A popular video game retailer develops apps. The total profit, in dollars per day, after x days can be modeled by the function R(x) = 3x3 – 21x2 + 21x + 45. The total cost, in dollars per day, after x days, can be modeled by the function A(x) = x2 – 2x – 3. After how many days does the retailer start making money?

Answer:

Try and start at the line?

Step-by-step explanation:

The width of the new road x = 1/3 yards.

Since the road is a parabola with H = -(12x - 2)² + 4 represents the height of the road over the swamp. The width of the new road is gotten when H = 0.

So, H = -(12x - 2)² + 4

0 = -(12x - 2)² + 4

-(12x - 2)² + 4 = 0

Expanding the bracket, we have

-(144x² - 48x + 4) + 4 = 0

-144x² + 48x - 4 + 4 = 0

-144x² + 48x + 0 = 0

-144x² + 48x = 0

144x² - 48x = 0

Factorizing, we have

48x(3x - 1) = 0

48x = 0 or 3x - 1 = 0

x = 0 or 3x = 1

x = 0 or x = 1/3

So, the width of the new road x = 1/3 yards.

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A recursive arithmetic sequence a₁ = -4 and aₙ = aₙ₋₁ + 11,

a) the value of a₁ = -4

b) The value of d = 11

c) The iterative sequence of the arithmetic sequence is -4, 7, 18, 29, 40, 51, 62, 73, 84, 95...….

d) The value of a₁₀ is 95.

Arithmetic Sequence:

Arithmetic sequence terms refer the difference between consecutive terms is always the same.

Given,

Here we have the recursive arithmetic sequence a₁ = -4 and aₙ = aₙ₋₁ + 11.

And we need to find the following:

a) a₁

b) d

c) iterative sequence rule

d) a₁₀

While we looking into the given expression we have easily identified the value f a₁ is -4 and the value of common difference d is 11.

Now we have to find the iterative sequence, that is non other than the consecutive values of the arithmetic sequence,

it can be calculated as follows:

a₂ = a₂₋₁ + 11 = a₁ + 11 = -4 + 11 = 7

Similarly, the next value is calculated by adding 11 to its previous values,

So, the value of

a₃ = 7 + 11 = 18

a₄ = 18 + 11 = 29

a₅ = 29 + 11 = 40

a₆ = 40 + 11 = 51

a₇ = 51 + 11 = 62

a₈ = 62 + 11 = 73

a₉ = 73 + 11 = 84

a₁₀ = 84 + 11 = 95

Therefore, the iterative sequence is calculated by adding 11 to it and the sequence is -4, 7, 18, 29, 40, 51, 62, 73, 84, 95...…..

And the value of a₁₀ is 95.

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Vertex , focus , and directrix of the given equation of parabola

12y = x² -6x +45 are as follow:

Vertex = ( 3, 3)

Focus = ( 3, 6)

Directrix is y =0.

As given in the question,

Given equation of the parabola is:

12y = x² -6x +45

Simplify the equation of parabola to get standard form:

12y = x² - 6x + 9 - 9 + 45

⇒ 12 y = ( x - 3 )² + 36

⇒ 12y - 36 = ( x - 3 )²

⇒ 12( y - 3 ) =  ( x - 3 )²

⇒ ( x - 3 )² = 4(3)(y - 3)

Standard form is :

( x - h )² = 4a ( y - k)

Here h = 3, k = 3 and a = 3

Vertex are ( h , k ) =  ( 3, 3 )

Focus is ( h, k + a) =  ( 3, 6)

Directrix is

   y = k -p

⇒y = 3 - 3

⇒y = 0

Therefore, for the given parabola the vertex is (3,3) , focus is (3,6) and directrix is y = 0.

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2 + √3 is  a polynomial root,  then another root will be  2 - √3 for the polynomial .

For a given polynomial equation, where the degree of the equation is  1, it is said to be a quadratic equation. Since the given polynomial root is  2 + √3 ,the formula we refer to for  calculating the solution of  a quadratic formula  is  x = -b ± √(b² - 4ac) / 2a, which implies the roots be e [ -b + √(b² - 4ac) / 2a] and [ -b - √(b² - 4ac) / 2a].So, using the formula the root will be  2 - √3, here we can say that a complex root comes with conjugate pairs.

