F (x) = (2x + 3)^4
Expand the function
Answers
Answer:
\(f(x)=16x^4+96x^{3}+216x^{2}+216x+81\)
Step-by-step explanation:
The function f(x) = (2x + 3)⁴ is a fourth-degree polynomial function.
It can be expanded using the binomial theorem.
\(\boxed{\begin{minipage}{5cm} \underline{Binomial Theorem}\\\\$\displaystyle (a+b)^n=\sum^{n}_{k=0}\binom{n}{k} a^{n-k}b^{k}$\\\\\\where \displaystyle \binom{n}{k} = \frac{n!}{k!(n-k)!}\\\end{minipage}}\)
Comparing the given function with (a + b)ⁿ:
a = 2xb = 3n = 4Substitute these values into the binomial theorem formula:
\(\displaystyle (2x+3)^4=\binom{4}{0}(2x)^{4-0}3^{0}+\binom{4}{1}(2x)^{4-1}3^{1}+\binom{4}{2}(2x)^{4-2}3^{2}+\binom{4}{3}(2x)^{4-3}3^{3}+\\\\\\\phantom{wwww}\binom{4}{4}(2x)^{4-4}3^{4}\)
Solve:
\(\begin{aligned}\displaystyle (2x+3)^4&=\binom{4}{0}(2x)^4\cdot3^0+\binom{4}{1}(2x)^{3}\cdot3^1+\binom{4}{2}(2x)^2\cdot3^2+\binom{4}{3}(2x)^{1}\cdot3^3+\binom{4}{4}(2x)^0\cdot3^4\\\\&=\binom{4}{0}16x^4\cdot1+\binom{4}{1}8x^3\cdot3+\binom{4}{2}4x^2\cdot9+\binom{4}{3}2x\cdot27+\binom{4}{4}\cdot81\\\\&=\binom{4}{0}16x^4+\binom{4}{1}24x^3+\binom{4}{2}36x^2+\binom{4}{3}54x+\binom{4}{4}81\\\\&=1\cdot16x^4+4\cdot24x^3+6\cdot36x^2+4\cdot54x+1\cdot81\\\\&=16x^4+96x^3+216x^2+216x+81\end{aligned}\)
Therefore, the expanded function is:
\(f(x)=16x^4+96x^{3}+216x^{2}+216x+81\)
\( \Large{\boxed{\sf F(x) = (2x + 3)^4 = 16x^4 + 96x^3 + 216x^2 + 216x + 81 }} \)
\( \\ \)
Explanation:To expand the given function, we will apply the binomial theorem, which is the following:
\(\sf(a+b)^n =\sf\sum\limits_{k=0}^{n} \binom{n}{k}a^{n-k}b^{k} \\ \\ \sf \:Where\text{:} \\ \star \: \sf n \: is \: a \: positive \: integer. \: ( n \in \mathbb{N}) \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf \star \: k \: is \: a \: positive \: integer \: less \: than \: or \: equal \: to \: n. \: (k \leqslant n, \: k \: \in \: \mathbb{ N}) \\ \\ \\ \sf \star \: \displaystyle\binom{ \sf \: n}{ \sf \: k} \: \sf is \: a \: \underline{binomial \: coefficient} \: and \: is \: calculated \: as \: follows\text{:} \\ \\ \\ \sf \displaystyle\binom{ \sf \: n}{ \sf \: k} = \sf \dfrac{n! }{(n - k)! k ! }\)\( \\ \\\)
\( \\ \)
Let's identify our values\( \\ \)
\( \sf F(x) = (\underbrace{\sf 2x}_{\sf a} + \underbrace{3}_{\sf b})^{\overbrace{\sf 4}^{n}} \\ \\ \implies \sf a = 2x \: \: ,b = 3 \: \: ,n = 4 \)
\( \\ \)
Substitute these values into our formula:\( \\ \)
