we need to determine the limits of integration for each variable. We are given the limits for x, y, and z, so we can set up the triple integral as follows:

V =  ∫₂⁰ ∫₃⁰ r2-2y 1 dz dy dx

We integrate first with respect to z, then y, and finally x.

Integrating with respect to z, we get:

V = ∫₂⁰ ∫₃⁰[r2-2y - 1] dy dx

Simplifying the integral, we get:

V =  ∫₂⁰ [r2y - y2]dy dx

Integrating with respect to y, we get:

V = ∫₂⁰ [(r2/2)y2 - (1/3)y3]dy

Simplifying the integral, we get:

V = [(r/2)(3)2 - (1/3)(3)3] - [(r2/2)(0)2 - (1/3)(0)3]

V = 9r/2 - 3

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Evan saved $1.75 per book by buying them on sale.

To find out how much Evan saved per book by buying them on sale, you can use the following formula:

Savings per book = (Regular price per book - Sale price per book)

First, you need to find the regular price per book:

Regular price per book = (Total regular price of 7 books) / 7

Regular price per book = 57.75 / 7

Regular price per book = 8.25

Next, you need to find the sale price per book:

Sale price per book = (Total sale price of 7 books) / 7

Sale price per book = 45.50 / 7

Sale price per book = 6.50

Now, you can find the savings per book:

Savings per book = (Regular price per book - Sale price per book)

Savings per book = (8.25 - 6.50)

Savings per book = 1.75

Therefore, Evan saved $1.75 per book by buying them on sale.

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The volume of the solid is 32/15 cubic units.

To find the volume of the solid, we need to integrate the area of each square cross section perpendicular to the x-axis over the interval [a, b], where a and b are the x-coordinates of the intersection points of f(x) and g(x):

First, we find the intersection points of the two functions:

x²+1 = 2x+1

x² - 2x = 0

x(x-2) = 0

x = 0 or x = 2

So, a = 0 and b = 2.

Next, we find the side length of each square cross section. Since the cross section is a square, the side length is equal to the difference between the y-coordinates of the functions f(x) and g(x) at each x:

Side length = f(x) - g(x) = (x²+1) - (2x+1) = x² - 2x

Finally, we integrate the area of each square cross section over the interval [0, 2] to get the volume of the solid:

V = ∫[0,2] (x² - 2x)² dx

V = ∫[0,2] (x⁴- 4x³ + 4x²) dx

V = [1/5 x⁵ - 1 x⁴ + 4/3 x³] [0,2]

V = (1/5 x⁵ - 1 x⁴ + 4/3 x³)|[0,2]

V = (1/5(2⁵) - 1(2⁴) + 4/3(2³)) - (1/5(0⁵) - 1(0⁴) + 4/3(0³))

V = (32/5 - 16/3)

V = 32/15

Therefore, the volume of the solid is 32/15 cubic units.

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The scale factor used is 2.

It is one of the simplest polygon shapes and is commonly used in mathematics and geometry. The sum of the internal angles of a triangle is always 180 degrees.

The vertices of a triangle are the three points in a two-dimensional (2D) or three-dimensional (3D) space that define the corners or corners of the triangle. In a 2D plane, the vertices are typically denoted as A, B, and C, and in a 3D space, they can be represented as (x1, y1, z1), (x2, y2, z2), and (x3, y3, z3) where (x, y, z) are the coordinates of each vertex along the x, y, and z axes respectively. The vertices of a triangle are connected by three line segments, known as edges, to form the sides of the triangle. The combination of the three vertices and the edges connecting them determines the shape and size of the triangle.

To find the scale factor used to dilate triangle ABC to A'B'C', we can compare the corresponding side lengths of the two triangles.

The distance between A(-3, -3) and B(3, 3) is √((3-(-3))^2 + (3-(-3))^2) = 6√2.

The distance between A'(-6, -6) and B'(6, 6) is √((6-(-6))^2 + (6-(-6))^2) = 12√2.

So the scale factor used to dilate triangle ABC to A'B'C' is:

scale factor = length of corresponding side in A'B'C' / length of corresponding side in ABC

            = (12√2) / (6√2)

            = 2

Therefore, the scale factor used is 2.

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If length of rectangle is 5 more than twice it's width and having perimeter as 88 feet, then the dimensions of the rectangle are, length is 31 feet, width is 13 feet.

Let width of the rectangle be represented as = "w" feet.

It is given that, the length of rectangle is 5 more than twice it's width,

So, Length can be represented as "2w + 5" in feet;

We use formula for perimeter of rectangle, which is "P = 2Length + 2Width", where P = perimeter, L = length, and W = width.

