You are asked by your teacher to arrange the letters in the word probability regardless of each word 's meaning. in how many ways can you arrange the letter in the word?

[tex]\color{blue}{analysis}[/tex] : the problem involve permutation or combination) of objects

[tex]\color{red}{required}[/tex] : the value that is to be solved in the problem is the____

[tex]\color{pink}{given}[/tex]: the given value is____ which is the_____ of the word probability

[tex]\color{cyan}{formula}[/tex]: we will use the formula______ to soive for the unknown.


solution

The value of x that maximizes revenue is x 28.99 and maximum revenue is $1680 (thousand).

The revenue function for a company that sells x (thousand) items is R(x) = x² + 840. To find the value of x that maximizes revenue, we need to differentiate the revenue function, set it equal to zero and solve for x. The maximum revenue can then be calculated by substituting the value of x into the revenue function.

Revenue function: R(x) = x² + 840

To find the value of x that maximizes revenue, we differentiate the revenue function with respect to x:

dR/dx = 2x

Setting this equal to zero, we get:

2x = 0

x = 0

However, this value does not make sense in the context of the problem, as the company cannot sell 0 items. Therefore, we need to consider the critical points of the function, which occur when dR/dx = 0 or is undefined.

dR/dx = 0 when 2x = 0, so x = 0 is a critical point.

dR/dx is undefined when x = ±√840, so these are also critical points.

We can use the second derivative test to determine which critical point corresponds to a maximum. The second derivative of the revenue function is:

d²R/dx² = 2

At x = 0, d²R/dx² = 2 > 0, so this critical point corresponds to a minimum.

At x = ±√840, d²R/dx² = 2 > 0, so these critical points correspond to a minimum.

Therefore, the value of x that maximizes revenue is x = √840 ≈ 28.99 (thousand items).

To find the maximum revenue, we substitute x = √840 into the revenue function:

R(√840) = (√840)² + 840 = 1680

So the maximum revenue is $1680 (thousand).

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Event A' has a probability of 50% (25% for freshmen + 25% for seniors).

To determine Event A', we need to first identify what Event A represents. Event A is the probability that a student is a sophomore or junior. Since students have equal probabilities of being freshmen, sophomores, juniors, or seniors, the probability of Event A is 50% (25% for sophomores + 25% for juniors).
Event A' is the complement of Event A, which means it includes the other two grade levels not included in Event A, in this case, freshmen and seniors. Therefore, Event A' is the probability that a student is a freshman or a senior. Since students have equal probabilities of being in each grade level, Event A' also has a probability of 50% (25% for freshmen + 25% for seniors).

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Answer:

[tex]-2a^{4}b^{6} c^{3}[/tex]

Step-by-step explanation:

(2a² b c³) (-a² b⁵)

distribute amongst each other

-2a²bc³a²b⁵

rearrange

-2a²a²bb⁵c³

combine like terms

[tex]-2a^{4}b^{6} c^{3}[/tex]

Hope this helps!

Answer:  -2[tex]a^{4}[/tex][tex]b^{6}[/tex]c³  

Step-by-step explanation:

Given:

(2a² b c³) (-a² b⁵)

Rules:

>Multiply numbers first

>If a number with an exponent is multiplied to another, you can combine if the bases are the same(lower number)  

>If they have the same base, you can add the exponents

Solution:

(2a² b c³) (-a² b⁵)       >multiply the - and 2

-2(a² b c³) (a² b⁵)      > combine the a's by adding the expnents 2+2

-2[tex]a^{4}[/tex]( b c³) ( b⁵)           > combine b, there is an invisible 1 by the first b 1+5

-2[tex]a^{4}[/tex][tex]b^{6}[/tex]c³                   >c stays as is because nothing to combine with

Answer:

--------------------------

Let the equal sides be both marked as ?

Use the perimeter formula to determine one of the equal sides.

Substitute the expression for the perimeter and find the value of ?

Hence the length of each of equal sides is 4x² - x + 2.

Answer:

Length: 22 inches × 3.5 = 77 inches

Width: 10 inches × 3.5 = 35 inches

Answer:

The three equivalent equations are x+1=4, -5+x=-2, and 2+x=5. The x equals 3.

Step-by-step explanation:

To find the derivative of y = cos(sin(14x-13)), we will use the chain rule.