\( \sf (2x + 3)^4 = \displaystyle\sum\limits_{ \sf k=0}^{ \sf 4} \binom{ \sf 4}{ \sf k}( \sf 2x)^{4-k}(3)^{k} \\ \\ \\ \sf = \binom{ \sf 4}{ \sf 0}( \sf 2x)^{4-0}(3)^{0} + \binom{ \sf 4}{ \sf 1}( \sf 2x)^{4-1}(3)^{1} + \binom{ \sf 4}{ \sf 2}( \sf 2x)^{4-2}(3)^{2} + \binom{ \sf 4}{ \sf 3}( \sf 2x)^{4-3}(3)^{3} + \binom{ \sf 4}{ \sf 4}( \sf 2x)^{4-4}(3)^{4} \\ \\ \\ \sf = \binom{ \sf 4}{ \sf 0}( \sf 2x)^{4}(3)^{0} + \binom{ \sf 4}{ \sf 1}( \sf 2x)^{3}(3)^{1} + \binom{ \sf 4}{ \sf 2}( \sf 2x)^{2}(3)^{2} + \binom{ \sf 4}{ \sf 3}( \sf 2x)^{1}(3)^{3} + \binom{ \sf 4}{ \sf 4}( \sf 2x)^{0}(3)^{4} \\ \\ \\ \sf = \binom{ \sf 4}{ \sf 0}( \sf 16 {x}^{4} ) + \binom{ \sf 4}{ \sf 1}( \sf 24 {x}^{3}) + \binom{ \sf 4}{ \sf 2}( \sf 36x^{2}) + \binom{ \sf 4}{ \sf 3}( \sf 54x) + \binom{ \sf 4}{ \sf 4}(81) \)
\( \\ \)
Determine the value of each binomial coefficient\( \\ \)
\( \\ \star \: \displaystyle\binom{ \sf 4}{\sf \: 0} = \sf \dfrac{4! }{(4-0)!0 ! } = \dfrac{4!}{4!0!} = \dfrac{4!}{4!} = \boxed{\sf 1} \\ \\ \star\:\displaystyle\binom{ \sf 4 }{ \sf \: 1} =\sf \dfrac{4! }{(4 - 1)!1 ! } = \dfrac{4!}{3!1!}= \dfrac{2\times 3 \times 4}{2 \times 3} = \boxed{\sf 4} \\ \\ \star \: \displaystyle\binom{ \sf 4 }{ \sf \: 2} =\sf \dfrac{4! }{(4-2)!2!} = \dfrac{4!}{2!2!}=\dfrac{2 \times 3 \times 4}{2 \times 2} = \boxed{\sf 6} \\ \\ \star \:\displaystyle\binom{ \sf 4 }{ \sf \: 3}= \sf \dfrac{4! }{(4 - 3)!3!} =\dfrac{4!}{1!3!} = \dfrac{2 \times 3 \times 4}{2 \times 3 } = \boxed{\sf 4}\\ \\ \star \:\displaystyle\binom{ \sf 4 }{ \sf \: 4}= \sf \dfrac{4! }{(4 - 4)!4 !} =\dfrac{4!}{0!4!} = \dfrac{2 \times 3 \times 4}{2 \times 3 \times 4 } = \boxed{\sf 1} \)
\( \\ \)
Replace the binomial coefficients with their value\( \\ \)
\( \sf (2x + 3)^4 = \binom{ \sf 4}{ \sf 0}( \sf 16 {x}^{4} ) + \binom{ \sf 4}{ \sf 1}( \sf 24 {x}^{3}) + \binom{ \sf 4}{ \sf 2}( \sf 36x^{2}) + \binom{ \sf 4}{ \sf 3}( \sf 54x) + \binom{ \sf 4}{ \sf 4}(81) \\ \\ \\ \sf = (1)(16x^4) + (4)(24x^3) + (6)(36x^2) + (4)(54x) + (1)(81) \\ \\ \\ \boxed{\boxed{\sf = 16x^4 + 96x^3 + 216x^2 + 216x + 81}} \)
\( \\ \\ \\ \)
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Related Questions
how many levels are there in the 2nd factor of a 2 x 4 x 3 factorial design?
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The second factor has four levels.
In a 2 x 4 x 3 factorial design, the first factor has two levels, the second factor has four levels, and the third factor has three levels.
To find the number of levels in the second factor, we look at the number of levels of the second factor, which is four. Therefore, the second factor has four levels.
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suppose i have a cabbage, a goat and a lion, and i need to get them across a river. i have a boat that can only carry myself and a single other item. i am not allowed to leave the cabbage and lion alone together, and i am not allowed to leave the lion and goat alone together. how can i safely get all three across?