In this case, we know that the perimeter is 88 feet, so we substitute the values,

We get,

⇒ 88 = 2(2w + 5) + 2w;

⇒ 88 = 4w + 10 + 2w,

⇒ 88 = 6w + 10,

⇒ 78 = 6w,

⇒ w = 13,

So the width is 13 feet. We use this value of width to find length of the rectangle:

⇒ L = 2w + 5,

⇒ L = 2(13) + 5,

⇒ L = 31

Therefore, the dimensions of the rectangle are 31 feet by 13 feet.

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In 2002, the average price of a two-bedroom apartment in the uptown area was approximately $0.316 million, and it was increasing at a rate of about $0.0328 million per year.

To find the average price of a two-bedroom apartment in 2002 (t=8), you need to evaluate the given function p(t) = 0.17e^(0.10t) at t=8:

p(8) = 0.17e^(0.10 * 8)

p(8) = 0.17e^0.8 ≈ 0.316 million dollars

To find the rate at which the price was increasing in 2002, you need to find the derivative of the function p(t) with respect to t, and then evaluate it at t=8:

p'(t) = d/dt (0.17e^(0.10t))
p'(t) = 0.17 * 0.10 * e^(0.10t)

Now, evaluate p'(t) at t=8:

p'(8) = 0.17 * 0.10 * e^(0.10 * 8)
p'(8) ≈ 0.0328 million dollars per year

So, in 2002, the average price of a two-bedroom apartment in the uptown area was approximately $0.316 million, and it was increasing at a rate of about $0.0328 million per year.

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The odds in favor of the event are 1.

The probability of an event is the ratio between the total number of favorable outcomes and the total number of outcomes.

The odds in favor of an event are the ratio of the total number of favorable outcomes and the total number of unfavorable outcomes. If the probability of the event is given, we can find the odds in favor by using the formula of odds in favor:

odds in favor = probability of event/probability of not event

In this case question, the probability of an event is given as 0.5 which means that the total number of favorable outcomes is 50 out of 100

So the probability of not having an event is also 0.5 (the other half of the part.)

So, odds in favor = 0.5/0.5 = 1

The probability of the happening of an event is the same as the probability of not happening of an event. This means that the odds in favor of the event are 1 to 1, or simply 1.

Therefore,  the odds in favor of the event are 1.

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We know that both np and nq are greater than or equal to 10, it is appropriate to use a normal distribution to model the sampling distribution of p^ in this case.

(a) To calculate the mean and standard deviation of the sampling distribution of p^, we'll use the formulas:

Mean (μ) = p
Standard deviation (σ) = √(p*q/n)

where p is the proportion of individuals with allergies (0.12), q is the proportion of individuals without allergies (1 - p = 0.88), and n is the sample size (100).

Mean (μ) = 0.12
Standard deviation (σ) = √(0.12 × 0.88 / 100) = 0.02887

(b) The standard deviation of 0.02887 in this context represents the variability in the proportions of individuals with allergies that we would expect to observe in different random samples of 100 adult patients each. It quantifies the dispersion of p^ around the true population proportion.

(c) To determine if it's appropriate to use a normal distribution to model the sampling distribution of p^, we can check if both np and nq are greater than or equal to 10:

np = 100 × 0.12 = 12
nq = 100 × 0.88 = 88

Since both np and nq are greater than or equal to 10, it is appropriate to use a normal distribution to model the sampling distribution of p^ in this case.

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The total number of bikes that can be painted in an hour would be 60 bikes.

With three identical machines,

the number of bikes machine can paint per hour = 20,

the number of machines bought again = 2,

so the total number of machines will be = 3,

when there are two same machines the productivity will be = 20 * 3 = 60 bikes.

This is because each machine works independently and can paint bikes simultaneously.

By adding two additional machines to the existing one,

the productivity of the painting process can be significantly increased. The new machines will not only increase the overall capacity but also reduce the turnaround time required for painting a large number of bikes.

By investing in additional machines,

the business can increase its output and generate more revenue,

which can be used to expand the operations further.

It's important to note that the investment in additional machines needs to be justified by the demand for painted bikes and the expected return on investment.

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The expression for the area of the triangle is given as follows:

A = 6x² - 7x - 3.

The area of a rectangle of base b and height h is given by half the multiplication of dimensions, as follows:

A = 0.5bh.

The dimensions for this problem are given as follows:

Hence the expression for the area of the triangle is given as follows:

A = 0.5(4x - 6)(3x + 1)

A = 0.5(12x² - 14x - 6)

A = 6x² - 7x - 3.

