The diameter of a spherical basketball is 10 inches. What is the volume of the basketball? Give the exact solution. ​

To show that `(n + 3)7 ∈ Θ(n7)`, we need to prove that `(n + 3)7 = Θ(n7)`.This can be done by showing that `(n + 3)7 = O(n7)` and `(n + 3)7 = Ω(n7)` .Now, let's prove the two parts separately:

Proof for `(n + 3)7 = O(n7)`.

We want to prove that there exists a positive constant c and a non-negative constant k such that `(n + 3)7 ≤ cn7` for all `n ≥ k`.Using the Binomial theorem, we can expand `(n + 3)7` as:```
(n + 3)7
= n7 + 7n6(3) + 21n5(3)2 + 35n4(3)3 + 35n3(3)4 + 21n2(3)5 + 7n(3)6 + 37
≤ n7 + 21n6(3) + 21n5(3)2 + 35n4(3)3 + 35n3(3)4 + 21n2(3)5 + 7n(3)6 + n7
≤ 2n7 + 21n6(3) + 21n5(3)2 + 35n4(3)3 + 35n3(3)4 + 21n2(3)5 + 7n(3)6
≤ 2n7 + 84n6 + 441n5 + 2205n4 + 10395n3 + 45045n2 + 153609n + 729
```Thus, we can take `c = 153610` and `k = 1` to satisfy the definition of big-Oh notation. Hence, `(n + 3)7 = O(n7)`.Proof for `(n + 3)7 = Ω(n7)`We want to prove that there exists a positive constant c and a non-negative constant k such that `(n + 3)7 ≥ cn7` for all `n ≥ k`.Using the Binomial theorem, we can expand `(n + 3)7` as:```
(n + 3)7
= n7 + 7n6(3) + 21n5(3)2 + 35n4(3)3 + 35n3(3)4 + 21n2(3)5 + 7n(3)6 + 37
≥ n7
```Thus, we can take `c = 1` and `k = 1` to satisfy the definition of big-Omega notation. Hence, `(n + 3)7 = Ω(n7)`.

As we have proved that `(n + 3)7 = O(n7)` and `(n + 3)7 = Ω(n7)`, therefore `(n + 3)7 = Θ(n7)`.Thus, we have shown that `(n + 3)7 ∈ Θ(n7)`.From the proof, we can see that we used the Binomial theorem to expand `(n + 3)7` and used algebraic manipulation to bound it from above and below with suitable constants. This technique can be used to prove the time complexity of various algorithms, where we have to find the tightest possible upper and lower bounds on the number of operations performed by the algorithm.

Hence, we have shown that `(n + 3)7 ∈ Θ(n7)` for non-negative integer n.

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Using the cosine rule, the distance the plane has flown during Jack's phone call can be calculated by taking the square root of the sum of the squares of the initial and final distances, minus twice their product, multiplied by the cosine of the angle difference.

To determine the distance the plane has flown during Jack's phone call, we can use the cosine rule in trigonometry.

The cosine rule relates the lengths of the sides of a triangle to the cosine of one of its angles.

Let's denote the initial distance from Jack to the plane as d1 and the final distance as d2.

We know that the altitude of the plane remains constant at 2400 meters.

According to the cosine rule:

\(d^2 = a^2 + b^2 - 2ab \times cos(C)\)

Where d is the side opposite to the angle C, and a and b are the other two sides of the triangle.

For the initial angle of elevation (70°), we have the equation:

\(d1^2 = (2400)^2 + a^2 - 2 \times 2400 \times a \timescos(70)\)

Similarly, for the final angle of elevation (50°), we have:

\(d2^2 = (2400)^2 + a^2 - 2 \times 2400 \times a \times cos(50)\)

To find the distance the plane has flown, we subtract the two equations:

\(d2^2 - d1^2 = 2 \times 2400 \times a \times (cos(70) - cos(50))\)

Now we can solve this equation to find the value of a, which represents the distance the plane has flown.

Finally, we calculate the square root of \(a^2\) to find the distance in meters.

It's important to note that the angle of elevation assumes a straight-line path for the plane's movement and does not account for any changes in altitude or course adjustments that might occur during the phone call.

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Answer:

Step-by-step explanation:

You just combine common variables.