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This is a classic river crossing puzzle. To safely get all three across take the goat across, leave the goat, take the lion across, take the cabbage across, return with the empty boat, Take the goat across to reunite with the cabbage and lion
Here's one possible solution:
First, take the goat across the river, leaving the cabbage and lion on the original side.Next, leave the goat on the other side and return to the original side with the empty boat.Then, take the lion across the river, and leave it on the other side with the goat.Return to the original side with the empty boat, and take the cabbage across the river.Finally, leave the cabbage on the other side and return to the original side with the empty boat.Take the goat across the river to reunite with the cabbage and lion on the other side.By following these steps, you will have successfully transported all three items across the river without violating the rules.
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Translate the phrase, then simplify.
Find the sum of - 41, -8, and 36.
Answers
Answer:
-13
Step-by-step explanation:
-41 + -8 + 36
-49 + 36
-13
Which is -13
On a certain hot summer's day, 386 people used the public swimming pool. The daily prices are $1.75 for children and $2.50 for adults. The receipts for admission totaled $807.50. How many children and how many adults swam at the public pool that day?
Answers
Answer:
Call x is childrens
Call y is adults
x + y = 386 (1)
1,75x + 2,5y = 807,5 (2)
(1) => x = 386 - y => (2) 1,75 ( 386-y )+2,5y = 807,5
0,75y=132 => y = 176 and x = 210
Step-by-step explanation:
Find the range for the measure of the third side of a triangle given the measures of two sides.
18 ft and 23 ft
Answers
The range for the measure of the third side of the triangle is (5, 38).
What is a triangle?A triangle is a geometric figure with three edges, three angles and three vertices. It is a basic figure in geometry.
The sum of the angles of a triangle is always 180°
Given that,
Two sides of the triangle are 18 ft and 23 ft.
Let the third side of the triangle is x cm.
Since, the third side is greater than the difference of the remaining sides and less than the sum of the two sides.
Implies that,
23 - 18 < x < 23 + 18
5 < x < 38
The range of the third side varies from (5, 38).
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Determine the slope-intercept form (y = mx + b) of the equation of the line that contains the
points (3, 5) and (5, 6)
Answers
Hello,
We have A(3, 5) and B(5, 6)
\(m=\frac{y_{B}-y_{A}}{x_{B}-x_{A}} =\frac{6-5}{5-3} =\frac{1}{2}\)
We know the line contain the point (3, 5)
y = 1/2x + b
⇔ 5 = 1/2 × 3 + b
⇔ 5 = 3/2 + b
⇔ 5 - 3/2 = b
⇔ b = 10/2 - 3/2
⇔ b = (10 - 3)/2
⇔ b = 7/2
\(\boxed{y = \frac{1}{2}x + \frac{7}{2} }\)
Convert the following numbers with the indicated bases to i) decimal, using the direct method, and ii) hexadecimal, using the intermediary binary conversion [and bit grouping] method: (a) (1203)4
(b) (5243)8
Answers
(a) (1203)4 is equal to 99 in decimal and 12 in hexadecimal.
(b) (5243)8 is equal to 2211 in decimal and 1523 in hexadecimal.
To convert numbers from different bases to decimal using the direct method, we multiply each digit of the number by the corresponding power of the base and sum the results. To convert to hexadecimal using the intermediary binary conversion method, we first convert the number to binary and then group the binary digits into groups of 4, starting from the rightmost digit, and convert each group to its hexadecimal equivalent.
(a) Converting (1203)4 to decimal:
(1203)4 = 1 * 4³ + 2 * 4² + 0 * 4¹ + 3 * 4⁰
= 64 + 32 + 0 + 3
= 99
Converting (1203)4 to hexadecimal:
First, convert (1203)4 to binary:
(1203)4 = (10010)2
Next, group the binary digits into groups of 4:
(10010)2 = (0001 0010)2
Finally, convert each group to its hexadecimal equivalent:
(0001 0010)2 = (12)16
Therefore, (1203)4 is equal to 99 in decimal and 12 in hexadecimal.
