Answers
Answer:
-12x⁴-10x³+3x²+3x+2
Step-by-step explanation:
(3x²+x−2)(−4x²−2x−1)=3x²(−4x²−2x−1)+x(−4x²−2x−1)−2(−4x²−2x−1)
{ ( a+b ) ( a+b )=a ( a+b )+b ( a+b ) by distributive property }
= -12x⁴-6x³-3x²-4x³-2x²-x+8x²+4x+2
= -12x⁴-10x³+3x²+3x+2
Related Questions
what us the order of 5 x 10^4, 7 x 10^-5, 3 x 10^-9, 8 x 10^4 from the least to greatest
Answers
Answer: 5x10^4, 8x10^4, 10^-5, 10^4
Step-by-step explanation: calculator
PLEASE HELP ASAP!! i really really need it
Answers
Answer:
The number of hamburgers is 136.
Step-by-step explanation:
What we know:
We got cheeseburgers and hamburgers
The total: 408
The number of cheese burgers was 2 times the number of hamburgers
This is pretty much all the info we need to know.
Lets define our variables:
x = hamburgers
y = cheeseburgers
Now, we need our two equations to make a system.
[tex]x+y=408[/tex]
[tex]y=2x[/tex]
To find hamburgers, x, we plug the value of y into our first equation.
[tex]x+2x=408[/tex]
Add like terms
[tex]3x=408[/tex]
Divide by 3
[tex]x=136[/tex]
The number of hamburgers is 136.
Just to check our work, lets find the cheeseburgers.
[tex]y=2(136)[/tex]
[tex]y=272[/tex]
[tex]272+136=408[/tex]
Which of the following represent a linear inequality in open-sentence form? (Select all that apply.) 2x + 8 = 10y 2x - 7y > 10 8 + x > 5 3 < 5
Answers
The linear inequality in open-sentence form are
2x - 7y > 10 8 + x > 5 What are open sentences?An open sentence is a mathematical statement that contains at least one blank or unknown and that becomes true or false when the blank is filled or a quantity is substituted for the unknown.
Now, since we have the expressions,
2x + 8 = 10y 2x - 7y > 10 8 + x > 5 and3 < 5First, we note that there are 3 open sentences and only two of them are inequalities.
The 3 open sentences are
2x + 8 = 10y 2x - 7y > 10 8 + x > 5The two inequalities are
2x - 7y > 10 8 + x > 5So, the linear inequality in open-sentence form are
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f(x)=3x^2-7x-4 determine el valor de f(0)
Answers
The numeric value of f(x) = 3x² - 7x - 4 at x = 0 is given as follows:
f(0) = -4.
How to obtain the numeric value of a function or of an expression?To obtain the numeric value of a function or of an expression, we substitute each instance of the variable in the function or in the expression by the value at which we want to find the numeric value.
The function for this problem is defined as follows:
f(x) = 3x² - 7x - 4
We want to find the numeric value of the function at x = 0, hence both instances of x in the definition of the function are replaced by zero and then we add the terms, as follows:
f(0) = 3(0)² - 7(0) - 4
f(0) = -4.
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penny would love to open a coffee shop at her local shopping centre, so she decides to hand out a questionnaire at the shopping center's entrance in order to do market research. draw up a questionnaire that penny could use to gather the information she requires
Answers
Penny plans to distribute a questionnaire at the entrance to the coffee shopping center in order to do market research because she would love to build a coffee shop there which are explained.
Explain about the term questionnaire?Market research is crucial for helping you understand the present status of the coffee shop business as well as for figuring out how to shape various areas of your company to draw in the suitable clients.
Penny plans to distribute a questionnaire at the entrance to the shopping center in order to do market research because she would love to build a coffee shop there.
