A rectangular prism has a base area of 400 square inches. The volume of the prism is 2,400 cubic inches. What is the height of the prism? (7.9A)

The sum from i = 1 to n of 2^i is 2(2^n - 1), the sum from i = 1 to n of 1/(i(i+1)) is n/(n+1), n! > 2^n for n ≥ 4, and therefore, sqrt(2) is irrational. The intersection of sets A and B has |A ∩ B| elements, the union of sets A and B has |A ∪ B| elements, and the number of students on the CS team is 6.

Let's break down the questions and provide the proofs and solutions step by step:

Sum of powers of 2: We want to find the sum from i = 1 to n of 2^i for each n ≥ 1. We can use the formula for the sum of a geometric series to simplify the expression:

The sum of a geometric series is given by the formula Sn = a(r^n - 1)/(r - 1), where a is the first term, r is the common ratio, and n is the number of terms. In this case, a = 2, r = 2, and we need to find Sn.

Plugging in the values, we get Sn = 2(2^n - 1)/(2 - 1) = 2(2^n - 1).

Therefore, the sum from i = 1 to n of 2^i is 2(2^n - 1).

Sum of fractions: We want to find the sum from i = 1 to n of 1/(i(i+1)) for each n ≥ 1. We can rewrite the expression as follows:

1/(i(i+1)) = 1/i - 1/(i+1).

Now, we can observe that the terms cancel out in pairs when we sum them. The first term 1/1 remains, and the last term 1/(n+1) remains as well.

Therefore, the sum from i = 1 to n of 1/(i(i+1)) is 1 - 1/(n+1) = n/(n+1).

Proof of n! > 2^n: We will prove this by induction. The base case is n = 4: 4! = 24 > 2^4 = 16.

Now, assume the inequality holds for some k ≥ 4, i.e., k! > 2^k.

We need to prove it for k + 1: (k + 1)! = (k + 1) * k! > (k + 1) * 2^k (since k! > 2^k by the induction hypothesis).

It suffices to show that (k + 1) * 2^k > 2^(k + 1), which simplifies to k + 1 > 2.

Since k ≥ 4, the inequality holds.

Therefore, by induction, we can conclude that n! > 2^n for each n ≥ 4.

Proof that sqrt(2) is irrational: We will prove this by contradiction. Assume that sqrt(2) is rational, i.e., sqrt(2) can be expressed as a ratio of two integers p and q in its simplest form, where q ≠ 0.

sqrt(2) = p/q.

Squaring both sides, we get 2 = p^2/q^2.

Rearranging, we have p^2 = 2q^2.

This implies that p^2 is even, and thus p must be even.

Let p = 2k, where k is an integer.

Substituting back, we have (2k)^2 = 2q^2, which simplifies to 4k^2 = 2q^2.

Dividing by 2, we get 2k^2 = q^2.

This implies that q^2 is even, and thus q must be even.

However, if both p and q are even, then p/q is not in its simplest form, contradicting our initial assumption.

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After considering the given data we conclude that the generated two year discount factor is 0.9132, under the condition that three year annuity factor is 2.6065.

Applying the formula for the annuity factor, we can find the two year discount factor:
[tex]annuity factor = (1 - discount factor^n) / r[/tex]
Here,
n = number of years
r = annual interest rate.
We are given that the three year discount factor is 0.7773 and the one year discount factor is 0.9434. We are also given that the three year annuity factor is 2.6065.
Applying the formula for the annuity factor, we can evaluate the annual interest rate:
[tex]2.6065 = (1 - 0.7773^3) / r[/tex]
r = 0.1
Now, we can evaluate the two year discount factor:
[tex]annuity factor = (1 - discount factor^2) / 0.1[/tex]
[tex]2.6065 = (1 - discount factor^2) / 0.1[/tex]
[tex]discount factor^2 = 0.89335[/tex]
discount factor = [tex]\sqrt(0.89335)[/tex]
discount factor = 0.9132
Therefore, the two year discount factor is 0.9132.
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Answer:

Any x-value other than 4, 8, or 9 would make this set of points a function.

Answer:

18.2 miles

Step-by-step explanation:

simple multiplication:

We know that 1 inch of distance on the map is equal to 5.2 miles.

Using this knowledge we can multiply the amount of inches by the miles per inch.

5.2*3.5=18.2 miles apart

Hope this helps!!

Answer:

Step-by-step explanation:

The  direction of the magnetic field at P is perpendicular to both the current direction and the direction of the curl of our fingers.

The direction of the magnetic field at point P due to the current-carrying wire can be determined using the right-hand rule. If we point our right thumb in the direction of the current (which is along the positive x-axis), and curl our fingers toward the point P, then the direction of the magnetic field at P is perpendicular to both the current direction and the direction of the curl of our fingers.

In this case, since the wire is straight and lies in the x-y plane, the direction of the magnetic field at point P will be perpendicular to the plane of the page, and will either be pointing into or out of the page. To determine which direction, we need to know the orientation of point P relative to the wire. If point P is above the wire, then the magnetic field will be pointing into the page, and if it is below the wire, then the magnetic field will be pointing out of the page.

