Answer:
Step-by-step explanation:
Consider the following four matrices:
(1000),(0010),(0100),(0001).
See what happens when you solve the equation AB=BA
for each of those four (let B
be each one of those four). To facilitate it, write A=(acbd)
You will get a set of equations for the entries of a
which are easily solved. This trick is quite general.
write the standard form of nine-ninths
Answer:
Step-by-step explanation:40
41
+
9
i
41
Please answer this question for me
The given two triangles are similar to each other by the postulate called SAS.
What is triangle ?
Triangle can be defined as in which it consists of three sides, three angles and sum of the three angles are 180 degrees always.
Given,
The triangles are RST and UVT
So
From the given figure we can say that,
RT = TU
RS = VU
As two sides are already equal and vertically opposite angles are always equal .
so we can say that By postulates SAS we can say that they are similar to each other.
Hence, The given two triangles are similar to each other by the postulate called SAS.
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A bag contains 2 gold marbles, 10 silver marbles, and 30 black marbles. You randomly select one marble from the bag. What is the probability that you select a gold marble? Give your answer as a reduced fraction.
The probability that you select a gold marble is, 1/21.
What is mean by Probability?The term probability refers to the likelihood of an event occurring. Probability means possibility. It is a branch of mathematics that deals with the occurrence of a random event. The value is expressed from zero to one.
We have to given that;
A bag contains 2 gold marbles, 10 silver marbles, and 30 black marbles.
Here, Number of gold marbles = 2
And, Number of total marbles = 2 + 10 + 30
= 42
Hence, The probability that you select a gold marble is,
⇒ Number of gold marbles / Total number of marbles
⇒ 2 / 42
⇒ 1 / 21
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on a variable ratio schedule, reinforcement is given for the . group of answer choices first correct response after varying amounts of time have passed next correct response after a varying number of responses have been made next correct response after a fixed number of responses have been made
On a variable ratio schedule, reinforcement is given for the next correct response after varying numbers of responses have been made.
What is variable ?
A variable is a value or quantity that can change or take on different values in a given context or system. In mathematics and science, variables are often used to represent unknown or changing values in an equation or model. In programming, a variable is a named container or storage location that holds a value that can be modified during the execution of a program.
On a variable ratio schedule, reinforcement is given for the group of correct responses that occur after varying amounts of time have passed. This means that the reinforcement is not given after a fixed number of responses or after a fixed amount of time has passed. Instead, the reinforcement is given after a varying number of correct responses have been made, making it unpredictable for the individual or organism being reinforced. This type of schedule is often used in gambling and other situations where the reinforcement is desired to be unpredictable and maintain the individual's interest or motivation.
On a variable ratio schedule, reinforcement is given for the next correct response after varying numbers of responses have been made.
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In the 30-60-90 triangle below, side s has a length of________ and side q has a length of_______
A. 16√3, 5
B. 8-5, 16
C. 4, 8√3
D. 4√2, 4√2
E. 4, 4.5
F. 16√3, 16√3
In the 30-60-90 triangle below, side s has a length of 4 and side q has a length of [tex]4\sqrt{3}[/tex] (Option E).
RIGHT TRIANGLEA triangle is classified as a right triangle when it presents one of your angles equal to 90º. The greatest side of a right triangle is called the hypotenuse. And, the other two sides are called legs.
The math tools applied for finding angles or sides in a right triangle are the trigonometric ratios or the Pythagorean Theorem. The Pythagorean Theorem says (hypotenuse)²= (leg1)²+(leg2)². And the main trigonometric ratios are: sin (x) , cos (x) and tan (x) .
The question asks the value of sides s and q of the given right triangle. If you have the value of the hypotenuse and all angles of the triangle, you can solve this exercise by applying the trigonometric ratios. See below.
[tex]sin 30\° = \frac{opposite \; leg}{hypotenuse}\\ \\ \frac{1}{2}=\frac{s}{8} \\ \\ 2s=8\\ \\ s=4[/tex]
[tex]cos 30\° = \frac{adjacent\; leg}{hypotenuse}\\ \\ \frac{\sqrt{3} }{2}=\frac{q}{8} \\ \\ 2q=8\sqrt{3} \\ \\ q=4\sqrt{3}[/tex]
Therefore, the solution to this question is the letter E.