Penny may utilize a questionnaire to collect the data she needs, which would include the following:
What are the current consumption patterns of customers at coffee shops? Do they stop by for a quick take-out latte on their way to work or come in for a sit-in coffee plus snack at lunchtime?What qualities define a normal coffee shop patron? Are families as well as teenage groups of friends much more prevalent than offices?What is the typical client budget?What are the best-selling products? What about flapjacks and scones for a snack or a flat white or an americano?know more about the questionnaire
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Please help and show a method, I have no idea how to do this :/
Answers
Answer:
30.4 meters
Step-by-step explanation:
Just plug in the given values of
V = 24.4 and
g = 9.8
in the given equation and you can solve for H
[tex]H = \dfrac{V^2}{2g}\\\\= \dfrac{(24.4)^2}{2\times 9.8}\\\\= 30.37551 \dots\\\\= 30.4\\\\[/tex]correct to 3 significant figures
Revise the suggested questionnaire so that it is more suitable. Include 5 reasons for taking a GAP YEAR. Submit a copy of a new revised questionnaire
Answers
The revised questionnaire that is more suitable that includes reasons for taking a GAP year is given below:
The QuestionnaireThe new questionnaire includes four questions aimed at gaining insight into the participant's motivations, goals, and timeframe for taking a gap year:
Have you considered taking a gap year before starting your next academic or professional endeavor?
What are your top 5 reasons for considering a gap year? (Please list in order of importance)
What are your goals for your gap year? (e.g., travel, volunteer work, internships, personal development, etc.)
How long do you plan to take for your gap year?
Reasons for taking a gap year may vary from person to person, but here are five common motivations for doing so:
To gain life experience and personal growth
To travel and explore new cultures
To gain work experience or internships in a desired field
To take a break from academic or professional responsibilities
To pursue passions or hobbies that may not be feasible during a regular academic or work schedule.
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Revise the suggested questionnaire so that it is more suitable. Include 5 reasons for taking a GAP YEAR. Submit a copy of a new revised questionnaire
Have you considered taking a gap year before starting your next academic or professional endeavor?
What are your reasons for considering a gap year?
What are your goals for your gap year?
How long do you plan to take for your gap year?
Suppose that the local government of Tulsa decides to institute a tax on soda producers. Before the tax, 35,000 liters of soda were sold every week at a price of $11 per liter. After the tax, 30,000 liters of soda are sold every week; consumers pay $14 per liter, and producers receive $6 per liter (after paying the tax).
The amount of the tax on a liter of soda is
$
per liter. Of this amount, the burden that falls on consumers is
$
per liter, and the burden that falls on producers is
$
per liter.
Answers
- Before the tax, 35,000 liters of soda were sold every week at a price of $11 per liter.
- After the tax, 30,000 liters of soda are sold every week; consumers pay $14 per liter, and producers receive $6 per liter (after paying the tax).
The total revenue from soda sales before the tax was:
35,000 liters × $11/liter = $385,000 per week
The total revenue from soda sales after the tax is:
30,000 liters × $14/liter = $420,000 per week
The difference in revenue is due to the tax. Let's call the amount of the tax per liter of soda "t". Then we have:
(30,000 liters × $6/liter) + (30,000 liters × t/liter) = $420,000 per week
Simplifying and solving for t, we get:
t = ($420,000 per week - $180,000 per week) / 30,000 liters
t = $8 per liter
Therefore, the amount of the tax per liter of soda is $8.
To find the burden that falls on consumers and producers, we can use the price of soda before and after the tax. Before the tax, consumers paid $11 per liter and producers received $11 per liter. After the tax, consumers paid $14 per liter and producers received $6 per liter. The difference in price is due to the tax. Let's call the burden that falls on consumers "bc" and the burden that falls on producers "bp". Then we have:
bc + bp = $14 - $11
bc + bp = $3 per liter
And we know that:
bp = $11 - $6 = $5 per liter
Substituting this value into the previous equation, we get:
bc + $5 per liter = $3 per liter
bc = $3 per liter - $5 per liter
bc = -$2 per liter
Since the burden that falls on consumers is negative, we can say that the burden falls entirely on producers, who bear the full $3 per liter increase in price due to the tax.
Therefore, the burden that falls on consumers is $0 per liter, and the burden that falls on producers is $3 per liter.
Solve the equation 3* = 8 for x.