Let's start by defining two functions:

u = sin(14x-13)
v = cos(u)

We can now apply the chain rule:

dy/dx = dv/du * du/dx

First, let's find dv/du:

dv/du = -sin(u)

Next, let's find du/dx:

du/dx = 14*cos(14x-13)

Now we can put it all together:

dy/dx = dv/du * du/dx
dy/dx = -sin(u) * 14*cos(14x-13)

But we still need to substitute u = sin(14x-13) back in:

dy/dx = -sin(sin(14x-13)) * 14*cos(14x-13)

So the derivative of y = cos(sin(14x-13)) is:

dy/dx = -14*sin(sin(14x-13)) * cos(14x-13)

To find the derivative of the function y = cos(sin(14x - 13)), we can use the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

Let u = sin(14x - 13), so y = cos(u). Now we find the derivatives:

1. dy/du = -sin(u)
2. du/dx = 14cos(14x - 13)

Now, using the chain rule, we get:

dy/dx = dy/du × du/dx

dy/dx = -sin(u) × 14cos(14x - 13)

Since u = sin(14x - 13), we can substitute back in:

dy/dx = -sin(sin(14x - 13)) × 14cos(14x - 13)

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Answer:

48 calories per (each) minute.

Step-by- Step

=

The inequality that can be used to determine the number of outfits Jason can purchase while staying within his budget is 68.54 o ≤ 274.16.

Solving the inequality gives:

o ≤ 4

The total cost of items that Jason spends on :

= 217. 34 + 36. 32 + 12. 18

= $ 265. 85

The amount that Jason has left from the $ 540 is:

= 540 - 265. 85

= $ 274. 16

If each outfit costs $ 68. 54 then the inequality that would help stay in budget is:

68.54 o ≤ 274.16

Solving this, gives:

o ≤ 274. 16 / 68.54

o ≤ 4

In conclusion, Jason can purchase up to 4 biking outfits while staying within his budget.

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Answer:

Step-by-step explanation:

Answer:

Step-by-step explanation:

5 6 7

The expression to find the number of events is: (1 event) / (1/8 side) = (n events) / (1/4 sides).

Devonte is studying for a history test and uses 1/8 of a side of one sheet of paper to write notes for each history event. He fills 2 full sides of one sheet of paper. To find out how many events he wrote notes for, you can set up an expression using the given information.

Since Devonte uses 1/8 of a side for each event, and he fills 2 sides, you can calculate the total amount of space he used by multiplying the fractions: (1/8) * 2. This simplifies to 2/8 or 1/4. Now, you can set up a proportion to find the number of events (n) that Devonte wrote notes for:

(1 event) / (1/8 side) = (n events) / (1/4 sides)

Cross-multiply to solve for n:

1 * (1/4) = n * (1/8)

1/4 = n/8

To find n, multiply both sides by 8:

(8) * (1/4) = n

2 = n

So, Devonte wrote notes for 2 history events using the 2 full sides of one sheet of paper. The expression to find the number of events is: (1 event) / (1/8 side) = (n events) / (1/4 sides).

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Complete Question:

Devonte is studying for a history test. He uses 1/8 of a side of one sheet of paper to write notes for each history event. He fills 2 full sides of one sheet of paper. Which expression could be used to find how many events Devonte makes notes for?

Answer:

147 mm^2

Step-by-step explanation: The surface area has to be more, but not double the smaller figures surface area therefore the answer is 147 mm^2

The requried surface area of the larger solid is approximately 147 mm². Option A is correct.

The surface area of any shape is the area of the shape that is faced or the Surface area is the amount of area covering the exterior of a 3D shape.

Let's call the larger solid's surface area S.

Since the two solids are similar, their volumes have a ratio of (side length)³. Let's call the ratio of the side lengths of the larger to the smaller solid as k. Then:

[tex](k^3)(540 mm^3) = 857.5 mm^3[/tex]

Simplifying the above equation, we get:

[tex]k = (857.5/540)^{(1/3)}[/tex] =7/6

So, the larger solid is about 7/6=1.183 times bigger than the smaller solid in terms of side length. Since the surface area has a ratio of [tex](side length)^2[/tex], we can find the surface area of the larger solid by:

[tex]S = (1.183^2)(108 mm^2) \approx 147 mm^2[/tex]

Therefore, the surface area of the larger solid is approximately 147 mm². Answer: A.

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The scale factor used in the dilation of the triangles is 1/2

From the question, we have the following parameters that can be used in our computation:

The scale factor is calculated as

Scale factor  = X'/X

Substitute the known values in the above equation, so, we have the following representation

Scale factor  = (4, 1)/(8, 2)

Evaluate

Scale factor = 1/2

Hence, the scale factor is 1/2

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The measure of angle ABC is 30 degrees.