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4) 7(5v − 2)(7v + 3) = 0
e) A student spent 50 minutes doing her homework. She spent m minutes doing Geography. 2m minutes doing Mathematics and the remaining (m + 7) minutes studying History. How many minutes did she spend doing Mathematics?
Answer: 22 minutes
Step-by-step explanation: m + 2m + m + 7 = 4m + 7
4m + 7 = 50
4m = 44
m = 11
2m = 11 x 2 = 22 minutes
Please help
Use the FOIL method to find the terms of the following multiplication problem
(4 + 4i) • (5 - 4i)
Using the FOIL method, the product of the first terms is the product of the outside terms is and the product of the inside terms is The product of the last terms in terms of i^2 is _ which simplifies to _
View image
Worth 100 points and person who gets it right gets brainliest!
Answer:
6% = $4000
Step-by-step explanation:
6%x + 7%(8500 - x) [tex]\geq[/tex] 555
0.06x + 0.07(8500 - x) [tex]\geq[/tex] 555
0.06x + 595 - 0.07x [tex]\geq[/tex] 555
-0.01x + 595 [tex]\geq[/tex] 555
-0.01x [tex]\geq[/tex] -40
x [tex]\leq[/tex] 4000
Containers of gochujang, a Korean chili sauce, are sold at a market. The scatter plot shows the relationship between the cost of different containers and the amount of sauce in each container. A line of best fit is drawn through the data. Based on the line of best fit, what is the approximate cost per ounce of a container of gochujang at the market?
The cost per ounce of a container of gochujang at the market can be calculated using the equation of the line of best fit.
The line of best fit is given by the equation y = 0.14x + 1.3, where y is the cost of the container and x is the amount of sauce in the container (in ounces). To calculate the approximate cost per ounce of a container of gochujang, we can use this equation. We set y = 1, since we want the cost of one ounce of sauce. The equation of this line is y = 0.25x + 3.2, where x is the amount of sauce in the container and y is the cost of the container. To find the cost per ounce, we need to divide the y-intercept, 3.2, by the slope, 0.25. This gives us 12.8. Therefore, the approximate cost per ounce of a container of gochujang at the market is $12.8.
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The beginning balance on the monthly bank statement for Dorsey's checking
account was $452.76, and the ending balance was $197.59. What can be said
about Dorsey's transactions for the month?
A. He had $650.35 more in credits than in debits.
B. He had $255.17 more in credits than in debits.
C. He had
$650.35 more in debits than in credits.
OD. He had $255.17 more in debits than in credits.
SUBMIT
He has 255.17 more in debits than in credits. Thus, option D is correct.
How to Dorsey's transaction for the month?Debits and credits in double-entry bookkeeping are entries made in account ledgers to reflect changes in value brought on by business transactions. A debit entry, not a credit entry, is used to show the amount of money that has been transferred into or out of an account. Value is exchanged with each transaction from credited to debited accounts.
Let's call B the amount in credit that Ignatius had, and C the amount in debits that Ignatius had during the month.
We know that the initial amount in his account was 124.82, while the final balance was 659.27.
Final amount = the initial amount + amount of credits - the amount of debits
$255.17= $255.17 (B-C)
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Determine the order of the given differential equation and state whether the equation is linear or nonlinear: d^4u/dr^4 + du/dr + 5u = cos(r + u) You have only one attempt for credit. (a) The order of this differential equation is 4 (b) The equation is Linear
CHOSE THE LETTER of the correct equation for the graph PLS HELP
The trigonometric function that represent the equation in the graph is y = sin(x + π/2).
What is the trigonometric function?The period of a function is the distance between the repetitions of any function. A period is the length of one complete cycle of a trigonometric function.The graph of y = sin(x) is like a wave that oscillates between -1 and 1, repeating itself every 2 units.The key components visible on the graph are:
y = π/2 is the midpoint.
We can conclude that the correct answer is either b or d.
at x = 0, f(x) = 0.
Function y = sin(x + π/2)
Therefore the correct answer is option d that is y = sin(x + /2) is the trigonometric function that represents the equation in the graph.