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The statement that is true is (LS – ES) >= 0. Option C

The latest possible time to begin an activity is denoted as LS in project management, while the earliest possible time is indicated by ES.

The distinction between LS and ES signifies the flexibility an activity possesses, allowing it to be postponed without affecting the ultimate timeframe of the project.

The calculation for Slack involves subtracting the earliest possible finish time (LS) from the latest possible finish time (LF). One must note that (LS - ES) will never be negative as the earliest start time (ES) can never be greater than the latest start time (LS).

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Answer:

(4x - 3)(4x + 3)

Step-by-step explanation:

16x² - 9 ← is a difference of squares and factors in general as

a² - b² = (a - b)(a + b) , then

16x² - 9

= (4x)² - 3²

= (4x - 3)(4x + 3)

Answer:

You basically grow rep

Step-by-step explanation:

Brainly points you need for asking questions and people are more likely to trust you with their questions if you have alot!

Answer:

i actually dont know

Step-by-step explanation:

Answer: 2,3,4

Step-by-step explanation:

Apple’s is represented by 8 and oranges is represented by 5. So the ratio 8:5 means every 8 apples their are 5 oranges

Answer:

ayla had 45 cupcakes for the class jayla gave a number out to the cladd Jayla took 15 cupcakes back home what is the does n equal to

In a case whereby sheri’s cab fare was $32, with a 20% gratuity and no taxes. sheri's write a check to the cab driver for $40, this can be considered as being reasonable amount because it is  $1.60 more to the cab driver.

A gratuity is a sum of money that customers typically give to specific service sector employees, including those in the hotel industry, in addition to the service's base charge for the work they have completed.

Given ; Sheri’s cab fare was $32 and the percentage of gratuity is 20%

amount of gratuity = 20% 0f 32 = 6.40

The fare of the cab  + gratuity = 32 + 6.40 = 38.40

Check to the cab driver for $40 ,  implies ($40 - $38.40)= $1.60 more to the cab driver.

Hence, it is reasonable.

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Rounding this to two decimal places, the equivalent nominal interest rate compounded monthly is approximately 2.53%.

To determine the equivalent nominal interest rate compounded monthly for a given nominal interest rate compounded quarterly, we can use the formula for nominal interest rate conversion.

The formula is:

r = (1 + i/n)^n - 1

Where:

r is the nominal interest rate compounded annually (we're looking for this),

i is the nominal interest rate compounded quarterly (2.50% in this case),

n is the number of compounding periods per year (4 for quarterly compounding).

Plugging in the values, we have:

r = (1 + 0.025/4)^4 - 1

Calculating this expression, we find:

r ≈ 0.025335

Rounding this to two decimal places, the equivalent nominal interest rate compounded monthly is approximately 2.53%.

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Answer:

1) D

2) B

Step-by-step explanation:

Hope this Helps!

:)

Answer:

The recipe will make 7.2 quarts of punch.

Step-by-step explanation:

A punch recipe calls for 3 liters of ginger ale and 1.5 liters of tropical fruit juice.

Then the recipe will make 4.5 liters of punch, obtained by the sum of ginger ale and tropical fruit juice.

The ration is the comparison of two quantities and is measured from the division of two values, then: [tex]\frac{a}{b}[/tex].

The proportion is the equality between two or more ratios. That is, [tex]\frac{a}{b}=\frac{c}{d}[/tex] equals a proportion.

In this case, being 1 L= 1.6 quarts, you have:

[tex]\frac{1.6 quarts}{1 L}=\frac{x}{4.5 L}[/tex]

Solving:

[tex]x=4.5 L*\frac{1.6 quarts}{1 L}[/tex]

x= 7.2 quarts

So, the recipe will make 7.2 quarts of punch.

Answer:

9526/87

Step-by-step explanation:

87·x=9,526

x=9526/87

Answer:

B. and D.

Step-by-step explanation:

If you folded that shape along the lines o and m, it would match up.

There are two lines of symmetry m and o. Options B and D are correct.

From the given figure we have to determine the axes of symmetries.

A line is a straight curve connecting two points or more showing the shortest distance between the initial and final points.

Here,
As of the given figure,
There are 2 axes of symmetry first line m and the other is line o.
Line l and line n are not symmetrical lines because the reflection of one portion will not be superimposed with the other.

Thus, there are two lines of symmetry m and o. Options B and D are correct.

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Vectors along the lines:
To find vectors along the lines, we need two points on each line. Let's denote the points on Line A as A₁ and A₂, and the points on Line B as B₁ and B₂.
Vector along Line A: v₁ = A₂ - A₁
Vector along Line B: v₂ = B₂ - B₁.

Define the vectors along the lines:
Let's say we have two lines represented by the following equations:
Line 1: r₁(t) = a₁ + t * u₁
Line 2: r₂(t) = a₂ + t * u₂
Here, a₁ and a₂ are points on the lines, and u₁ and u₂ are the direction vectors of the lines.