Answers
Answer:
x = log(3) (8)
or log 8 base 3
Step-by-step explanation:
some number^x = another number
some number = base (the lower case number)
another number = the big number next to it.
put log infront in these situations.
Ball Bearings, Inc., faces costs of production as follows:
Quantity
Total Fixed Costs
Total Variable Costs
(Dollars)
(Dollars)
0 120 0
1 120 60
2 120 80
3 120 100
4 120 140
5 120 200
6 120 320
Complete the following table by calculating the company's total cost, marginal cost, average fixed cost, average variable cost, and average total cost at each level of production.
Quantity
Total Cost
Marginal Cost
Average Fixed Cost
Average Variable Cost
Average Total Cost
(Dollars)
(Dollars)
(Dollars)
(Dollars)
(Dollars)
0
1
2
3
4
5
6
The price of a case of ball bearings is $60. Seeing that the company can't make a profit, the company's chief executive officer (CEO) decides to shut down operations.
The firm's profit in this case is
$
. (Note: If the firm suffers a loss, enter a negative number in this cell.)
True or False: This was a wise decision.
True
False
Vaguely remembering an introductory economics course, the chief financial officer tells the CEO it is better to produce 1 case of ball bearings, because marginal revenue equals marginal cost at that quantity.
At this level of production, the firm's profit is
$
. (Note: If the firm suffers a loss, enter a negative number in this cell.).
True or False: This is the best decision the firm can make.
True
False
Answers
The firm's profit in this case is = $360.
What is the total cost?
The total economic cost of production is made up of two types of costs: the variable cost, which changes depending on how much of a good is produced and includes inputs like labour and raw materials, and the fixed cost, which is unaffected by how much of a good is produced and includes inputs like buildings and machinery that cannot be changed in the short term, possibly including sunk costs.
In economics, the whole cost of each unit of production, whether fixed or variable, includes the total opportunity cost (benefits from the next-best alternative).
Example:
Q Fixed Variable Total Marginal Aver. Aver. Aver.
Costs Costs Cost Cost FC VC TC
0 100 0 100 - - - -
1 100 50 150 150 100 50 150
2 100 70 170 20 50 35 85
3 100 90 190 20 33.33 30 63.33
4 100 140 240 50 25 35 60
5 100 200 300 60 20 40 60
6 100 360 460 160 16.67 60 76.67
The firm's profit in this case is = $360.
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Find a 5th degree polynomial with real coefficients that has zeros 0, -7i, and 2-i. show work
Answers
Answer:
f(x) = x^5 - 4x^4 + 54x^3 - 196x^2 + 245x.
Step-by-step explanation:
If a polynomial has a complex root, then its conjugate is also a root. Therefore, if 0 and -7i are zeros of the polynomial, then so are 7i and the conjugate of 2-i, which is 2+i.
The factors of the polynomial are then:
(x-0)(x-7i)(x+7i)(x-(2-i))(x-(2+i))
Simplifying the factors, we get:
x(x^2+49)(x^2-4x+5)
Expanding the last factor, we get:
x^5 - 4x^4 + 5x^3 + 49x^3 - 196x^2 + 245x
Finally, we have the 5th degree polynomial:
f(x) = x^5 - 4x^4 + 54x^3 - 196x^2 + 245x
Therefore, the polynomial with zeros 0, -7i, and 2-i and 5th degree with real coefficients is f(x) = x^5 - 4x^4 + 54x^3 - 196x^2 + 245x.
The solution is, f(x) = x^5 - 4x^4 + 54x^3 - 196x^2 + 245x, is a 5th degree polynomial with real coefficients that has zeros 0, -7i, and 2-i.
What are Polynomials?Polynomials are sums of k-xⁿ terms, where k can be any number and n can be any positive integer.
here, we have,
If a polynomial has a complex root, then its conjugate is also a root. Therefore, if 0 and -7i are zeros of the polynomial, then so are 7i and the conjugate of 2-i, which is 2+i.