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1. You are the manager of a small store that specializes in hats, sunglasses, and other accessories. You are considering a sales promotion of a new line of hats and sunglasses. You will offer the sunglasses only to those who purchase two or more hats, so you will sell at least twice as many hats as pairs of sunglasses. Moreover, your supplier tells you that, due to seasonal demand, your order of sunglasses cannot exceed 100 pairs. To ensure that the sale items fill out the large display you have set aside, you estimate that you should order at least 210 items in all. Assume that you will lose $3 on every hat and $2 on every pair of sunglasses sold. Given the constraints above, how many hats and pairs of sunglasses should you order to lose the least amount of money in the sales promotion? [Using Graphic method]
You should order 197.5 hats and 12.5 pairs of sunglasses to lose the least amount of money in the sales promotion.
How to solve the equation?This is a linear programming problem, which can be solved using a system of equations.
Let x be the number of hats sold and y be the number of pairs of sunglasses sold.
From the problem, we know the following constraints:
x >= 2y (since for every pair of sunglasses sold, two hats must be sold)
x + y >= 210 (to ensure that the items fill out the display)
y <= 100 (the maximum number of pairs of sunglasses that can be ordered)
We also know that the profit per hat sold is -$3 and the profit per pair of sunglasses sold is -$2.
So, the objective function is:
-3x - 2y (to minimize the loss)
Now we can use the equations and inequalities to find the optimal solution.
x >= 2y
x >= 4y
y <= 100
x + y >= 210
x >= 8y
y <= 100
x + y >= 210
Now we can use the equation to find the optimal solution
x = 8y
y <= 100
x + y >= 210
8y <=100
y <= 12.5
x + y >= 210
x >= 210 - y
Now we can find the value of x, y
y = 12.5
x = 210 - y
x = 210 - 12.5
x = 197.5
So the optimal solution is to order 197.5 hats and 12.5 pairs of sunglasses.
You should order 197.5 hats and 12.5 pairs of sunglasses to lose the least amount of money in the sales promotion.
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K
Each of the following three data sets represents the IQ scores of a random sample of adults. IQ
scores are known to have a mean and median of 100
105
105
111
105
111
102
106
91
99
99
109
99
109
111
115
112
Sample of Size 5
91
91
111
Sample of Size 12
888888 888
100
110
97
104
Sample of Size 30
91
111
113
113
108
113
108
91
118
110
Full data set o
101
109
101
109
91
97
95
CELES
m
For each data set, compute the mean and median
What is the mean of the sample of size 5?
(Type an integer or decimal rounded to one decimal place as needed
A 1.5 standard deviations above the mean IQ score is 122.5.
How do equations work?An expression that depicts the relationship between two or more numbers and variables is called an equation.
A dependent variable is one that depends on another variable, whereas an independent variable does not depend on any other variable for its value.
Giving the z score are:
z is calculated as (raw score - mean) / standard deviation.
with a standard deviation of 15 and a mean of 100. The z-score of 1.5 is equal to the IQ score that is 1.5 standard deviations above the mean, so:
1.5 = (x - 100)/15
x = 122.5
A 1.5 standard deviations above the mean IQ score is 122.5.
The Complete Question is IQ score.
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Show and explain how to find the equation of the plane which contains the point P(1,1,3) and the line x=3t,y=5ât,z=1+2t.
The equation of the plane is:-6x + 0y + 6z - 18 = 0
The equation of a plane containing a point P(x1, y1, z1) and a line defined by x = x0 + at, y = y0 + bt, z = z0 + ct, is given by:
Ax + By + Cz + D = 0
Where:
A = (y1 - y0)a - (x1 - x0)b
B = (z1 - z0)a - (x1 - x0)c
C = (z1 - z0)b - (y1 - y0)c
D = -(A*x1 + B*y1 + C*z1)
Plugging in the given values, we have:
A = (1 - 5)3 - (1 - 3)5 = -6
B = (3 - 1)3 - (1 - 3)2 = 0
C = (3 - 1)5 - (1 - 5)2 = 6
D = -(-6*1 + 0*1 + 6*3) = -18
Therefore, the equation of the plane is:
-6x + 0y + 6z - 18 = 0
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10) The table below shows the year and the number of households in a building
that had high-speed broadband internet access.
Number of 11 16 23 33 42 47
Households
Year
2002 2003 2004 2005 2006 2007
For which interval of time was the average rate of change the smallest?
(1)2002-2004
(3) 2004-2006
(2) 2003-2005
(4) 2005-2007
The interval of time that has the average rate of change being the smallest is 1. 2002 - 2004.