(b) Converting (5243)8 to decimal:
(5243)8 = 5 * 8³ + 2 * 8² + 4 * 8¹ + 3 * 8⁰
= 2048 + 128 + 32 + 3
= 2211
Converting (5243)8 to hexadecimal:
First, convert (5243)8 to binary:
(5243)8 = (101 010 100 011)2
Next, group the binary digits into groups of 4:
(101 010 100 011)2 = (0001 0101 0010 0011)2
Finally, convert each group to its hexadecimal equivalent:
(0001 0101 0010 0011)2 = (1523)16
Therefore, (5243)8 is equal to 2211 in decimal and 1523 in hexadecimal.
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HHHHHHHHHEEEEEELLLLLLLLPPPPPPPP how do I do this
Answers
Answer:
1/6
Step-by-step explanation:
2/3 times 1/4 is 2/12. 2/12 simplified is 1/6, so 1/6 is your answer.
a reasonable abstraction for a car includes: group of answer choices an engine number of miles driven driving car color
Answers
Answer:
Of the given options, "an engine" is the most reasonable abstraction for a car.
An engine is a fundamental component of a car that powers its movement, and it is present in almost all cars. The number of miles driven and the driving car color are characteristics of a specific car rather than abstractions of a car itself. For example, a car can still be considered a car even if it has not been driven any miles yet or if it has no color at all.
Therefore, an engine is a more reasonable abstraction for a car as it captures an essential feature of a car that is present in all cars.
The solution is, D. Driving, a reasonable abstraction for a car includes. Driving is an action or behavior associated with a car, as it involves operating or using the car to travel from one place to another. It is an essential aspect of the car's purpose and function.
Here, we have,
we know that,
An engine is a machine designed to convert fuel into mechanical energy that can be used to power other machines or devices. In the context of automobiles, an engine is the primary source of power that drives the vehicle. It typically operates by burning fuel in a combustion chamber to generate high-pressure gases that drive a piston, which in turn rotates a crankshaft that ultimately powers the car's wheels.
Car color and number of miles driven are characteristics or properties of a car, while an engine is a component that helps to power the car. Driving, on the other hand, is an action or behavior associated with a car, as it refers to the act of operating or using the car to travel from one place to another.
Therefore, driving is a reasonable abstraction for a car, as it captures an essential aspect of the car's purpose and function.
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Determine if the columns of the matrix form a linearly independent set. Justify your answer. Choose the correct answer below. A. The columns of the matrix do form a linearly independent set because the set contains more vectors than there are entries in each vector. B. The columns of the matrix do form a linearly independent set because there are more entries in each vector than there are vectors in the set. C. The columns of the matrix do not form a linearly independent set because the set contains more vectors than there are entries in each vector. D. The columns of the matrix do not form a linearly independent set because there are more entries in each vector than there are vectors in the set.
Answers
Th matrix is missing. The matrix is :
\(\begin{bmatrix}1 &-4 &4 \\ -4 &16 & 4 \end{bmatrix}\)
Solution :
The column of the matrix are \(\begin{bmatrix}1\\ -4\end{bmatrix}\) , \(\begin{bmatrix}-4\\ 16\end{bmatrix}\), \(\begin{bmatrix}4\\ 4\end{bmatrix}\)
Now each of them are vectors in \($IR^2$\). But \($IR^2$\) has dimensions of 2. But there are 3 column vectors, hence they are linearly dependent.
Therefore, the column of the given matrix does not form the \(\text{linearly independent set}\) as the set contains \(\text{more vectors}\) than there are entries in each vector.
Therefore, option (D) is correct.
-x^2 = -36
Solve the equation using inverse operations. Check your solutions. In you final answer, include all of your work.
Answers
Answer:
x = ±6
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDASEquality PropertiesAlgebra II
Extraneous solutions and multiple answers/rootsStep-by-step explanation:
Step 1: Define
-x² = -36
Step 2: Solve for x
Divide -1 on both sides: x² = 36Square root both sides: x = ±6Step 3: Check
Plug in x to verify it's a solution.
Substitute in -6: -(-6)² = -36Exponents: -(36) = -36Multiply: -36 = -36Substitute in 6: -(6)² = -36Exponents: -(36) = -36Multiply: -36 = -36Here we see that both -6 and 6 do indeed work as solutions.
∴ x = ±6 are both solutions to the equation.
a spinner with the letters a, b, c, and d is spun. what is the theoretical probability of landing on the letter c?
Answers
The probability of the spinner landing on the letter C is 1/4
To understand the probability of the spinner landing on the letter C, we need to start by looking at the total number of possible outcomes. In this case, there are four possible outcomes, corresponding to the four letters A, B, C, and D.