The factors of the polynomial are then:
(x-0)(x-7i)(x+7i)(x-(2-i))(x-(2+i))
Simplifying the factors, we get:
x(x^2+49)(x^2-4x+5)
Expanding the last factor, we get:
x^5 - 4x^4 + 5x^3 + 49x^3 - 196x^2 + 245x
Finally, we have the 5th degree polynomial:
f(x) = x^5 - 4x^4 + 54x^3 - 196x^2 + 245x
Therefore, the polynomial with zeros 0, -7i, and 2-i and 5th degree with real coefficients is f(x) = x^5 - 4x^4 + 54x^3 - 196x^2 + 245x.
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What is the total payment
required to pay off a promissory
note issued for $600.00 at 10%
ordinary interest and a 180-day
term?
Answers
Answer:
$630.00
Step-by-step explanation:
IF THIS IS WRONG I AM SORRY BUT I KNOW THIS IS RIGHT AND IF YOU NEED MORE ANSWERS I AM ALWAYS OPEN!
I will mark you brainiest!
Which of the following choices is matched with ∠MLA to make alternate interior angles? A) ∠GAL
B) ∠FAH
C) ∠LMB
D) ∠CLH
Answers
Answer:BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB TRUST ME
Step-by-step explanation:
15. Find the missing side lengths. Answers are in simplest radical form with the denominator rationalized.
60°
2√6
Oz=4√6.y=6√2
Oz= 8√3,y=6
= 4√6, y=6
Oz=6√2y = 8√3
Answers
Thus, the length of the missing side for the given Right triangles using the Trigonometric Ratios is found as :x = 18.47 units.
What is meant by Trigonometric Ratios?In order to examine the sides of both a right triangle, which will depend on an acute angle, trigonometric ratios are used. Right triangles what about those produced by the intersection of the legs, which have an internal angle of 90 degrees. We can determine the right triangle's missing sides or sharp angles using trigonometric ratios.The leg, denoted as, that is next to the 30 degree angle θ is represented by Side BC. To get the length of side BD, the hypotenuse of triangle BCD, we use the cosine ratio.
cos θ = Adjacent leg / Hypotenuse = BC/BD
cos 30° = 8√2 / BD
Now, we know.
cos 30° = √3/2
√3/2 = 8√2 / BD
BD = 8√2 / √3/2
BD = 16√2 / √3
On rationalising;
BD = 16√2*√3 / √3*√3
BD = 16√6 / 3
Sin A = Opposite leg /Hypotenuse = BD / x
Sin 45° = (16√6 / 3) / x = 16√6 / 3x
We know that:
Sin 45° = 1/√2
(1/√2) = 16√6 / 3x
(1/√2) * x = 16√6 /3
Or,
√2/2 * x = (16√6 /3) * (2/√2)
x = 32√3 / 3
x = 18.47
Thus, the length of the missing side for the given Right triangles using the Trigonometric Ratios is found as :x = 18.47 units.
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The correct question is-
Find the missing side length. Give the answer as radicals in simplest form.
find the domain of the function f(x)=7x+5/3x-1
Answers
The domain of the function f(x)=7x+5/3x-1 is all real numbers except x = 1/3. This means that the function will produce a real number for any value of x except for x = 1/3.
What is a function?Function in mathematics is a relation which assigns each element of a set (called the domain) to exactly one element of another set (called the codomain). It can also be thought of as a type of mapping from one set to another. Functions are used to model real-world problems and to simplify the expression of a certain problem. They are widely used in algebra, trigonometry and other areas of mathematics.
The domain of a function is the set of all possible inputs for the function. In other to determine the domain of a function, we must first identify the independent variable (x) and then determine which values of x can be used in the function.
In the given function, f(x)=7x+5/3x-1, the independent variable is x. To find the domain of this function, we must identify the values of x that will produce results that are real numbers. To do this, we must set the denominator (3x-1) equal to zero and solve for x.
We can solve this equation by factoring and setting each factor equal to zero:
3x - 1 = 0
3x = 1
x = 1/3
Therefore, the domain of the function f(x)=7x+5/3x-1 is all real numbers except x = 1/3. This means that the function will produce a real number for any value of x except for x = 1/3.