8a^5 -4

3a^5 + a -2

11a^5 + a -6

The solution to the first is 11a^5+a-6.

The second:

3x^2 -2x + 1

-x^2 + 3x + 1

2x^2 + x + 2

2x^2+x+2

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see this attachment

Step-by-step explanation:

age of grandpa=72

age of grandson =16

For any function, we call x the numerator value and y is the denominator value.

A function can be defined as a mathematical expression which is used to define and represent the relationship that exists between two or more variables. This ultimately implies that, a function is typically used for mapping an input variable to an output variable.

In Mathematics, a fraction comprises two (2) main parts and these include the following:

For any function in Mathematics, we call x the numerator value while y is the denominator value.

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Answer:

the data is constructed around 6 and 1 is the outlier

Step-by-step explanation:

\(3\times \dfrac 7{12} = \dfrac{3 \times 7}{12}=\dfrac{3 \times 7}{3 \times 4} = \dfrac 74\)

Answer:

399.6 square miles

Step-by-step explanation:

The Area of a sector =

x/360° × πr²

r = 13.4 miles

x = 255°

Hence,

255°/360° × π × 13.4²

= 399.5739336 square miles

Approximately = 399.6 square miles

Answer:

399.6

Step-by-step explanation:

It will take approximately 39.532 days for $9500 to earn $800 at an annual interest rate of 8.25%.

To find the number of days it will take for $9500 to earn $800 at an annual interest rate of 8.25%, we need to use the formula for simple interest:

Interest = Principal * Rate * Time

In this case, we are given the interest ($800), the principal ($9500), and the annual interest rate (8.25%). We need to solve for time.

Let's denote the time in years as "t". Since we're looking for the number of days, we'll convert the time to a fraction of a year by dividing by 365 (assuming a standard 365-day year).

$800 = $9500 * 0.0825 * (t / 365)

Simplifying the equation:

800 = 9500 * 0.0825 * (t / 365)

Divide both sides by (9500 * 0.0825):

800 / (9500 * 0.0825) = t / 365

Simplify the left side:

800 / (9500 * 0.0825) ≈ 0.1083

Now, solve for t by multiplying both sides by 365:

0.1083 * 365 ≈ t

t ≈ 39.532

Therefore, it will take approximately 39.532 days for $9500 to earn $800 at an annual interest rate of 8.25%.

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The best way to analyze their data for significance is to Conduct a two-way variable analysis.

As we can see, there is a 2x3 factorial design. Thus, we know that there will be a two level or two way analysis of variables, which is referred to as two way variables analysis. Factorial design is a research method that allows for the inquiry of a main and interaction effects of two or more independent variables on one or more outcome variables (s).

It has been argued that factorial designs represent the true beginning of modern behavioral research and have resulted in a significant paradigm shift in how social scientists conceptualize their research questions and produce objective results (Kerlinger & Lee, 2000).

Factorial design is an experimental methodology that goes beyond standard single-variable experimentation. Previously, social scientists were fixated on single independent variable experiments, foreshadowing the significance of extraneous variables that can attenuate or diminish research findings.

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Answer:

1.) 27a^11 + 18a^9 -72a^7

2.) 6p^4q^3 - 10p^3q + 4p^2q3

Step-by-step explanation:

1.) Distribute and do the math

-9a^5 x (-3a^6) - 9a^5 X (-2a^4) - 9a^5 x 8a^2

27a^11 + 18a^9 -72a^7

2.)  Distribute and do the math

2pq^2q x 3p^2q^2 - sp^2q x 5p + 2p^2q x 2q^2

6p^4q^3 - 10p^3q + 4p^2q3

The x is a specific value that the variable can take on.

