A number increased by 14?

If 4n is a factor of 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9, then the largest value of n is 90,720.

Factors of a number or an algebraic expression are the numbers or expressions which divides the given number evenly.

For example, 12 = 2 × 6

So we can say that 2 and 6 are factors of 12.

The product of the given numbers from 1 to 9 is given.

1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9

Given a multiple of 4 is a factor of this product, where that is the largest factor.

So, we have to find the greatest factor of this product which is also a multiple of 4.

Any number multiplied by 4 is a multiple of 4.

1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 = 4 × (1 x 2 x 3 x 5 x 6 x 7 x 8 x 9)

So the greatest n such that 4n is a factor of 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 is 1 x 2 x 3 x 5 x 6 x 7 x 8 x 9 = 90,720

Hence the greatest n is 90,720.

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The 90% confidence interval for the population standard deviation \($\sigma$\)   is (2971.08, 5447.54).

What is the confidence interval?

A confidence interval is a range of values, derived from a sample of data, that is likely to contain an unknown population parameter with a certain level of confidence.

To construct a confidence interval for the population variance \($\sigma^2$\) and the population standard deviation \($\sigma$\) with a 90% level of confidence, we will use the Chi-Square distribution. The formula for the Chi-Square distribution is:

\($$(n-1)\frac{S^2}{\sigma^2} \sim \chi^2_{n-1}$$\)

\($$\frac{(n-1)S^2}{\chi^2_{n-1,0.95}} \leq \sigma^2 \leq \frac{(n-1)S^2}{\chi^2_{n-1,0.05}}$$\)

Substituting the values from the problem, we get:

\($$\frac{(21-1)(3687)^2}{32.852} \leq \sigma^2 \leq \frac{(21-1)(3687)^2}{10.856}$$\)

Simplifying, we get:

\($$8823846 \leq \sigma^2 \leq 29745123$$\)

So the 90% confidence interval for the population variance \($\sigma^2$\) is (8823846, 29745123).

To find the confidence interval for\($\sigma$\), we take the square root of both sides of the inequality:

\($$\sqrt{\frac{(n-1)S^2}{\chi^2_{n-1,0.95}}} \leq \sigma \leq \sqrt{\frac{(n-1)S^2}{\chi^2_{n-1,0.05}}}$$\)

Substituting the values from the problem, we get:

\($$\sqrt{\frac{(21-1)(3687)^2}{32.852}} \leq \sigma \leq \sqrt{\frac{(21-1)(3687)^2}{10.856}}$$\)

Simplifying, we get:

\($$2971.08 \leq \sigma \leq 5447.54$$\)

Interpretation: We can say with 90% confidence that the population variance \($\sigma^2$\) lies within the interval (8823846, 29745123) and the population standard deviation \($\sigma$\) lies within the interval (2971.08, 5447.54). This means that if we were to take many random samples of size 21 from this population and calculate the sample variance and sample standard deviation for each sample, we would expect 90% of the intervals we construct using these formulas to contain the true population variance and standard deviation, respectively.

Hence, the 90% confidence interval for the population standard deviation \($\sigma$\)   is (2971.08, 5447.54).

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x(t) = \(c_1e^1^4^t+c_2e^-^t\)  , y(t) = \(2c_1e^1^4^t-c_2e^-^t\)   is the general solution of the sytem of differention eqution dx/dt = 4x+ 5y , dy/dt = 10x + 9y .

given  dx/dt = 4x+ 5y  and dy/dt = 10x + 9y

X'(t) = \(\left[\begin{array}{ccc}4&5\\10&9\end{array}\right]\)  X

where X = \(\left[\begin{array}{ccc}x\\y\\\end{array}\right]\)

so A = \(\left[\begin{array}{ccc}4&5\\10&9\end{array}\right]\)

now we need to find eigen value of the matrix and the corresponding eigen vector.

| A- λI | = 0

so after equation the value to zero

λ = 14 and λ = -1

now for the  λ = 14 corresponding eigen vector = \(\left[\begin{array}{ccc}1\\2\\\end{array}\right]\)

for λ = -1 corresponding eigen vector =  \(\left[\begin{array}{ccc}1\\-1\\\end{array}\right]\)

So general equation is given by:

X(t) = \(c_1e^1^4^t\)  \(\left[\begin{array}{ccc}1\\2\\\end{array}\right]\)  +  \(c_2e^-^t\)   \(\left[\begin{array}{ccc}1\\-1\\\end{array}\right]\)

after solving this

x(t) = \(c_1e^1^4^t+c_2e^-^t\)

y(t) = \(2c_1e^1^4^t-c_2e^-^t\)

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Answer:

  (B)  123.2 yd²

Step-by-step explanation:

You want the area of the decagon shown with side length 4 yd and apothem 6.16 yd.