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i need help pleaseee
Answers
The dataset is evenly distributed, so the appropriate measure of variability is the range
The dataset is evenly has no outliers, so the appropriate measure of center is the mean
How to determine the appropriate measuresMeasure of variability
When the data set is evenly distributed, a measure of variability appropriate for stem and leaf plot would be the range
The range is simply the difference between the largest and smallest value in the data set.
It is a simple measure of spread that can be easily calculated from a stem and leaf plot.
Measure of center
When there are no outliers in a dataset, the appropriate measure of center to use in a stem and leaf plot is the mean
The mean, also known as the average, is calculated by summing all the values in the dataset and then dividing by the total number of values.
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Bridgette and her friends, Jill and Barb, are talking about how much money they earn. Bridgette makes $615 biweekly, Jill makes $670 semimonthly, and Barb makes $300 a week. Who earns the most?
Answers
Answer: Jill earns the most money.
Step-by-step explanation:
Bridgette:1230
Jill: 1340
Barb:300
You currently have $9,300 (Present Value) in an account that has an interest rate of 5% per year compounded annually (1 times per year). You want to withdraw all your money when it reaches $15,810 (Future Value). In how many years will you be able to withdraw all your money?
Answers
Answer:
heueuehehhrahfzngzjfsjtzjg
The number 72 lies between the perfect squares. So the square root of 72 lies between the numbers
Answers
Answer:
To find the numbers between which the square root of 72 lies, we need to determine the perfect squares that are closest to 72.
The perfect squares closest to 72 are 64 (8^2) and 81 (9^2). Since 72 is closer to 81 than to 64, we know that the square root of 72 is closer to 9 than to 8.
Therefore, the square root of 72 lies between the numbers 8 and 9. We can write this as:
8 < √72 < 9y-step explanation:
Anybody It's easy!! if correct answered, will give crown.. I'm stuck at this for sooo long.. Anyone please helpppppppp.
What kind of transformation converts the graph of f(x)=2|x|+4 into the graph of g(x)=|x|+2?
reflection across the y-axis
reflection across the x-axis
Vertical stretch
Vertical Shrink
Answers
Answer:
The transformation that converts the graph of f(x)=2|x|+4 into the graph of g(x)=|x|+2 is a Vertical Shrink.
To see this, note that the 2 in f(x) causes a vertical stretch of the absolute value graph. Removing the 2 in g(x) results in a Vertical Shrink of the absolute value graph. The +4 and +2 only cause a vertical shift of the graph.
Answer: Vertical Shrink
Step-by-step explanation:
The transformation that converts the graph of f(x)=2|x|+4 into the graph of g(x)=|x|+2 is a vertical shrink.
To see why, let's examine each function separately:
The function f(x) = 2|x| + 4 has an absolute value in its equation, which means it has a "V" shape. The coefficient 2 in front of the absolute value means that the "V" is stretched vertically by a factor of 2. The constant 4 added to the end of the function moves the entire graph up by 4 units.The function g(x) = |x| + 2 also has an absolute value in its equation, but without a coefficient in front of it. This means that the "V" shape is not stretched or shrunk vertically. The constant 2 added to the end of the function moves the entire graph up by 2 units.Therefore, the transformation from f(x) to g(x) involves removing the vertical stretch by dividing the absolute value term by 2. This results in a smaller "V" shape for the graph of g(x), or a vertical shrink.
The answer is:
[tex]$\boxed{\text{Vertical Shrink}}$[/tex]
Martin recorded data on the temperature one day in two of his favorite places.
Temperature
-12 degrees F
82 degrees F
He was surprised when he calculated the difference as 94 degrees F. Do you think he was correct?
Answers
The actual difference between the twο temperatures is: 94 degrees F
What is Temperature?Temperature is measure οf degree οf hοtness οr cοldness οf οbject οr substance. It is physical quantity that describes average kinetic energy οf particles that make up οbject οr substance.