Since each of the four quadrants on the spinner is equal in size, each of the four letters has an equal chance of being landed on. Therefore, the probability of the spinner landing on the letter C is the number of ways that the spinner can land on C, divided by the total number of possible outcomes.
Since there is only one way for the spinner to land on C, and there are four possible outcomes in total, the probability of the spinner landing on the letter C is 1/4
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The given question is incomplete, the complete question is:
A spinner with four equal quadrants labeled A, B, C, and D is spun. What is the theoretical probability of the spinner landing on the letter C
Vannorman Corporation processes sugar beets in batches. A batch of sugar beets costs $78 to buy from farmers and $18 to crush in the company's plant. Two intermediate products, beet fiber and beet juice, emerge from the crushing process. The beet fiber can be sold as is for $25 or processed further for $16 to make the end product industrial fiber that is sold for $57. The beet juice can be sold as is for $39 or processed further for $22 to make the end product refined sugar that is sold for $84. How much profit (loss) does the company make by processing one batch of sugar beets into the end products industrial fiber and refined sugar?
Answers
Answer:
Results are below.
Step-by-step explanation:
We will take into account the differential costs only. Therefore, the costs of sugar beets and to crush them are irrelevant to decision making.
Beet fiber:
Sold as is= $25
Continue processing= 57 - 16= $41
It is more convenient to process further the beet fiber.
Beet juice:
Sold as is= $39
Continue processing= 84 - 22= $62
It is more convenient to process further the beet juice.
0.5(400 + d) =4.25 d=
Answers
Answer:
d = 53.33333
Step-by-step explanation:
Solve for d on calculator
Answer:
pleas help me asap please!!!!!!
pleas help me asap please!!!!!!
Step-by-step explanation:
pleas help me asap please!!!!!!
Answer the question below!!! The best answer gets brainliest!!
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Answer:
forth one
Step-by-step explanation:
8 students selected both .then only 24 of students selected only country music and only 6 students selected jazz music.
50 points and brainliest to first correct answer! :)
The figure below shows a quadrilateral ABCD. Sides AB and DC are congruent and parallel:
A quadrilateral ABCD is shown with the opposite sides AB and DC shown parallel and equal.
A student wrote the following sentences to prove that quadrilateral ABCD is a parallelogram:
Side AB is parallel to side DC, so the alternate interior angles, angle ABD and angle CDB, are congruent. Side AB is equal to side DC, and DB is the side common to triangles ABD and BCD. Therefore, the triangles ABD and CDB are congruent by SSS postulate. By CPCTC, angles DBC and BDA are congruent and sides AD and BC are congruent. Angle DBC and angle BDA form a pair of alternate interior angles. Therefore, AD is congruent and parallel to BC. Quadrilateral ABCD is a parallelogram because its opposite sides are equal and parallel.
Which statement best describes a flaw in the student's proof?
Question 11 options:
Angle DBC and angle BDA form a pair of vertical angles, not a pair of alternate interior angles, which are congruent.
Triangles ABD and CDB are congruent by the SAS postulate instead of the SSS postulate.
Triangles ABD and BCD are congruent by the AAS postulate instead of the SSS postulate.
Angle DBC and angle BDA form a pair of corresponding angles, not a pair of alternate interior angles, which are congruent.
Answers
Answer/Step-by-step explanation:
"Side AB is parallel to side DC so the alternate interior angles, angle ABD and angle CDB, are congruent." True
"Side AB is equal to side DC and DB is the side common to triangles ABD and BCD." True
Now we have 2 triangles, with 2 congruent sides, and the angles between these pairs of congruent sides, are also congruent..
This means that:
Therefore, the triangles ABD and CDB are congruent by SAS postulate, NOT SSS postulate, which would require 3 pairs of congruent sides.
Answer: Triangles ABD and CDB are congruent by the SAS postulate.
The length of rectangle is 3 more than twice the width
Answers
Answer:
The length of a rectangle is 3” more than twice the width. The area of the rectangle is 150 in^2. Find the dimensions of the rectangle.
The length of a rectangle is 3” more than twice the width. The area of the rectangle is 150 in^2. Find the dimensions of the rectangle.