A random variable is a variable that has a numerical value, which is determined by the result of a random experiment. The numerical value is determined by a set of probabilities, which are associated with each possible outcome of the experiment. Random variables can be classified as discrete or continuous, depending on the type of outcomes that they can have. Examples of random variables include the number of heads in a series of coin tosses, the time until a traffic light changes from red to green, and the height of a randomly selected person.The randomness of a random variable comes from the fact that the outcome of the experiment that it represents is uncertain. The uncertainty arises because the experiment is subject to the effects of chance, rather than being completely determined by fixed physical laws or other deterministic factors. Because of this, it is not possible to predict with certainty what the value of a random variable will be, even if we know the probabilities associated with each possible outcome.There are many different notations that can be used to represent random variables, depending on the context and the preferences of the person using the notation. Some of the most common notations include X, Y, and Z, as well as Greek letters such as µ, σ, and ρ. When a random variable is discrete, it is often represented using a probability mass function, which is denoted by P(X=x), where x is a specific value that the variable can take on. When a random variable is continuous, it is often represented using a probability density function, which is denoted by f(x), where x is a specific value that the variable can take on.

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Answer:

A is the correct answer

Step-by-step explanation:

Hope that helps

Answer:

279

Step-by-step explanation:

Answer:

The y-intercept is (0,-3

Step-by-step explanation:

Step-by-step explanation:

to find the y intercept you simply equate x to 0. therefore, the y intercept is -4.5

and its coordinate is:

(0, -4.5)

Answer:

0.75

Step-by-step explanation:

9+3 = 12

a half is 50%, a quarter is 25% so a half quarter would be 75%.

75 ÷ 100 = 0.75

Answer:

35

Step-by-step explanation:

f(x)=5x^3-2x^2+3

f(2)=5x2^3-2x2^2+3

=5x8-2x4+3

=40-8+3

=40-5

=35

Answer:

35

Step-by-step explanation:

F(x) = 5x^3 -2x^2 + 3

Let x=2

F(2) = 5(2)^3 -2(2)^2 + 3

       = 5*8 -2(4) +3

       = 40 -8+3

       = 32+3

      = 35

Balloon's altitude = 43.7ft as

change in altitude/500 = tan 5°  

change in altitude = .0875*500

                              =43.7ft

A flow network with supplies is a directed capacitated graph where each edge has a capacity indicating the maximum flow that can pass through it. The flow in the network represents the movement of a certain resource (e.g., water, electricity, goods) from sources to sinks.

In a flow network with supplies, there may be multiple sources, which are nodes that generate the resource and have outgoing edges, and multiple sinks, which are nodes that consume the resource and have incoming edges.

The sources and sinks can have different supply or demand values, indicating the amount of resource they generate or consume.

The edges in the flow network have capacities that restrict the maximum flow that can pass through them. The capacity represents the limit on the amount of resource that can traverse the edge. The flow through an edge cannot exceed its capacity.

The objective in a flow network is to determine the maximum flow that can be sent from sources to sinks while respecting the capacities of the edges. This is typically solved using algorithms such as the Ford-Fulkerson algorithm or the Edmonds-Karp algorithm.

The concept of a flow network with supplies is important in various applications, such as transportation networks, communication networks, and supply chain management, where resources need to be efficiently distributed from multiple sources to multiple sinks, taking into account the capacity constraints of the network.

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Using linear functions, it is found that both snowman's will have the same height of 21 inches after 5 hours.

A linear function is modeled by:

y = mx + b

In which:

For this problem, we have that:

Hence the functions are given as follows:

They will have the same height when:

A(t) = B(t)

51 - 6t = 36 - 3t

3t = 15

t = 5.

The height will be of:

B(3) = A(3) = 51 - 6(5) = 21 inches.

Both snowman's will have the same height of 21 inches after 5 hours.

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Answer:

Sum is

Step-by-step explanation:

Answer:

x = 7

Step-by-step explanation:

The proportion is

14/6 = 21/(3x - 12)                 Cross Multiply

14*(3x - 12) = 6*21                 Remove the brackets and combine

42x - 168 = 126                     Add 168 to both sides

42x = 126 + 168

42x =  294                           Divide by 42

x = 294/42

x = 7

Answer:

4/5

Step-by-step explanation:

The expected value with a sample size of 20 and a probability of 0.25 will be 5.

The anticipated value is an extension of the weighting factor in statistical inference. Informally, the anticipated value is the simple average of a significant number of outcomes of a randomly selected variable that was separately chosen.

The expected value is given below.

E(x) = np

Where n is the number of samples and p is the probability.

If x is a binomial random variable with n = 20 and p = 0.25. Then the expected value is given below.

E(x) = 20 x 0.25

E(x) = 5

The expected value with n = 20 and p = 0.25 will be 5.

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