The area is given by the formula ...

  A = 1/2Pa

where P is the perimeter and 'a' is the apothem, the distance from the center to the middle of one side of the regular polygon.

The perimeter is the sum of the side lengths. Each of the 10 sides has a length of 4 yd, so that sum is 10·4 yd = 40 yd.

Using the known values in the formula, we find the area to be ...

  A = 1/2(40 yd)(6.16 yd)

  A = 123.2 yd²

__

Additional comment

Effectively, you are computing 10 times the area of the triangle that is the area of one sector. That triangle has a base of 4 yd and a height of 6.16 yd. Its area is ...

  A = 1/2bh = 1/2(4 yd)(6.16 yd) = 12.32 yd²

The 10 sectors of the decagon will have an area of ...

  A = 10 × 12.32 yd² = 123.2 yd²

Answer:

u=122

Step-by-step explanation:

hey there!

32+u-47+u-49=180 (sum of angle of triangle)

2u-64=180

2u=180+64

u=244/2

u=122

hope it helps!!

thank u!

Answer:

So the answer is 12cm square

Step-by-step explanation:

Base is 6cm and hight is 4 so 6×4 is 24÷2 is 12

Answer:

Yes.

Step-by-step explanation:

Answer:

yes, they are congruent

Step-by-step explanation:

Answer:

y=2X

Answer:

I think is B.

Step-by-step explanation:

hope its help

Answer:

Option B

Step-by-step explanation:

The pressure of a gas in a container does not depend on the number of atoms in the container.

In the ideal gas law, pressure (P) is determined by the gas constant (R), temperature (T), and the number of moles of gas (n). It is given by the equation P = (nRT) / V, where V is the volume of the container. This equation shows that pressure is directly proportional to temperature and the number of moles of gas, and inversely proportional to the volume of the container.

Therefore, the pressure of the gas in the container does not depend on the number of atoms in the container, but rather on the number of moles of gas present. The number of atoms in a gas depends on the molecular formula and the Avogadro's constant, but it does not directly affect the pressure of the gas.

A real-world situation that would be described by this graph is a gas cylinder used for storage or transportation. The pressure inside the cylinder would depend on the number of moles of gas present, which can be controlled by adjusting the volume of the container or adding/removing gas. The temperature of the gas would also affect the pressure, as an increase in temperature would increase the pressure. However, the number of atoms in the gas would not directly affect the pressure in this scenario.

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Answer:

1.8125 lb(pounds)

Step-by-step explanation:

there are 16  oz in 1 lb

so 29/16 = 1.8125

The rate of change of the area is equal to -3 * 8^2 = -12.694 in2/min.

The rate of change of the area is the rate at which the area is increasing or decreasing over a given period of time. It is calculated by taking the difference between the initial and final area values and dividing it by the amount of time it took for the change to occur.

For example, if the area of a square increases from 10 square meters to 12 square meters over a period of two days, then the rate of change of the area would be (12-10)/2 = 1 square meter per day.

The rate of change of the area of a circle is equal to the negative of the product of the rate of change of the radius and the square of the radius. The rate of change of the radius is given as -3 inches per minute, and the radius is given as 8 inches. So, the rate of change of the area is equal to -3 * 8^2 = -12.694 in2/min.

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Answer:You set it up as y=mx+b

Step-by-step explanation:

Answer:

27n + 66p and 12x +75y +21

Answer:

the second one and the fourth one

Answer:

6. No. See explanation below.

7. 18 months

8. 16

Step-by-step explanation:

6. To rewrite a sum of two numbers using the distributive property, the two numbers must have a common factor greater than 1.

Let's find the GCF of 85 and 99:

85 = 5 * 17

99 = 3^2 + 11

5, 3, 11, and 17 are prime numbers. 85 and 99 have no prime factors in common. The GCF of 85 and 99 is 1, so the distributive property cannot be used on the sum 85 + 99.

Answer: No because the GCF of 85 and 99 is 1.

7.

We can solve this problem with the lest common multiple. We need to find a number of a month that is a multiple of both 6 and 9.

6 = 2 * 3

9 = 3^2

LCM = 2 * 3^2 = 2 * 9 = 18

Answer: 18 months

We can also answer this problem with a chart. We write the month number and whether they are home or on a trip. Then we look for the first month in which both are on a trip.

Month                Charlie          Dasha

1                          home             home

2                          home             home

3                          home             home

4                          home             home

5                          home             home

6                         trip                home

7                          home             home

8                          home             home

9                          home             trip

10                         home             home

11                         home             home

12                         trip               home

13                         home             home

14                         home             home

15                         home             home

16                         home             home

17                         home             home

18                         trip                 trip

Answer: 18 months

8.