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Hence, the cone is 8/3 cm tall as we can get the height using the following formula for a cone's volume.

A three-dimensional object's volume is a measurement of how much space it takes up. It is a real-world physical number that can be expressed in cubic measurements like cubic metres (m3), cubic centimetres (cm3), or cubic feet (ft3). Physics, chemistry, architecture, and mathematics all use the idea of volume extensively. Volume is frequently used to refer to the amount of space that an object or substance takes up, for instance the amount of a container, the volume of either a liquid, or the quantity of a gas. Depending on an object's shape, a different formula is required to determine its volume.

given

The formula V = (1/3)r2h, where V is the volume, r is the radius of the base, and h is the height, can be used to determine the volume of a cone.

Hence, by multiplying the circumference by two, we can determine the radius of the base:

12π / 2π = 6

Thus, the base's radius is 6 cm.

Also, we are informed that the cone's volume is 96. As a result, we can get the height using the following formula for a cone's volume:

V = (1/3)r2h

96 = (1/3)(6/2)h

96 = 36 h

96 / 36 = 8/3

Hence, the cone is 8/3 cm tall as we can get the height using the following formula for a cone's volume.

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Nora will be able to cut 52 equal pieces of yarn for her Science project.

To find out how many equal pieces of yarn Nora can cut for her Science project, we need to divide the total length of the yarn by the length of each piece.

Total length of yarn: 67.6 inches
Length of each piece: 1.3 inches

Step 1: Divide the total length by the length of each piece.
67.6 inches ÷ 1.3 inches = 52

Nora will have 52 equal pieces of yarn, each 1.3 inches long, for her Science project.

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Answer:

6

Step-by-step explanation:

A=hbb/2=4·3/2=6

To find a triangle's area, use the formula area = 1/2 * base * height. Choose a side to use for the base, and find the height of the triangle from that base. Then, plug in the measurements you have for the base and height into the formula

or

The area of a triangle is the space enclosed within the three sides of a triangle. It is calculated with the help of various formulas depending on the type of triangle and is expressed in square units like, cm2, inches2, and so on.

2 cos x - 1 = 0, cos x = 1/2, and x = π/3 or 5π/3. These are the critical numbers of g(x) on (0,7). the absolute maximum value of g(x) on (0,7) is √3 - π/3 and the absolute minimum value is -√3 - 5π/3.

To find the critical numbers for g(x) = 2sin(r) - r on the interval (0, 7), follow these steps:
1. Find the derivative of g(x): g'(x) = 2cos(r) - 1
2. Set the derivative equal to zero: 2cos(r) - 1 = 0
3. Solve for r: r = cos^(-1)(1/2)
Now, find the absolute maximum and minimum values for g(x) on the interval (0, 7):
1. Evaluate g(x) at the critical numbers: g(cos^(-1)(1/2)) = 2sin(cos^(-1)(1/2)) - cos^(-1)(1/2)
2. Evaluate g(x) at the endpoints of the interval: g(0) = 2sin(0) - 0, g(7) = 2sin(7) - 7
3. Compare the values from steps 1 and 2 to find the maximum and minimum values.
The critical numbers for g(x) = 2sin(r) - r on the interval (0, 7) are r = cos^(-1)(1/2). The absolute maximum and minimum values for g(x) on the interval (0, 7) can be found by comparing g(cos^(-1)(1/2)), g(0), and g(7).

To find the critical numbers of g(x) = 2 sin x - x on (0,7), we first need to find its derivative:
g'(x) = 2 cos x - 1
Setting g'(x) = 0, we get:
2 cos x - 1 = 0
cos x = 1/2
x = π/3 or 5π/3
These are the critical numbers of g(x) on (0,7).

To find the absolute maximum and minimum values of g(x) on (0,7), we need to evaluate g(x) at the critical numbers and at the endpoints of the interval (0,7).
g(0) = 0
g(π/3) = 2 sin(π/3) - π/3 = √3 - π/3
g(5π/3) = 2 sin(5π/3) - 5π/3 = -√3 - 5π/3
g(7) = 2 sin(7) - 7
To determine the absolute maximum and minimum values, we compare these values:
The absolute maximum value is √3 - π/3, which occurs at x = π/3.
The absolute minimum value is -√3 - 5π/3, which occurs at x = 5π/3.
Therefore, the absolute maximum value of g(x) on (0,7) is √3 - π/3 and the absolute minimum value is -√3 - 5π/3.

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