How to calculate the average rate?From the information given, the table shows the year and the number of households in a building that had high-speed broadband internet access.
In this case, the average rate of change from 2002-2004 will be:
= 23 - 11
= 12
The average rate of change from 2004-2006 will be:
= 47 - 23
= 24
The average rate of change from 2003-2005 will be:
= 33 - 16
= 17
The smallest is from 2002 to 2004.
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Abby is cycling to her school from home .on her way , she meets her friend Betty, who says that the roads were blocked and the school closed .Abby turned to go back home ..Abby's distance from home is given by function : s= -100t2 +400t
How many minutes after leaving home did Abby meet Betty ?
t is time in minutes . and s i s distance .
The amount of time that it took for Abby to meet Betty is given as follows:
2 minutes.
How to obtain the amount of time that it took for Abby to meet Betty?The quadratic function giving Abby's distance from home is defined as follows:
s(t) = -100t² + 400t.
The coefficients of the function are given as follows:
a = -100, b = 400.
The vertex of the quadratic function is the turning point of the graph, that is, the moment that Abby meet Betty.
The t-coordinate of the vertex is given as follows:
t = -b/2a
t = -400/-200
t = 2 minutes.
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Use the Spider Tool located on page 1 of this activity to recreate the eight-pointed logo. Start by asking the spider to move 50 units, rotate 135 degrees, and repeat the sequence until she returns to her starting point. Remember that you can use "Zoom out" or "Fit to window" if the spider leaves the viewing area. In the space below,
sketch the eight-pointed logo you created with the Spider Tool. (4 points)
PLEASE HELP
the star of the mentioned, figure is attached below.
What is a star?
A polygon, which is a form with many sides, is a type of star. In fact, five-pointed stars are categorised as decagons or polygons with ten sides, although the number of sides changes depending on the number of points on the star, such as six, ten, etc. An equilateral and equiangular polygon that self-intersects is referred to as a regular star shape or regular star polygon. It was made by connecting the vertices of a straightforward, regular, n-sided polygon to one another in a nonadjacent direction until the original vertices were once again reached.
How are stars with five points made?
All stars are formed by lines at particular angles. The basic pattern is as follows in the case of five-pointed stars, which are the most prevalent:
Use lines of the same length and rotate 135° each time.
Hence the star of the mentioned figure is attached below.
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anyone know this ? please
The artist needs 78,835.2 cm² of metal to make the sculpture.
How much metal does the artist need to make the sculpture?The artist needs to calculate the total volume of the sculpture in order to determine how much metal is required to make the sculpture.The volume of a rectangular prism can be calculated by multiplying the length, width, and height. In this case, the length is 21.8 cm, the width is 24.4 cm, and the height is 39.6 cm. Multiplying these three measurements gives a total volume of 25,942.56 cm3 (cubic centimeters).Now that the artist knows the total volume of the sculpture, they can calculate how much metal they need to make it.To do this, they need to know the density of the metal they want to use. Let’s assume they are using steel, which has a density of 7,860 kg/m3 (kilograms per cubic meter).To find out how much metal the artist needs, they need to convert the volume of the sculpture from cm3 to m3, by dividing it by 1,000,000.This gives a volume of 0.02594256 m3. Multiplying this by the density of steel gives a total mass of 206.48 kg of steel required to make the sculpture.Therefore, the artist needs 206.48 kg of steel to make their rectangular prism sculpture.To learn more about the total volume of the sculpture refer to:
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dominic is making meatballs. he uses 3/4 cup of breadcrumbs
In a case whereby Dominic is making and uses 3/4 cup of breadcrumbs for every 11/4 pounds of ground beef the number of cups of breadcrumbs he need when he uses 1 pound of ground beef is 3/5.
How can the cups of breadcrumbs he used be calculated?The concept that will be used here is fraction. fraction: A number that is stated as a quotient in mathematics, when the numerator and denominator are divided. Both are integers in a simple fraction. A fraction appears in the numerator or denominator of a complex fraction. The numerator of a proper fraction is less than the denominator.
Since Dominic is using 3/4 cup of breadcrumbs for every 11/4 pounds of ground beef
The amount that will be needed for 1 pound of ground beef will now be:
11/4 pounds = 3/4 cup
we can convert to improper fraction as
5/4 * 3/4
12/ 20= 3/5
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complete question:
Dominic is making meatballs. He uses 3/4 cup of breadcrumbs for
every 11/4 pounds of ground beef. How many cups of breadcrumbs
does he need when he uses 1 pound of ground beef? *
There are 24 students in a class. The teacher divided the students into 4 teams. Then, she will divided the teams in half. How many students will be on each team.