Temperature is a usually measured using thermοmeter, which cοnsists οf narrοw, unifοrm tube filled with liquid, like mercury οr alcοhοl
Nο, Martin's calculatiοn is nοt cοrrect. The difference between twο temperatures cannοt be greater than the sum οf the twο temperatures. In this case, the difference Martin calculated is 94 degrees F, which is greater than the sum οf the twο temperatures (-12 degrees F + 82 degrees F = 70 degrees F). Therefοre, Martin must have made an errοr in his calculatiοn οr recοrded the temperatures incοrrectly.
The actual difference between the twο temperatures is:
82 degrees F - (-12 degrees F) = 94 degrees F
This cοnfirms that Martin must have recοrded οne οf the temperatures incοrrectly οr made a mistake in his calculatiοn.
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graph the equation f(x)=4x-3
Answers
Answer:
Step-by-step explanation:
SEE THE GRAPH
-3-(-2)(5)-(-4)³ show your calculations
Answers
Answer:71
Step-by-step explanation:
-3-(-10)-(-64)=-3+10+64=71
What is the slope of the line
m=?
Answers
Answer:
Proofs attached to answer
Step-by-step explanation:
Proofs attached to answer
Find a formula for the linear function depicted in the following graph.
Answers
The equation of the line passing through the points (0, 7) and (2, -5) is y = -6x + 7
What is an equation?An equation is an expression that shows how two or more numbers and variables are related using mathematical operations of addition, subtraction, multiplication, division, exponents and so on.
The standard form of a linear equation is:
y = mx + b
where m is the slope(rate of change) and b is the y intercept
As shown in the graph, the line passes through the points (0, 7) and (2, -5). Hence:
[tex]y-y_1=\frac{y_2-y_1}{x_2-x_1} (x-x_1)\\\\substituting:\\\\y-7=\frac{-5-7}{2-0} (x-0)\\\\y=-6x+7[/tex]
The equation of the line is y = -6x + 7
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Weights of female cats of a certain breed are normally distributed with mean 4.1 kg and standard deviation 0.6 kg.
Six female cats are chosen at random. What is the probability that exactly one of them weights more than 4.5 kg.
(I did the first part to this question already, the probability that one chosen at random will weigh more than 4.5 kg is 0.2514.)
Answers
The prοbabiIity that exactIy οne οf the six chοsen femaIe cats weighs mοre than 4.5 kg is apprοximateIy 0.2834 οr 28.34%.
The binοmiaI distributiοn fοrmuIa:
The binοmiaI distributiοn fοrmuIa gives the prοbabiIity οf οbtaining exactIy k successes in n independent BernοuIIi triaIs, where each triaI has a prοbabiIity οf success p.
The formuIa is:
[tex]P(X = k) = ^{n}C_{k} P^{k} \times(1-P)^{n-k}[/tex]
Here we have
Weights οf female cats οf a certain breed are nοrmally distributed with a mean οf 4.1 kg and a standard deviatiοn οf 0.6 kg.
The prοbability that οne chοsen at randοm will weigh mοre than 4.5 kg is 0.2514.
Tο find the prοbability that exactly οne οf the six chοsen female cats weighs mοre than 4.5 kg, we can use the binοmial distributiοn with parameters
Using the binοmial distributiοn fοrmula
[tex]P(X = k) = ^{n}C_{k} P^{k} \times(1-P)^{n-k}[/tex]
where:
n = 6 and p = 0.2514,
The probabiity of exacty one success in six trias can be cacuated as
P(X = 1) = (6 choose 1) × 0.2514¹ × (1 - 0.2514)⁵
P(X = 1) = 6 × 0.2514 × 0.7486⁵
P(X = 1) = 0.2834
Therefore, The probabIIty that exacty one of the sIx chosen femae cats weIghs more than 4.5 kg Is approxImatey 0.2834 or 28.34%.
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Consider the following function.
h(x)=(x−4)2−1
Step 3 of 4 : Find two points on the graph of the parabola other than the vertex and x-intercepts.