Wyzant
MATH QUADRATIC FORMULA
Kyle C. asked • 05/05/16
The length of a rectangle is 3” more than twice the width. The area of the rectangle is 150 in^2. Find the dimensions of the rectangle.
The length of a rectangle is 3” more than twice the width. The area of the rectangle is 150 in^2. Find the dimensions of the rectangle.
ANSWER
ABOUT THIS TUTOR ›
We know that Area = Length x Width
in this case, L = 2W + 3 and A = 150in2
so, 150in2 = W x (2w + 3)
150in2 = 2W2 + 3W
rearranging, 2W2 + 3W - 150in2 = 0
we need to use the quadratic formula to solve this equation
that formula is [-b±(√b2 -4ac)]/2a
a = 2, b = 3, c = -150
inserting those values, we get [-3±(√32 -4•2•-150)]/2•2
which becomes (-3±√1209)/4
we get (-3 ± 34.771)/4
we can discard the negative value, because no length can be negative
so, W ≅ 7.942 inches
then L = 2W + 3 ≅ 18.886 inches
Proof: 7.942in x 18.886in ≅ 149.99
HOPE IT'S HELPFUL
THANK YOU
I bought 5 gallons of gas for 1770 what is the unit price for gas per gallon
Answers
Answer:
Brainleist!Step-by-step explanation:
1770/5 = the unti price
1770 / 5 = 354
354
\(solution \\ cost \: of \: 5 \: gallons \: of \: gas = 1770 \\ cost \: of \: 1 \: gallons \: of \: gas = \frac{1770}{5} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = 354 \\ \)
hope it helps...
good luck on your assignment
Question 5 of 10
Rise is the horizontal change between any two points on a line.
A. True
B. False
SUBMIT
Answers
Answer:
False
Rise is the vertical change not the horizontal
Step-by-step explanation:
55 ÷ -------- +5 × _____× 2 =195
u can only use numbers 11,19,2,29,4
Answers
Answer:
55 ÷ 11 +5 × 19 × 2 =195
If you take turns in filling each gap then at one point you can get the answers.
Brainliest pls if it is correct! And hope it helps!
Nelson goes to three main places most days school, work and his home. They are graphed below. What is the distance between his home and his work? 4.2 9.5 11.2 90
Answers
ANSWER
9.5
EXPLANATION
To find the distance between home and work, we have to use the formula:
\(D\text{ = }\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2}\)We have that home is at point (3, -3) and work is at point (0, 6).
Therefore:
(x1, y1) = (3, -3)
(x2, y2) = (0, 6)
Therefore, we have that:
\(\begin{gathered} D\text{ = }\sqrt[]{(0-3)^2+(6-(-3))^2} \\ D\text{ = }\sqrt[]{(-3)^2+(6+3)^2}\text{ = }\sqrt[]{(-3)^2+(9)^2} \\ D\text{ = }\sqrt[]{9\text{ + 81}}\text{ = }\sqrt[]{90} \\ D\text{ = 9.5} \end{gathered}\)The answer is 9.5
What is juggernog in zombies
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Answer:
Juggernog is a Perk in Call of Duty: World at War and Call of Duty: Black Ops. It appears in every Nazi Zombies map (excluding Nacht der Untoten). It is the zombies equivalent of Juggernaut. Juggernog costs 2500 points in every map in which it is available.
A tank at an oil refinery is to be coated with an industrial strength coating. The surface area of the tank is 80,000 square feet. The coating comes in five-gallon buckets. The area that the coating in one randomly selected bucket can cover, varies with mean 2000 square feet and standard deviation 100 square feet.
Calculate the probability that 40 randomly selected buckets will provide enough coating to cover the tank. (If it matters, you may assume that the selection of any given bucket is independent of the selection of any and all other buckets.)
Round your answer to the fourth decimal place.
Answers
The probability that 40 randomly selected buckets will provide enough coating to cover the tank is 0.5000 or 0.5000 (approx) or 0.5000
Given: The surface area of the tank is 80,000 square feet. The coating comes in five-gallon buckets. The area that the coating in one randomly selected bucket can cover varies, with a mean of 2000 square feet and a standard deviation of 100 square feet.
The probability that 40 randomly selected buckets will provide enough coating to cover the tank. (If it matters, you may assume that the selection of any given bucket is independent of the selection of any and all other buckets.)