A company that ships glass for a glass manufacturer claimed that its shipping boxes are constructed so that no more than 8 percent of the boxes arrive with broken glass. The glass manufacturer believed the actual percent is greater than 8 percent. The manufacturer selected a random sample of boxes and recorded the proportion of boxes that arrived with broken glass. The manufacturer tested the hypotheses H0:p=0.08 versus Ha:p>0.08 at the significance level of α=0.01. The test yielded a p-value of 0.001. Assuming all conditions for inference were met, which of the following is the correct conclusion?
A
The p-value is greater than α, and the null hypothesis is rejected. There is convincing evidence that the proportion of all boxes that contain broken glass is greater than 0.08.
B
The p-value is greater than α, and the null hypothesis is rejected. There is not convincing evidence that the proportion of all boxes that contain broken glass is greater than 0.08.
C
The p-value is greater than α, and the null hypothesis is not rejected. There is not convincing evidence that the proportion of all boxes that contain broken glass is greater than 0.08.
D
The p-value is less than α, and the null hypothesis is rejected. There is convincing evidence that the proportion of all boxes that contain broken glass is greater than 0.08.
E
The p-value is less than α, and the null hypothesis is not rejected. There is not convincing evidence that the proportion of all boxes that contain broken glass is greater than 0.08.
The correct answer is D. the proportion of boxes with broken glass is greater than 0.08.
The p-value is 0.001 which is less than the significance level of α = 0.01; therefore, the null hypothesis (H0: p = 0.08) can be rejected. We can conclude that there is convincing evidence that the proportion of all boxes that contain broken glass is greater than 0.08.
To calculate the p-value, we can use the formula: p-value = P(x ≥ X|H0)
Where X is the sample mean, and H0 is the null hypothesis that the proportion of boxes with broken glass is equal to 0.08.
Using this formula, we get: p-value = P(x ≥X|H0) = P(x ≥ 0.08) = 0.001
Since the p-value is less than the significance level of 0.01, the null hypothesis can be rejected and we can conclude that there is convincing evidence that the proportion of boxes with broken glass is greater than 0.08.
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maria wants to rent a kayak she has two options to choose from surf bargains charges 31.54 for 2 hours kayak central charges 43.96 for 3.5 hours which shop offers the lower hourly rate
Answer: Kayak Central
Step-by-step explanation:
Surf Bargains is 31.54 for 2 hours, if you want to find the hourly cost your going to do 31.54/2= 15.76 So Surf Bargains is 15.76 per hour.
Kayak Central you will do the same thing, 43.96/3.5= 12.56 per hour.
This means kayak central offers the lowest rate of $12.56 per hour.
distribute (-3u⁴+4u+6) (-5u²v)
NEED HELP ASAP
Answer: 4(u−2)−3u=2(u−6)
Step-by-step explanation:
Height vs. Age
Height (cm)
160
160
160
160
160
160
160
Function? (Yes or No)
no
If it is a Function what is the:
Domain
Range
****
Age (years)
20
25
30
35
40
45
We can see that height is the dependent variable and age is independent.
Therefore the graph of the relation, if connected, is a line segment parallel to the x-axis.
If we prove it with the vertical line test it will pass the test. Hence we can state it is a function.
The domain is:
x ∈ {20, 25, 30, 35, 40, 45}The range is:
y = 160you could at least answer the question not say why should anyone care about a school bus
Answer:
Step-by-step explanation:
I think anyone needs to care about the school bus, it's just a bus, and maybe the people inside are like you, so relax, keep calm on a school bus
If someone tells us that a random variable is described by a Binomial or Poisson distribution, it becomes a very simple matter (presuming we know the key parameters) to be able to then predict the likelihood that any particular combination of values will be seen in a dataset containing that variable. Sometimes, though, we don’t know what kind of distribution our variable will fit, or we don’t know enough to be able to identify the key parameters needed to make accurate predictions.
The binomial distribution has one possible outcome, success, and two possible outcomes, failure. In contrast, the Poisson distribution has an infinite number of potential outcomes.