Answers
Two points on the graph of the parabola h(x) = (x-4)^2 - 1 other than the vertex and x-intercepts are (1, 8) and (6, 3).
How to determine the points on the functionTo find two points on the graph of the parabola other than the vertex and x-intercepts
We can simply choose two values of x and find their corresponding y values using the function h(x).
Let's choose x = 1:
h(1) = (1-4)^2 - 1 = 9 - 1 = 8
So one point on the graph is (1, 8).
Now let's choose x = 6:
h(6) = (6-4)^2 - 1 = 4 - 1 = 3
So another point on the graph is (6, 3).
Hence, the two points are (1, 8) and (6, 3)
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A certain airline requires that rectangular packages carried on an airplane by passengers be such that the sum of the three dimensions is at most 270 centimeters. Find the dimensions of the square-ended rectangular package of greatest volume that meets this requirement.
Answers
The dimensions of the square-ended rectangular package of greatest volume that meets the airline's requirement are x = y = 90 cm and z = 90 cm.
What is dimensions?Dimensions refer to measurements or quantities that describe the size, shape, or extent of an object or space, often expressed in length, width, and height.
Let x, y, and z represent the dimensions of the rectangular package. Since the package has square ends, we know that x = y.
We want to maximize the volume of the package, which is given by V = x² × z.
The airline requires that the sum of the three dimensions is at most 270 centimeters, so we have the constraint x + y + z ≤ 270.
Substituting x = y, we get 2x + z ≤ 270.
We can solve for z in terms of x: z ≤ 270 - 2x.
Substituting this inequality into the expression for V, we get V = x^2 * (270 - 2x) = 270x² - 2x³.
To maximize V, we take the derivative of V with respect to x, set it equal to zero, and solve for x:
dV/dx = 540x - 6x² = 0
6x(90 - x) = 0
x = 0 or x = 90
Since x represents a length, it must be positive, so x = 90.
Since x = y, y = 90 as well.
Substituting x = y = 90 into the inequality z ≤ 270 - 2x, we get z ≤ 270 - 2(90) = 90.
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How do I solve for the answer please?
Answers
The value of the 5 measurements according to the given function is 66.
What is summation of values?When a group of numbers, known as addends or summands, are added together in mathematics, the outcome is their sum or total. Functions, vectors, matrices, polynomials, and, generally, any sort of mathematical object on which an operation labelled "+" is specified can all be added together, in addition to integers. The position of the items in a sequence is frequently used to characterise them through a regular pattern. Long sequences can be summarised for basic patterns by replacing most summands with ellipses.
The function is given as:
[tex]\sum_{i = 1}^5 (x_i^2)\\\\\sum_{i = 1}^5 = x_1 + x_2 + x_3 +x_4 + x_5\\\\\sum_{i = 1}^5 (x_i^2) = 12 + 14 + 16 + 7 + 17 = 66[/tex]
Hence, the value of the 5 measurements according to the given function is 66.
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Find all critical points for the function
4x + 6
x² + x + 1
on (-∞, ∞) and then list them (separated by commas) in the box below.
List of critical points:
f(x) =
Answers
Answer: [tex]\frac{-3+\sqrt{7}}{2}, \ \frac{-3-\sqrt{7}}{2}[/tex]
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Explanation:
Let
g(x) = 4x+6h(x) = x^2+x+1Each derivative is,
g ' (x) = 4h ' (x) = 2x+1which will be useful in the next section.
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[tex]f(\text{x}) = \frac{4\text{x}+6}{\text{x}^2+\text{x}+1} = \frac{g(\text{x})}{h(\text{x})}\\\\f(\text{x}) = \frac{g(\text{x})}{h(\text{x})}\\\\[/tex]
Apply the derivative with respect to x. We'll use the quotient rule.