First, we find the prime factorizations of 96 an 80.

96 = 2^5 * 3

80 = 2^4 *5

GCF = 2^4 = 16

Answer: 16

The depth (d) in feet of the aquatic robot (t) minutes after it is launched from a ship 12,189 feet above the ocean floor is given by the formula he robot is about 6,351 feet away from the ocean floor.

The depth is negative, which means the robot is above the ocean floor, not below it. To find the distance of the robot from the ocean floor, we need to take the absolute value of the depth The distance of the robot from the ocean floor can be found by substituting t = 45 in the formula d = -412 + 12,189 d = 11,777 feet Therefore, the robot is approximately 11,777 feet from the ocean floor after 45 minutes. However, none of the answer choices provided match this value exactly.

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The values of ∂z/∂r and ∂z/∂s. These partial derivatives will depend on the specific function F(u, v, w) provided.

To find ∂z/∂r and ∂z/∂s, we need to differentiate z = F(u, v, w) with respect to r and s.
Given that u = r^2, v = -2s^2, and w = ln(r) + ln(s), we can substitute these values into z = F(u, v, w).
So, z = F(r^2, -2s^2, ln(r) + ln(s)).
To find ∂z/∂r, we differentiate z with respect to r while treating s as a constant. This gives us:
∂z/∂r = ∂F/∂u * ∂u/∂r + ∂F/∂w * ∂w/∂r.
Similarly, to find ∂z/∂s, we differentiate z with respect to s while treating r as a constant. This gives us:
∂z/∂s = ∂F/∂v * ∂v/∂s + ∂F/∂w * ∂w/∂s.
Since we don't have the specific function F(u, v, w) mentioned in the question, we cannot determine the values of ∂z/∂r and ∂z/∂s. These partial derivatives will depend on the specific function F(u, v, w) provided.

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In the basis step of the mathematical induction proof for P(n), we show that the equation is true for n = 1. The equation of the inductive hypothesis (IH) is P(k), where k is an arbitrary natural number. In the inductive step, we need to show that if P(k) is true, then P(k+1) is also true.

In the basis step, we substitute n = 1 into the equation 11⋅2+12⋅3+⋅⋅⋅+1n⋅(n+1)=nn+1. This gives us the equation 1⋅2 = 1+1, which is true.

The inductive hypothesis (IH) is denoted as P(k), where k is an arbitrary natural number. We assume that P(k) is true, meaning that 11⋅2+12⋅3+⋅⋅⋅+1k⋅(k+1)=kk+1 holds.

In the inductive step, we need to show that if P(k) is true, then P(k+1) is also true. This involves substituting n = k+1 into the equation 11⋅2+12⋅3+⋅⋅⋅+1n⋅(n+1)=nn+1 and demonstrating that the equation holds for this value. The specific equation we need to show in the inductive step is 11⋅2+12⋅3+⋅⋅⋅+1(k+1)⋅((k+1)+1)=(k+1)(k+1+1).

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In the basis step, we need to show that the equation P(1) is true. The equation of the inductive hypothesis (IH) is P(k), where k is any natural number greater than or equal to 1. In the inductive step, we need to show that if P(k) is true, then P(k+1) is also true.

To prove the equation P(n): 11⋅2 + 12⋅3 + ... + 1n⋅(n+1) = n(n+1) using mathematical induction, we follow the two-step process.

1. Basis Step:

We start by showing that the equation is true for the base case, which is n = 1:

P(1): 11⋅2 = 1(1+1)

Simplifying, we get: 2 = 2, which is true.

2. Inductive Step:

Assuming that the equation is true for some arbitrary value k, the inductive hypothesis (IH) is:

P(k): 11⋅2 + 12⋅3 + ... + 1k⋅(k+1) = k(k+1)

In the inductive step, we need to show that if P(k) is true, then P(k+1) is also true:

P(k+1): 11⋅2 + 12⋅3 + ... + 1k⋅(k+1) + 1(k+1)⋅((k+1)+1) = (k+1)((k+1)+1)

By adding the (k+1)th term to the sum on the left side and simplifying the right side, we can demonstrate that P(k+1) is true.

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Answer:

  7√2 ≈ 9.9 dm

Step-by-step explanation:

You want the radius of a circle when tangents from a point 14 dm from the center make a right angle.

The attached figure shows all of the angles between radii and tangents are right angles. Effectively, the tangents and radii make a square whose side length is the radius of the circle. The diagonal of the square is given as 14 dm. We know this is √2 times the side length, so the length of the radius is ...

  r = (14 dm)/√2 = 7√2 dm ≈ 9.8995 dm ≈ 9.90 dm

The radius is about 9.90 dm.