To find the measure of angle ABC, we need to use the fact that the sum of angles in a triangle is always 180 degrees. Since we know that angle ABD is a right angle (90 degrees) and angle CBD is 60 degrees, we can find the measure of angle ABC as follows:

angle ABD + angle CBD + angle ABC = 180 degrees

Substituting the given values, we get:

90 degrees + 60 degrees + angle ABC = 180 degrees

Simplifying, we get:

150 degrees + angle ABC = 180 degrees

Now we can solve for angle ABC by subtracting 150 degrees from both sides:

angle ABC = 180 degrees - 150 degrees

angle ABC = 30 degrees

Therefore, angle ABC measures 30 degrees.

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In a rectangle, the diagonals are equal in length. So we can write the equation: AC = BD or 16 = 2x + 4. Solving for x, we get x = 6.

The corresponding measure of the parameters are;

i. 1ml = 0. 001 liter a.

ii. 1000ml = 1 liter b.

iii. 1 cl = 0. 01 liter c.

iv. 10dcl = 1 liter d.

v. 1cl = 100ml e.

v. 0. 01 cl = 1ml f.

To convert the factors, we need to know the following conversion rates.

We have;

1 milliliter = 0. 001 liter

1 centiliter = 0. 01 liter

1 deciliter = 0. 1 liter

1 cubic centimeter = 1 millimeter

Hence, the sizes are determined by the corresponding factor.

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Complete question:

Convert the following to their equivalent measurement for each letter

i. 1 ml = a liters

ii) b ml = 1 liters

iii) 1 cl = c liters

iv)d cl = 1 liters

v) 1 cl = e ml

vi) f cl = 1 ml

z - 9 = 10(x - 5) - 8(y - 4) this is the equation of the tangent plane at the point (5, 4, 9). (x - 5)/10 = (y - 4)/(-8) = (z - 9)/(-1). These are the symmetric equations for the normal line to the surface at the given point.

(a) To find the equation of the tangent plane to the surface z = x^2 - y^2 at the point (5, 4, 9), we first need to find the partial derivatives with respect to x and y:

∂z/∂x = 2x
∂z/∂y = -2y
Now, we evaluate these at the given points (5, 4, 9):
∂z/∂x(5, 4) = 2(5) = 10
∂z/∂y(5, 4) = -2(4) = -8
Using the tangent plane equation:
z - z₀ = ∂z/∂x (x - x₀) + ∂z/∂y (y - y₀)
Plugging in the values:
z - 9 = 10(x - 5) - 8(y - 4)
This is the equation of the tangent plane at the point (5, 4, 9).

(b) The normal vector to the surface at the given point is given by the gradient vector (∂z/∂x, ∂z/∂y, -1) = (10, -8, -1). To find the symmetric equations for the normal line, we use the point-normal form:
(x - x₀)/a = (y - y₀)/b = (z - z₀)/c
Plugging in the point (5, 4, 9) and the normal vector components (10, -8, -1):
(x - 5)/10 = (y - 4)/(-8) = (z - 9)/(-1)
These are the symmetric equations for the normal line to the surface at the given point.

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Answer:

360 cubes

Step-by-step explanation:

A real-world situation that might involve three linear equations in three variables is trying to determine the ages of three siblings.

Let's call them A, B, and C. We know that the sum of all three ages is a certain value, let's say it's 60. We also know the sum of the first two ages is either twice the third age or the third age squared. For example, if the sum of the first two ages is twice the third age, we could write it as A + B = 2C.

Alternatively, if the sum of the first two ages is the third age squared, we could write it as A + B = C^2. Similarly, we know the sum of the first and third ages is either twice the second age or the square root of the second age. So, we could write it as A + C = 2B or A + C = sqrt(B).

We now have three linear equations in three variables that we can use to solve for the ages of the three siblings. By solving the system of equations, we can find out how old each sibling is.

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Answer:

Perry would need to place his ladder 6.63 ft away from the base of the basketball hoop in order to reach the hoop.

Hope this helps!

Step-by-step explanation:

The Pythagorean theorem is [tex]a^{2} +b^{2} =c^{2}[/tex].

It is given that c is 12 and b is 10, so that would be:

[tex]c^{2} -b^{2} =a^{2}[/tex]

[tex]12^{2} -10^{2} =a^{2}[/tex]

144 - 100 = [tex]a^{2}[/tex]

44 = [tex]a^{2}[/tex]

a = [tex]\sqrt{44}[/tex]

a = 6.6332...