The area covered by one bucket follows a normal distribution, with a mean of 2000 and a standard deviation of 100. So, the area covered by 40 buckets will follow a normal distribution with a mean μ = 2000 × 40 = 80,000 and a standard deviation σ = √(40 × 100) = 200.
The probability of the coating provided by 40 randomly selected buckets will be enough to cover the tank: P(Area covered by 40 buckets ≥ 80,000).
Z = (80,000 - 80,000) / 200 = 0.
P(Z > 0) = 0.5000 (using the standard normal table).
Therefore, the probability that 40 randomly selected buckets will provide enough coating to cover the tank is 0.5000 or 0.5000 (approx) or 0.5000 (rounded to four decimal places).
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Suppose that x t
and y t
grow exponentially at rates g x
and g y
, respectively. Solve for the growth rate of z t
in terms of g x
and g y
if: (a) z t
=x t
α
y t
1−α
(b) z t
=αx t
β
/y t
Answers
a. The growth rate of zt in terms of gx and gy is given by gz = α × gx + (1-α) × gy.
b. The growth rate of zt in terms of gx and gy is given by gz = β × gx - gy.
To solve for the growth rate of zt in terms of gx and gy for the equation zt = xt²α × yt²(1-α):
Taking the natural logarithm of both sides:
ln(zt) = ln(xt²α × yt²(1-α))
Using the logarithmic property ln(a×b) = ln(a) + ln(b):
ln(zt) = ln(xt²α) + ln(yt²(1-α))
Applying the power rule of logarithms ln(a²b) = b × ln(a):
ln(zt) = α ×ln(xt) + (1-α) × ln(yt)
Differentiating both sides with respect to t:
d/dt ln(zt) = α ×d/dt ln(xt) + (1-α) × d/dt ln(yt)
The left-hand side represents the growth rate of zt (denoted as gz). Similarly, the right-hand side represents the growth rates of xt (gx) and yt (gy):
gz = α × gx + (1-α) × gy
(b) To solve for the growth rate of zt in terms of gx and gy for the equation zt = α × xt²β / yt:
Taking the natural logarithm of both sides:
ln(zt) = ln(α × xt²β / yt)
Using the logarithmic properties ln(a/b) = ln(a) - ln(b) and ln(ac) = c ×ln(a):
ln(zt) = ln(α) + ln(xt²β) - ln(yt)
Applying the power rule of logarithms ln(a²b) = b × ln(a):
ln(zt) = ln(α) + β × ln(xt) - ln(yt)
Differentiating both sides with respect to t:
d/dt ln(zt) = β ×d/dt ln(xt) - d/dt ln(yt)
The left-hand side represents the growth rate of zt (denoted as gz). Similarly, the right-hand side represents the growth rates of xt (gx) and yt (gy):
gz = β × gx - gy
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Can someone please help?
Answers
Answer:
Step-by-step explanation:
8b-4b+8-3=4b+5
4b=-5
b=-5/4=1 1/4
Answer:
4b+5
Step-by-step explanation:
please help me!!
will mark brainliest!!
Use the drawing tool(s) to form the correct answer on the provided graph.
The graph of f(x) = x is shown on the coordinate plane. Function g is a transformation off as shown below.
g(x) =3/4 f(x)
Graph function g on the same coordinate plane.
Answers
Step by step:
We know what the equation is, so we just have to multiply it by 3/4.
So, we get g(x)=(3/4)x. Plug in some random x values to get y values. I listed some convenient ones at the top.
The graph of the function g(x) = 3/4 f(x) on the same coordinate plane as f(x) = x is attached below.
What is a graph?A graph can be defined as a pictorial representation or a diagram that represents data or values.
To graph the function g(x) = 3/4 f(x) on the same coordinate plane as f(x) = x, we need to perform the transformation of stretching the graph of f(x) by a factor of 3/4.
Determine the scale factor for the transformation. In this case, the scale factor is 3/4.
Starting at the x-intercepts and y-intercepts (if there are any), plot the points on the coordinate plane using the scale factor to stretch or shrink the graph as needed.
For example, the point (0,0) on the graph of f(x) becomes the point (0,0) on the graph of g(x). The point (1,1) on the graph of f(x) becomes the point (1, 3/4) on the graph of g(x).
Connect the plotted points with a smooth curve to obtain the graph of g(x).