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Answer:

C

Step-by-step explanation:

the area (A) of the parallelogram is calculated as

A = bh ( b is the base and h the perpendicular height )

here b = 9 \(\frac{1}{2}\) and h = 5 , then

A = 9 \(\frac{1}{2}\) × 5 = 47 \(\frac{1}{2}\) ft²

As per the volume of a rectangular prism, it is found that 34.76% of the container is empty.

Volume of a rectangular prism

The volume of the rectangle prism is calculated by multiplying the length, breadth and height of the prism.

The formula for the volume of rectangle prism is

V = l x w x h

where

l represents the length

w represents the width

h represents the height

Given,

A shipping container is in the form of a right rectangular prism, with dimensions of 30 ft by 8 ft by 7 ft 3 in.

Here we need to find if the container holds 1096 cubic feet of shipped goods, what percent is empty

According to the given question,

Let us consider the length is 30ft, Width is 8ft and Height is 7ft.

Here the value 3ft refers the depth.

That one is also multiplied with the volume.

Therefore, the volume of the container is

=> 8 x 7 x 30

=> 1680

Therefore, the volume of the container is 1680 cubic feet.

So, here we have to percentage of the empty space for that we have to find the difference of them and convert it into percentage,

=> 1680 - 1096

=> 584

The percent of empty space is,

=> x% of 1680 = 584

=> x% = 584/1680

=> x% = 0.3476

=> x = 34.76

Therefore, 34.76% of rectangular prism container is empty.

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To calculate the slope of the line, we have that its points are A(x₁, y₁) and B(x₂,y₂), we apply the following formula:

\(\boxed{\large\displaystyle\text{$\begin{gathered}\sf \bf{m=\frac{\Delta y}{\Delta x} \iff m=\frac{y_2-y_1}{x_2-x_1} } \end{gathered}$}}\)

To solve, we have that the points are:

\(\boxed{\large\displaystyle\text{$\begin{gathered}\sf \bf{A(x_1=\underbrace{2} \ \ , \ \ y_1=\overbrace{7} \ ) \ and \ B(x_2=\underbrace{-4} \ \ , \ \ y_2=\overbrace{19} } \end{gathered}$}}\)

What we do to solve is that we substitute this data in the formula provided above.

 \(\boxed{\large\displaystyle\text{$\begin{gathered}\sf \bf{m=\dfrac{y_2-y_1}{x_2-x_1} } \end{gathered}$}}\)

 \(\boxed{\large\displaystyle\text{$\begin{gathered}\sf \bf{m=\frac{19-7}{-4-2} } \end{gathered}$}}\)

 \(\boxed{\large\displaystyle\text{$\begin{gathered}\sf \bf{m=\frac{12}{-6}=-2 } \end{gathered}$}}\)

Answer:

Step-by-step explanation

First divide 29.6 by 8.  Your answer will be 3.7.

There is a 25% chance that a flight between New York City and Chicago will take between 97 and 98 minutes.

Since the time to fly between New York City and Chicago is uniformly distributed between 96 and 100 minutes, the probability of a flight being between any two times is proportional to the length of the time interval between those two times.

To find the probability that a flight is between 97 and 98 minutes, we need to calculate the length of the time interval between those two times, which is 1 minute.

Then, we divide the length of the interval by the length of the entire time range (100-96 = 4 minutes) to obtain the probability of a flight being between 97 and 98 minutes: Probability = 1 minute / 4 minutes = 0.25 or 25%.

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Answer: $492.03

Step-by-Step explanation: just multiply 546.7*.10 and that equals 54.67

subtract 54.67 from 546.7

the total is now 492.03

X = 700 m and Y = 350 m will require the least amount of fencing.