( c is the length of the ladder, b is the height of the hoop, and a is the distance between the ladder and the basketball hoop )

The number 71.38 can be written in word form as "seventy-one and thirty-eight hundredths." The correct answer is option D.

In decimal notation, the number 71.38 can be broken down into its whole number and decimal parts. The whole number part is 71, and the decimal part is 0.38.

In a decimal number, the digits to the right of the decimal point represent fractions of a whole. Each digit to the right of the decimal point has a place value that is a power of 10.

In word form, the decimal part 0.38 is read as "thirty-eight hundredths." Therefore, when combined with the whole number 71, the correct word form is "Seventy-one and thirty-eight hundredths."

Therefore option D is the correct answer.

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In Year 6 at Woodland Junior School, there are 90 children split into three classes of 14, 24, and 12 on the basis of there selection of sports. In 6M, 14 not chose hockey, and the rest of the class was split equally between football and netball. In 6P, 24 not chose netball, 16 chose football, and 8 choose hockey  In 6T, the ratio of football to netball to hockey was 1:2:3. The completed table is shown.

To complete the table, we need to distribute the remaining children who did not choose their sport in each class. Here's how we can calculate it

In 6M, the number of children who did not choose hockey is 27 - 13 = 14.

Since the rest of the class was split equally between football and netball, each of these two sports will have 14/2 = 7 children.

In 6P, the number of children who did not choose netball is 33 - 9 = 24.

Since twice as many children chose football as chose hockey, we can write the number of footballers as 2x, and the number of hockey players as x. Then we have 2x + x + 9 = 33, which gives x = 8. Therefore, we have 16 children who chose football, and 8 children who chose hockey. The number of children who did not choose any of these two sports is 33 - 16 - 8 - 9 = 0.

In 6T, the ratio of children who chose football to netball to hockey was 1:2:3. Let's call the number of children who chose netball as 2x, and the number of children who chose hockey as 3x. Then we have x + 2x + 3x = 30, which gives x = 6. Therefore, we have 6 children who chose football, 12 children who chose netball, and 18 children who chose hockey.

Thus, the completed table is shown.

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The price of the medium pizza without the coupon is approximately $14.74.

To find the original price of the pizza without the coupon, we can use the following formula:

Original Price = Discounted Price / (1 - Discount Percentage)

In this case, the discounted price is $10.32 and the discount percentage is 30% or 0.3. Plugging the values into the formula:

Original Price = $10.32 / (1 - 0.3)

Original Price = $10.32 / 0.7

Original Price ≈ $14.74

So, the price of the medium pizza without the coupon is approximately $14.74.

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The volume of foam in the model is approximately 27,211.5 cubic centimeters

The radius of the circular base can be found by dividing the diameter by 2:

radius = diameter / 2 = 48 cm / 2 = 24 cm

The formula for the volume of a cone is:

V = (1/3) * π * r² * h

where π is approximately 3.14, r is the radius of the circular base, and h is the height of the cone.

Substituting the values we have:

V = (1/3) * 3.14 * 24² * 12

V = 27,211.52 cm³

Rounding this to the nearest tenth, we get:

V ≈ 27,211.5 cm³

Therefore, the volume of foam in the model is approximately 27,211.5 cubic centimeters

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Answer:

P = (5.4, -2.6)

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{5 cm}\underline{Equation of a circle}\\\\$(x-h)^2+(y-k)^2=r^2$\\\\where:\\ \phantom{ww}$\bullet$ $(h,k)$ is the center. \\ \phantom{ww}$\bullet$ $r$ is the radius.\\\end{minipage}}[/tex]

As the given circle has a radius of 6 units and is centred at the origin, the equation of the circle is:

[tex]x^2+y^2=36[/tex]

The formula for the equation of the tangent line to a circle with the equation x² +  y² = a² is:

[tex]\boxed{y = mx \pm a \sqrt{1+ m^2}}[/tex]

where:

To find the slope of the equation of the tangent line to the circle that passes through the point (0, -14), substitute a = 6, x = 0 and y = -14 into the formula and solve for m:

[tex]\implies -14 = m(0) \pm 6 \sqrt{1+ m^2}[/tex]

[tex]\implies -14 = \pm 6 \sqrt{1+ m^2}[/tex]

[tex]\implies \pm\dfrac{14}{6} =\sqrt{1+ m^2}[/tex]

[tex]\implies \left(\pm\dfrac{14}{6}\right)^2 =1+m^2[/tex]

[tex]\implies m^2= \left(\pm\dfrac{14}{6}\right)^2-1[/tex]

[tex]\implies m^2=\dfrac{40}{9}[/tex]

[tex]\implies \sqrt{m^2}= \sqrt{\dfrac{40}{9}}[/tex]

[tex]\implies m=\pm\sqrt{\dfrac{40}{9}}[/tex]

[tex]\implies m=\pm\dfrac{2\sqrt{10}}{3}[/tex]

The slope-intercept form of a straight line is y = mx + b, where m is the slope and b is the y-intercept.