Find the cross product:
To find the unit vector n, we can take the cross product of the direction vectors:
n = u₁ × u₂
Note: × represents the cross product operator.

Define a vector spanning two points on each line:
Let's choose two points on each line, which we'll call b₁ and b₂. The vector spanning these points on Line 1 is given by:
v₁ = b₁ - a₁

Similarly, the vector spanning the points on Line 2 is given by:
v₂ = b₂ - a₂

Determine the minimum distance:
The minimum distance (d) between the lines can be found using the formula:
d = |(v₁ × u₂) · n| / |n|
In this formula, · represents the dot product operator, and | | represents the magnitude or length of the vector.

Alternatively, if you prefer using symmetric equations, you can substitute the equations of the lines directly into the formula to find the minimum distance.

Remember to substitute the appropriate values for a₁, a₂, u₁, u₂, b₁, and b₂ based on the specific problem you are solving.

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Answer:

9/30

Step-by-step explanation:

Answer:

The equations are

X + Y = 48 ----Eq (1)

Y = 2 X -----Eq (2)

Solution of the equations and the two number are

X = 16, Y = 32

Step-by-step explanation:

Let the two numbers be X and Y

Given -

The sum of two numbers is 48.

Thus, X + Y = 48 ----Eq (1)

The second number is twice the first

Thus, Y = 2 X -----Eq (2)

Substituting the value of Y in equation (1), we get -

X + 2X = 48

3 X = 48

X = 16

Y = 2 X = 32

Answer:

Step-by-step explanation:

15,13,11,9,1,3,5,7

Answer:

13

Step-by-step explanation:

Supplementary angles add up to 180 and angles on a straight line add up to 180 so 13

To evaluate the line integral ∫c xy⁴ ds, we need to parameterize the curve c, then substitute into the integrand, and integrate with respect to the parameter.

The right half of the circle x² + y² = 9 can be parameterized as x(t) = 3cos(t), y(t) = 3sin(t) for t in [0, pi]. Note that this parameterization traces out the right half of the circle oriented counterclockwise.

Now, we can express ds as ds = sqrt(dx/dt² + dy/dt²) dt. Using the parameterization x(t) = 3cos(t), y(t) = 3sin(t), we get dx/dt = -3sin(t) and dy/dt = 3cos(t). Thus, ds = sqrt((-3sin(t))² + (3cos(t))²) dt = 3dt.

Substituting x(t) = 3cos(t), y(t) = 3sin(t), and ds = 3dt into the integrand xy⁴, we get xy⁴ = (3cos(t))(3sin(t))⁴ = 81/4 sin⁴(t) cos(t).

So, the line integral becomes:

∫c xy⁴ ds = ∫₀ᴨ (81/4 sin⁴(t) cos(t))(3 dt)

Using trigonometric identities, we can simplify the integrand to:

(81/4)(1/5)(sin⁵(t))' = 81/20 sin⁵(t)

Evaluating the integral from t = 0 to t = pi, we get:

∫c xy⁴ ds = ∫₀ᴨ 81/20 sin⁵(t) dt = 81/16π

Therefore, the value of the line integral is 81/16π.

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To apply Rouché's Theorem, we consider two functions: f(z) = 2z⁸ + 3z⁵ - 9z³ + 2 and g(z) = 2z⁸. We want to analyze the number of roots of f(z) = 0 inside the region 1 < |z| < 2.

First, let's examine the behavior of f(z) and g(z) on the boundary of the region. When |z| = 1, the term 2z⁸ dominates over the other terms in f(z), so |f(z)| < |g(z)|. On the other hand, when |z| = 2, the term 2z⁸ is still dominant, and again |f(z)| < |g(z)|.

Since |f(z)| < |g(z)| on the boundary of the region, Rouché's Theorem guarantees that f(z) and g(z) have the same number of roots inside the region, counting multiplicities. In this case, g(z) = 2z⁸ has exactly eight roots, counting multiplicities.

Therefore, by Rouché's Theorem, we can conclude that the equation 2z⁸ + 3z⁵ - 9z³ + 2 = 0 has eight roots, counting multiplicities, inside the region 1 < |z| < 2.

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Answer:

x=41

Step-by-step explanation:

x/50=82/100

cross multiply

100x= 82×50

100x= 4100

x= 4100/100

x= 41

check our work

41/50 × 100= 82 percent

Answer: [tex]202.5\ in.^3[/tex]

Step-by-step explanation:

Given

The area of the base is [tex]A=40.5\ in.^2[/tex]

height of the triangular prism is [tex]h=15\ in.[/tex]

The volume of a triangular prism is  [tex]V=\dfrac{1}{3}Ah[/tex]

The volume of given Prism is

[tex]V=\dfrac{1}{3}\times 40.5\times 15=202.5\ in.^3[/tex]

The volume of the triangular prism is [tex]202.5\ in.^3[/tex]