Area of a rectangular field = length × breath =  245000 = XY

⇒ Y = 245000/X

Perimeter of a rectangular field = X + 2Y

⇒P = X+ 2Y

⇒P = X + 490000/X

⇒ X + 490000X⁻¹

⇒P [X] =  X + 490000X⁻¹

⇒P'[X] = 1+[-490000X⁻²]

⇒0 = 1 - 490000/X²

⇒X²= 490000

⇒X = 700 unit

As XY = 245000  [ area of a rectangle ]

⇒700Y = 245000

⇒Y = 350 unit.

Hence , P is minimum [ fencing ] when X = 700 unit and Y = 350 unit.

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Answer:

C. -4 = -12

Step-by-step explanation:

15x-4 = 3(5x-4)

15x - 4 = 15x - 12

-4 = -12

Answer: Using the Frobenius inner product, we have:

To find the corresponding induced norm, we first find the Frobenius norm of A:

||A||F = sqrt(|55|^2 + |-2|^2 + |-5|^2 + |-5|^2 + |-2|^2 + |-3|^2 + |1|^2 + |-3|^2 + |2|^2)

= sqrt(302)

Then, using the formula for the induced norm, we have:

||A|| = sup{||A||F * ||x|| / ||x||2 : x is not equal to 0}

= sup{sqrt(302) * sqrt(x12 + x22 + x32) / sqrt(x1^2 + x2^2 + x3^2) : x is not equal to 0}

Since we only need to find the value for A, we can let x = [1 0 0] and substitute into the formula:

||A|| = sqrt(302) * sqrt(1) / sqrt(1^2 + 0^2 + 0^2)

= sqrt(302)

Finally, to find the angle between A and B in radians, we can use the formula:

cos(theta) = (A,B) / (||A|| * ||B||)

where ||B|| is the Frobenius norm of B:

||B||F = sqrt(|-5|^2 + |-2|^2 + |-5|^2 + |-5|^2 + |-2|^2 + |-53|^2 + |3|^2)

= sqrt(294)

So, we have:

cos(theta) = -301 / (sqrt(302) * sqrt(294))

= -0.510

Taking the inverse cosine of this value, we get:

theta = 2.094 radians (rounded to three decimal places)

The frobenius inner product and the corresponding induced norm to determine the value of each of the following is Arccos((A,B) / ||A||F ||B||F) = arccos(-198 / (sqrt(305) * sqrt(54)))

≈ 1.760 radians

First, we need to calculate the Frobenius inner product of the matrices A and B:

(A,B) = tr(A^TB) = tr([55 -2 -5]^T [-5 -2 -5 3])

= tr([25 4 -25] [-5 -2 -5; 3 0 -2; 5 -5 -3])

= tr([-125-8-125 75+10+75 -125+10+15])

= tr([-258 160 -100])

= -258 + 160 - 100

= -198

Next, we can use the Frobenius norm formula to find the norm of each matrix:

||A||F = \(\sqrt(sum_i sum_j |a_ij|^2)\) = \(\sqrt(55^2 + (-2)^2 + (-5)^2) = \sqrt(305)\)

||B||F =\(sqrt(sum_i sum_j |b_ij|^2)\)=\(\sqrt(5^2 + (-2)^2 + (-5)^2 + (-3)^2 + 3^2) = \sqrt(54)\)

Finally, we can use these values to calculate the requested expressions:

(A,B) / ||A||F ||B||F = (-198) / (sqrt(305) * sqrt(54)) ≈ -6.200

||A - B||F = \(sqrt(sum_i sum_j |a_ij - b_ij|^2)\)

= \(\sqrt((55 + 5)^2 + (-2 + 2)^2 + (-5 + 5)^2 + (0 - (-3))^2 + (0 - 3)^2)\)

= \(\sqrt(680)\)

≈ 26.076

arccos((A,B) / ||A||F ||B||F) = arccos(-198 / (sqrt(305) * sqrt(54)))

≈ 1.760 radians

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Answer:

about 24.38 feet

Step-by-step explanation:

Graphing window: [0, 30] by [-20, 160]

y1 = 20 + 2x + √(400 + 4x^2)

y2 = 1.6(20 + x + √(400 + x^2))

y1 = y2 when x = 24.3798.

So the length of the smaller garden's longer leg is about 24.38 feet.