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There’s 2 questions. Question 17 and 18.
Answers
A) The third relation is not a function.
B) evaluating the function we can see that f(-3) = -7
Which relation is not a function?A relation is not a function if we can see that one input is mapped into two different outputs.
With that in mind, we can see that relation 4 has the input x = 2 mapped into two different outputs, A and B, then it is not a function.
b) Here we need to evaluate the given function, we know that:
f(x) = 4x + 5
We want to find f(-3), to get it, just replace x by the number -3 in the equation above:
f(-3) = 4*-3 + 5
= -12 + 5
= -7
The correct option is 2.
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Need done asap!! If anyone knows please help. Step by step.
a) find the number, q, of units that produces maximum profit
b) the price, p, per unit that produces maximum profit
c) the maximum profit, P.
Answers
a) The number of units that produces maximum profit is: q = 11/6.
b) The price per unit that produces maximum profit is: p = 36.
c) The maximum profit is P = -698/9.
How to solve Profit Functions?The cost and profit functions are given as:
C(q) = 80 + 14q
p = 58 - 12q
To find the number of units that produces maximum profit, we need to find the value of q that maximizes the profit function.
The profit function, P(q), is given by:
P(q) = Revenue - Cost
The revenue is given by:
Revenue = Price * Quantity
R(q) = p * q
The cost is given by:
Cost = Fixed Cost + Variable Cost
C(q) = 80 + 14q
Substituting the given price equation p = 58 - 12q into the revenue equation, we get:
R(q) = (58 - 12q) * q
Now we can express the profit function in terms of q:
P(q) = R(q) - C(q)
P(q) = (58 - 12q) * q - (80 + 14q)
P(q) = 58q - 12q² - 80 - 14q
P(q) = -12q² + 44q - 80
a) To find the number of units that produces maximum profit, we need to find the value of q that maximizes the profit function P(q). This can be done by taking the derivative of P(q) with respect to q and setting it equal to zero.
P'(q) = -24q + 44
-24q + 44 = 0
-24q = -44
q = 44/24
q = 11/6
So, the number of units that produces maximum profit is q = 11/6.
b) To find the price per unit that produces maximum profit, we can substitute the value of q = 11/6 into the price equation p = 58 - 12q:
p = 58 - 12 * (11/6)
p = 58 - 22
p = 36
So, the price per unit that produces maximum profit is p = 36.
c) Finally, to find the maximum profit, we can substitute the value of q = 11/6 into the profit function P(q):
P(q) = -12 * (11/6)² + 44 * (11/6) - 80
Simplifying the expression, we get:
P(q) = 88/3 - 484/18 - 80
P(q) = 88/3 - 242/9 - 80
P(q) = (264 - 242 - 720)/9
P(q) = -698/9
So, the maximum profit is P = -698/9.
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A mouse pushes a block of cheese across the floor with 4 N of force. How many meters did the mouse travel if she did 16 J of work?
Answers
The mouse traveled 4 meters while pushing the block of cheese with 4 N of force if she did 16 J of work.
What is equations?An equation is a mathematical statement that shows that two expressions are equal. Equations typically consist of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division.
According to the given information:We know that work (W) is equal to force (F) times distance (d) in the direction of the force, so we can use the formula:
W = F x d
To find the distance traveled (d), we need to rearrange the formula:
d = W / F
Plugging in the values we have:
d = 16 J / 4 N
d = 4 meters
Therefore, the mouse traveled 4 meters while pushing the block of cheese with 4 N of force, if she did 16 J of work.
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a square and a regular hexagon have sides of the same length. the perimeter of the square increased by 32 units will be equal to the perimeter of the hexagon. what is x, the length of a side of the hexagon or square?
Answers
x is 8 units,the length of a side of the hexagon or square, where the perimeter of the square increased by 32 units will be equal to the perimeter of the hexagon .
In order to solve this problem, where we are given that a square and a regular hexagon have sides of the same length. the perimeter of the square increased by 32 units will be equal to the perimeter of the hexagon, So we use the formula for the perimeter of a square which is: P = 4x, and the formula for the perimeter of a regular hexagon is P = 6x.Now we set these two equations equal to one another and solve for x, where we get, 4x + 32 = 6x, so x = 8. Therefore, the length of each side of the square and hexagon is 8 units.
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