__

Additional comment

The angles at A and O are supplementary, so both are 90°. The angles at the points of tangency are 90°, so the figure is at least a rectangle. Since adjacent sides (the radii, the tangents) are congruent, the rectangle must be a square. The given length is the diagonal of that square.

For side lengths s, the Pythagorean theorem tells you the diagonal length d satisfies ...

  d² = s² +s² = 2s²

  d = s√2

  d/√2 = s . . . . . . . . the relation we used above

This relationship between the sides and diagonal of a square is used a lot, so is worthwhile to remember.

Answer:

Step-by-step explanation:

The dimension of the basis is {[1 0 0 2], [-1 1 0 0]}.

To find a basis for the subspace S = span {[1 -1 0 2], [3 -5 4 8], [0 1 -2 -1]}, we can use the same method as in the example. First, we put the vectors in a matrix and row-reduce it:

[1 -1 0 2]
[3 -5 4 8]
[0 1 -2 -1]

R2 - 3R1 -> R2
R3 -> R3 + 2R1

[1 -1 0 2]
[0 -2 4 2]
[0 1 -2 -1]

-1/2R2 -> R2

[1 -1 0 2]
[0 1 -2 -1]
[0 1 -2 -1]

R3 - R2 -> R3

[1 -1 0 2]
[0 1 -2 -1]
[0 0 0 0]

We can see that the last row is all zeros, so we have only two pivots and one free variable. This means that the dimension of the subspace S is 2. To find a basis, we can write the pivots as linear combinations of the original vectors:

[1 -1 0 2] = [1 0 0 2] + [-1 1 0 0]
[0 1 -2 -1] = [0 1 -2 -1]

Therefore, a basis for S is {[1 0 0 2], [-1 1 0 0]}.

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Answer:

312,000,000

Step-by-step explanation:

Standard notation is when you move the decimal however many times the 10 is being squared by. In this case, it is 8.

Since the 8 is not negative, we move it 8 times to the right.

Since there are only 2 digits, we add 6 more zeros because 2+6=8

that is how you get 312,000,000

Step-by-step explanation:

great than 2 so the answer could never be 2

Answer:

Number of regular packs sent = 13

Step-by-step explanation:

Let,

x be the number of regular packs shipped

y be the number of super packs shipped

According to given statement;

x+y=32     Eqn 1

4x+6y=166    Eqn 2

Multiplying Eqn 1 by 4

4(x+y=32)

4x+4y=128     Eqn 3

Subtracting Eqn 3 from Eqn 2

(4x+6y)-(4x+4y)=166-128

4x+6y-4x-4y=38

2y=38

Dividing both sides by 2

\(\frac{2y}{2}=\frac{38}{2}\\y=19\)

Putting y=19 in Eqn 1

x+19=32

x=32-19

x= 13

Hence,

Number of regular packs sent = 13

The value of (f· g)(x) from the given functions is  -x^4 - x^2 - x.

Functions are the fundamental part of the calculus in mathematics. The functions are the special types of relations. A function in math is visualized as a rule, which gives a unique output for every input x.

Here,

The given functions are f(x)=x³+x+1 and g(x)=-x.

(f·g)(x)= f(x) × g(x)

= (x³+x+1) × (-x)

=  -x^4 - x^2 - x

so, e get,

(g·f)(x)= g(x) × f(x)

= (-x) × (x³+x+1)

= -x^4 - x^2 - x

Therefore, the value of (f· g)(x) from the given functions is  -x^4 - x^2 - x.

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By using the properties of parallelogram, it can be calculated that

Value of angle O is 105°

What is a parallelogram?

Parallelogram is a four sided two dimensional figure whose every pair of opposite sides are parallel.

x + 40 + 3x = 180 [Co interior angles are supplementary]

4x + 40 = 180

4x = 180 - 40

4x = 140

x = \(\frac{140}{4}\)

x = 35°

m\(\angle\)O = 35 \(\times\) 3 = 105°

The value of x in the parallelogram LMNO is 105°

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Complete Question

What is the measure of angle O in parallelogram LMNO? a) 35° b) 75°

c) 105° d) 155°

Step-by-step explanation:

G = 90° ( vertically opposite angles)

50 + 90 +x = 180 ( sum of angles on a straight line)

140 + x = 180

x= 40 °

Angles BCG = angle G + angle c

= 90 + 4

Answer:

Circle

Step-by-step explanation:

A cross section of a cone sliced parallel to iits bases is a circle and a rectangle when sliced perpendicular to its bases.

Answer:

its a circle

Step-by-step explanation:

got this from Khan and micaelaamisial :)