As the slope of the given tangent line is positive, and the y-intercept is (0, -14), the equation of the tangent line is:

[tex]\boxed{y=\dfrac{2\sqrt{10}}{3}x-14}[/tex]

As point P is the point of intersection of the circle and the tangent line, substitute the tangent line into the equation of the circle and solve for x:

[tex]x^2+\left(\dfrac{2\sqrt{10}}{3}x-14\right)^2=36[/tex]

Expand the brackets:

[tex]x^2 +\dfrac{40}{9}x^2-\dfrac{56\sqrt{10}}{3}x+196=36[/tex]

Subtract 36 from both sides of the equation:

[tex]\dfrac{49}{9}x^2-\dfrac{56\sqrt{10}}{3}x+160=0[/tex]

Multiply both sides of the equation by 9:

[tex]49x^2-168\sqrt{10}x+1440=0[/tex]

Rewrite the equation in the form a² - 2ab + b²:

[tex](7x)^2-2 \cdot 7 \cdot 12\sqrt{10}x+(12\sqrt{10})^2=0[/tex]

Apply the Perfect Square formula: a² - 2ab + b² = (a - b)²

[tex](7x-12\sqrt{10})^2=0[/tex]

Solve for x:

[tex]7x-12\sqrt{10}=0[/tex]

[tex]7x=12\sqrt{10}[/tex]

[tex]x=\dfrac{12\sqrt{10}}{7}[/tex]

To find the y-coordinate of point P, substitute the found value of x into the equation of the tangent line:

[tex]y=\dfrac{2\sqrt{10}}{3}\left(\dfrac{12\sqrt{10}}{7}\right)-14[/tex]

[tex]y=\dfrac{2\sqrt{10}\cdot 12\sqrt{10}}{3\cdot 7}\right)-14[/tex]

[tex]y=\dfrac{240}{21}-14[/tex]

[tex]y=\dfrac{80}{7}-\dfrac{98}{7}[/tex]

[tex]y=-\dfrac{18}{7}[/tex]

Therefore, the exact coordinates of point P are:

[tex]\left(\dfrac{12\sqrt{10}}{7}, -\dfrac{18}{7}\right)[/tex]

The coordinates of point P to 1 decimal place are:

[tex](5.4, -2.6)[/tex]

The most responsible choice for Nick regarding his birthday money would be option C, which is to use all the money to pay down his credit card debt.

This is because credit card debt  generally comes with high- interest rates, and it can be  grueling  to pay off the debt if the interest keeps adding up. By using the$ 500 to pay down his credit card debt, Nick can reduce the  quantum he owes and, as a result, pay  lower interest in the long run.   Options A and D suggest  unyoking the  plutocrat between paying down debt and investing/ saving, which may not be the stylish choice given Nick's current  fiscal situation.

Option B suggests putting all the  plutocrat into a savings  regard with a lower interest rate, which would not be as effective in reducing his credit card debt. thus, option C is the most responsible choice for Nick.

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The length of side AB is 6.3 units.

The length of side AB is solved using the distance formula below

AB = √((x₂ - x₁)² + (y₂ - y₁)²

where

(x₁, y₁) = (-9, 0) and

(x₂, y₂) = (-3, 2).

AB = √((-3 - (-9))² + (2 - 0)²)

AB = √(6² + 2²)

AB = √(40)

AB = 2√(10)

AB = 6.3245

AB = 6.3 to the nearest tenth

Therefore, the length of side AB is 6.3 units.

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. 0.587 in scientific notation is 5.87 x 10^-1.

Scientific notation is a way of writing numbers that are very large or very small in a compact and standardized way. In scientific notation, a number is expressed as a coefficient multiplied by 10 raised to some power.

For example, the number 0.587 can be written in scientific notation as 5.87 x 10^-1. The coefficient 5.87 is obtained by moving the decimal point one place to the right, while the negative exponent -1 indicates that the decimal point has been moved one place to the left, to the tenth place.

Scientific notation is commonly used in scientific and engineering fields where very large or very small numbers are often encountered, and it allows for easier calculation and comparison of these values.

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