[tex]f(\text{x}) = \frac{g(\text{x})}{h(\text{x})}\\\\\\f'(\text{x}) = \frac{g'(\text{x})h(\text{x})-g(\text{x})h'(\text{x})}{\big[h(\text{x})\big]^2}\\\\\\f'(\text{x}) = \frac{4(\text{x}^2+\text{x}+1)-(4\text{x}+6)(2\text{x}+1)}{(\text{x}^2+\text{x}+1)^2}\\\\[/tex]
The critical value(s) occur when either...
f ' (x) = 0f ' (x) doesn't exist, when x is in the domain of f(x)The first criteria will be handled in the next section.
The second criteria is handled in the section after that.
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f ' (x) is in the format A/B. It means f ' (x) = 0 leads to A/B = 0 and A = 0.
We set the numerator equal to zero and solve for x.
[tex]4(\text{x}^2+\text{x}+1)-(4\text{x}+6)(2\text{x}+1) = 0\\\\4(\text{x}^2+\text{x}+1)-(8\text{x}^2+16\text{x}+6) = 0\\\\4\text{x}^2+4\text{x}+4-8\text{x}^2-16\text{x}-6 = 0\\\\-4\text{x}^2-12\text{x}-2 = 0\\\\-2(2\text{x}^2+6\text{x}+1) = 0\\\\2\text{x}^2+6\text{x}+1 = 0\\\\[/tex]
From here we use the quadratic formula.
Plug in a = 2, b = 6, c = 1.
[tex]\text{x} = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\\text{x} = \frac{-6\pm\sqrt{(6)^2-4(2)(1)}}{2(2)}\\\\\text{x} = \frac{-6\pm\sqrt{28}}{4}\\\\\text{x} = \frac{-6\pm2\sqrt{7}}{4}\\\\\text{x} = \frac{2(-3\pm\sqrt{7})}{4}\\\\\text{x} = \frac{-3\pm\sqrt{7}}{2}\\\\\text{x} = \frac{-3+\sqrt{7}}{2} \text{ or } \text{x} = \frac{-3-\sqrt{7}}{2}\\\\[/tex]
If x is equal to either of those values, then f ' (x) = 0 would be the case. Therefore, these are the critical points of f(x).
There may be other critical values. We'll still need to check the second criteria.
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f ' (x) doesn't exist when we divide by zero.
Set the denominator equal to 0 and solve for x.
[tex](\text{x}^2+\text{x}+1)^2 = 0\\\\\text{x}^2+\text{x}+1 = 0\\\\[/tex]
Plug a = 1, b = 1, c = 1 into the quadratic formula.
[tex]x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\x = \frac{-1\pm\sqrt{(1)^2-4(1)(1)}}{2(1)}\\\\x = \frac{-1\pm\sqrt{-3}}{2}\\\\[/tex]
We have a negative number as the discriminant, which leads to complex number solutions in the form a+bi where [tex]i = \sqrt{-1}[/tex]
Therefore, there aren't any real number values for x that lead [tex](\text{x}^2+\text{x}+1)^2[/tex] to be zero.
No matter what we pick for x, the expression [tex](\text{x}^2+\text{x}+1)^2[/tex] will never be zero.
In short, the second criteria yields no real value critical points (assuming your teacher is only focused on real-valued functions and not complex-valued functions).
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Summary:
The first criteria f ' (x) = 0 led to [tex]\text{x} = \frac{-3+\sqrt{7}}{2} \text{ or } \text{x} = \frac{-3-\sqrt{7}}{2}[/tex] as the critical valuesThe second criteria, f ' (x) doesn't exist where x is in the domain of f(x), leads to no critical values (assuming your teacher is not focusing on complex-valued functions).Therefore, the only critical values are [tex]\text{x} = \frac{-3+\sqrt{7}}{2} \text{ or } \text{x} = \frac{-3-\sqrt{7}}{2}[/tex]
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Extra info:
A critical value is where a local/absolute min, a local/absolute max, or a saddle point would be at this x value. The 1st derivative test or 2nd derivative test would be used to determine the nature of each critical value.
A real world application of a critical value would be to determine the max revenue. Another example is to minimize the surface area while holding the volume constant. Linear regression relies on a similar concept.
To check the answer, you can type in "critical points of (4x+6)/(x^2+x+1)" without quotes into WolframAlpha.