Answer:

Well, I'm not the smartest person, nor do I exactly know what you mean, but I have a wild guess that the least you could rotate it is 100 degrees.

Step-by-step explanation:

Considering the fact that it's sides are all equal, you could compare this to a square, which also has equal sides. Its angles are equal, too, so you could measure it's angles (100 degrees) and turn it that exact amount,

I hope this helps you, like I said, I don't exactly know what you are talking about, I am not studying that sort of thing yet.

9514 1404 393

Answer:

  a  (3,-1)

Step-by-step explanation:

The number that "completes the square" is the square of half the x-coefficient, (-6/2)^2 = 9. Rearranging the given function to include the square trinomial, we have ...

  f(x) = x^2 -6x +9 -1 . . . . . . . here, we have 8 = 9 - 1

  f(x) = (x -3)^2 -1 . . . . . . . . . . vertex form

Comparing this to the generic vertex form ...

  f(x) = (x -h)^2 +k . . . . . . . vertex at (h, k)

we see that h=3 and k=-1.

The vertex is (h, k) = (3, -1).

Answer:

2

Step-by-step explanation:

Round the divisor to the nearest 10 or 100, depending on how many digits are in the divisor.

Round the dividend to the nearest 10, 100, or 1,000 so that all of the digits, except the first one are zeros.

Divide 600 by 30.

Compare your estimate to the exact answer to determine if the exact answer is reasonable.

Answer:

x = 16

Step-by-step explanation:

90 = x+8+66

x = 90-74

x = 16

Answer:

16

Step-by-step explanation:

:D

To show that the total charge contained in the charge distribution is Q, we integrate the charge density over the entire volume. The charge density is given by:

ρ(r) = ρ₀(1 - r/R) for r ≤ R,

ρ(r) = 0 for r > R,

where ρ₀ = 3Q/πR³.

To find the total charge, we integrate ρ(r) over the volume:

Q = ∫ρ(r) dV,

where dV represents the volume element.

Since the charge density is spherically symmetric, we can express dV as dV = 4πr² dr, where r is the radial distance.

The integral becomes:

Q = ∫₀ᴿ ρ₀(1 - r/R) * 4πr² dr.

Evaluating this integral gives:

Q = ρ₀ * 4π * [r³/3 - r⁴/(4R)] from 0 to R.

Simplifying further, we get:

Q = ρ₀ * 4π * [(R³/3) - (R⁴/4R)].

Simplifying the expression inside the parentheses:

Q = ρ₀ * 4π * [(4R³/12) - (R³/4)].

Simplifying once more:

Q = ρ₀ * π * (R³ - R³/3),

Q = ρ₀ * π * (2R³/3),

Q = (3Q/πR³) * π * (2R³/3),

Q = 2Q.

Therefore, the total charge contained in the charge distribution is Q.

To show that the electric field in the region r > R is identical to that created by a point charge Q at r = 0, we can use Gauss's law. Since the charge distribution is spherically symmetric, the electric field outside the distribution can be obtained by considering a Gaussian surface of radius r > R.

By Gauss's law, the electric field through a closed surface is given by:

∮E · dA = (1/ε₀) * Qenc,

where ε₀ is the permittivity of free space, Qenc is the enclosed charge, and the integral is taken over the closed surface.

Since the charge distribution is spherically symmetric, the enclosed charge within the Gaussian surface of radius r is Qenc = Q.

For the Gaussian surface outside the distribution, the electric field is radially directed, and its magnitude is constant on the surface. Hence, E · dA = E * 4πr².

Plugging these values into Gauss's law:

E * 4πr² = (1/ε₀) * Q,

Simplifying:

E = Q / (4πε₀r²).

This is the same expression as the electric field created by a point charge Q at the origin (r = 0).

To derive an expression for the electric field in the region r ≤ R, we can again use Gauss's law. This time we consider a Gaussian surface inside the charge distribution, such that the entire charge Q is enclosed.

The enclosed charge within the Gaussian surface of radius r ≤ R is Qenc = Q.

By Gauss's law, we have:

∮E · dA = (1/ε₀) * Qenc.