What are the distribution about?The distribution of binary data from a finite sample is described by the binomial distribution. As a result, it provides the likelihood that r events will occur in n trials. The distribution of binary data from an infinite sample is described by the Poisson distribution. It provides the likelihood that r events will occur in a population. Two discrete probability distributions are the Poisson distribution and the Binomial distribution.
In discrete trials, the binomial distribution counts discrete occurrences. In a continuous domain, the poisson distribution counts discrete occurrences. The poisson should ideally only be used when success could happen anywhere in a domain.
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A division of Chapman Corporation manufactures a pager. The weekly fixed cost for the division is $23,000, and the variable cost for producing x pagers/week in dollars is represented by the function V(x). $ V(x) = 0.000001x^3 - 0.01x^2 + 50x $ The company realizes a revenue in dollars from the sale of x pagers/week represented by the function R(x). $ R(x) = -0.02x^2 + 150x \; \quad \; (0 \leq x \leq 7500) $ (a) Find the total cost function C. C(x) = (b) Find the total profit function P. P(x) = (c) What is the profit for the company if 1,900 units are produced and sold each week? $
The total cost function C is T(x) = 0.000001x³ - 0.03x² + 200x + 15,000, the total profit function P is P(x) = 0.000001x³ - 0.01x² + 50x + 15,000 and the profit for the company if 1,900 units are produced and sold each week is $80,759
The term function in math is written as a rule that gives value of a dependent variable that corresponds to specified values of one or more independent variables.
Here we have given that the division of Chapman Corporation manufactures a pager.
Here we also know that the weekly fixed cost for the division is $15,000, and the variable cost for producing x pagers/week in dollars is represented by the function V(x).
Then the function is written as,
=> V(x)= 0.000001x³ - 0.01x² + 50x.
Here we have given that company realizes a revenue in dollars from the sale of x pagers/week represented by the function R(x).
Then the function is written as,
=> R(x)= -0.02x² + 150x Where the value of x lies between (0<x<7500)
Then the total cost is calculated as,
=> T(x) = Total Cost + fixed cost + variable cost
Then apply the values on it, then we get,
=> T(x) = 15,000 + 0.000001x³ - 0.01x² + 50x - 0.02x² + 150x
=> T(x) = 0.000001x³ - 0.03x² + 200x + 15,000
And the total profit function is written as,
=> P(x) = 0.000001x³ - 0.03x² + 200x + 15,000 - (-0.02x² + 150x)
=> P(x) = 0.000001x³ - 0.01x² + 50x + 15,000
If the is the profit for the company if 1,900 units are produced and sold each week, then the profit is calculated as,
=> P(1900) = 0.000001(1900)³ - 0.01(1900)² + 50(1900) + 15,000
=> P(1900) = 80759
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Noah had 151 stamps. He had 42 more stamps than Jonas.
a) How many stamps did Jonas have?
b) How many stamps must Noah give to Jonas so that they each have the same number of stamps?
A linear combination of the values of a1 = (1,-2,0), a2 = (0,1,2), and a3 = (5,-6,8) with the corresponding weights is b = (2,-1,6).
How to find the calculation?We may construct a system of equations using the coefficients of the linear combination and the elements a1, a2, and a3 to determine whether b is a linear combination of a1, a2, and a3.
It is possible to write the equation for a linear combination of the type b = k1a1 + k2a2 + k3a3 in the following way:
b = k1 × (1,-2,0) + k2 × (0,1,2) + k3 × (5,-6,8) (5,-6,8)
So,
x = k1 + 5k3
y = -2k1 + k2 - 6k3
z = 2k2 + 8k3
From b =, we know the values of x, y, and z. (2,-1,6)
Thus, we can construct the equations in the following system:
x = k1 + 5k3 = 2
y = -2k1 + k2 - 6k3 = -1
z = 2k2 + 8k3 = 6
To determine the values of k1, k2, and k3, we can solve the system of equations using techniques like substitution or elimination.
k1 = 1, k2 = 3, k3 = 1
As a result, we can write b as a linear combination of the variables a1, a2, and a3 with the respective weights of (1, 3, and 1) for k1, k2, and k3.
b = 1a1 + 3a2 + 1 × a3
This indicates that b is a linear combination of the three variables a1, a2, and a3 with the weights (1, 1, and 3, respectively).
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