Since the charge distribution is spherically symmetric, the electric field is radially directed, and its magnitude is constant on the Gaussian surface. Hence, E · dA = E * 4πr².

Plugging these values into Gauss's law:

E * 4πr² = (1/ε₀) * Q.

Simplifying:

E = Q / (4πε₀r²).

This expression represents the electric field inside the charge distribution for r ≤ R.

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Answer:

x equals 0

Step-by-step explanation:

it is a vertical line on zero and this is 1st grade level math

Answer:

see explanation

Step-by-step explanation:

Given 2 sides of a triangle then the third side x is between the limits

difference of 2 sides < x < sum of 2 sides, that is

7 - 5 < x < 7 + 5 , then

2 < x < 12

Choose any 2 values between 2 and 12 as third side, so

third side could be 6 or 9 for example

We need to know about rate of change to solve the problem. The rate of change of volume of the sphere is 5026.55 sq/m.

Rate of change of a quantity is the rate at which it increases or decreases. It is given that the radius of the sphere increases at a rate of 1m/second. We need to find out the rate of change of volume of the sphere given the radius is 20m. We can calculate the rate of change of volume by calculating the derivative of volume with respect to time.

V=4/3\(\pi r^{3}\)

dV/dt=4\(\pi r^{2}\) dr/dt =4 x \(\pi\)x20x20x1=5026.55 sq m/ second

Therefore the volume of the sphere increases at  a rate of 5026.55 sq/m.

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Answer:

x = 3; y = 2

Step-by-step explanation:

You would just set each side equal to its opposite so for x, it'll be like:

4x+6=7x-3

and for y:

4y-3=3y+1

I attached my work below as well to help you.

The probability that the a point will fall on the red-shaded square to nearest hundredth is 0.56

A probability is a number that reflects the chance or likelihood that a particular event will occur. The certainty of an event is 1 which is equivalent to 100% in decimal.

Probability = sample space / total outcome

sample space = area of shaded part

total outcome = area of the whole shape.

area of the shaded part = 3×3 = 9

area of the whole shape = 4×4 = 16

Therefore the probability that a point will fall in the shaded area = 9/16

= 0.56( nearest hundredth)

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Answer:

0.56

Step-by-step explanation:

just got it right

By using binomial distribution and formula, the probability of the indicated event (a) P(17 ≤ X ≤ 20) = 0.3040 (b) P(12 < X < 15) = 0.4675.

a) Given, the distribution is binomial X ~ B(n=22, p=0.8).

Let, X1= 17 and X2 = 20. Therefore, P(17 ≤ X ≤ 20) = P(X = 17) + P(X = 18) + P(X = 19) + P(X = 20).

By using binomial formula, P(X=k) = 22Ck (0.8)^k (0.2)^(22-k).

Thus, P(X=17) = 22C17 (0.8)^17 (0.2)^5

P(X=18) = 22C18 (0.8)^18 (0.2)^4  

P(X=19) = 22C19 (0.8)^19 (0.2)^3

P(X=20) = 22C20 (0.8)^20 (0.2)^2.

By putting the values, we get P(17 ≤ X ≤ 20) = 0.0040 + 0.0212 + 0.0784 + 0.2003.

The probability of the event, P(17 ≤ X ≤ 20) = 0.3039 ≈ 0.3040.

Therefore, P(17 ≤ X ≤ 20) = 0.3040

b) Given, the distribution is binomial X ~ B(n=21, p=0.6)

Let, X1= 12 and X2 = 15. Therefore, P(12 < X < 15) = P(X = 13) + P(X = 14)

By using binomial formula, P(X=k) = 21Ck (0.6)^k (0.4)^(21-k).

Thus, P(X=13) = 21C13 (0.6)^13 (0.4)^8

P(X=14) = 21C14 (0.6)^14 (0.4^)7.

By putting the values, we get P(12 < X < 15) = 0.1657 + 0.3018

The probability of the event, P(12 < X < 15) = 0.4675 ≈ 0.4675 (rounded to 4 decimal places).

Therefore, P(12 < X < 15) = 0.4675

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