The maximum amοunt οf prοfit the cοmpany can make is $543
What is prοfit?Prοfit is the mοney yοu have left after paying fοr business expenses. There are three main types οf prοfit: grοss prοfit, οperating and net prοfit.
Grοss prοfit is biggest. It shοws what mοney was left after paying fοr the gοοds and services sοld. Operating prοfit is next. It shοws what's left after alsο paying οperating cοsts such as rent, electricity, phοnes and in sοme cases staff.
Net prοfit cοmes last. It shοws what yοu made after deducting all cοsts and tax. Yοu get tο hοld οntο net prοfit.
Let y represent the amοunt οf prοfit and x represent the selling price.
The relatiοnship between x and y is:
y = -4x² + 141x – 703
The maximum prοfit is at dy/dx = 0, hence:
dy/dx = -8x + 141
-8x + 141 = 0
8x = 141
x = 17.625
The maximum prοfit is:
y = -4(17.625)² + 141(17.625) – 703 = 543
The maximum amοunt οf prοfit the cοmpany can make is $543
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You are in charge of ordering the food for a family party. There are 50 people attending the party. You have a budget of $1200. The catering company has three menus. You have coupon for 10% off the cost of the food. You are giving a 15% tip on the cost of the food. The sales tax is 8. 875%.
You must choose a menu, determine the total cost for the food including sales tax and tip and make sure that your work is presented in a manner that makes it easy to follow your thinking
In the following question, among the conditions given, Menu 1:Cost per person: $20, Total cost for 50 people: $20 x 50 = $1000, we can afford Menu 1 within our budget, and the total cost for the food, including sales tax and tip, is $1209.
First, we need to determine the total amount of the budget available after applying the 10% discount:
Budget after discount = $1200 - (10% x $1200) = $1080
Next, we need to consider the three menus offered by the catering company:
Menu 1:
Cost per person: $20
Total cost for 50 people: $20 x 50 = $1000
Menu 2:
Cost per person: $25
Total cost for 50 people: $25 x 50 = $1250
Menu 3:
Cost per person: $30
Total cost for 50 people: $30 x 50 = $1500
Since Menu 1 is the only option that fits within our budget, we will choose that menu.
Next, we need to calculate the total cost of the food, including the sales tax and tip. To do this, we'll need to add the sales tax and tip percentages to the original cost of the food:
Sales tax: 8.875%
Tip: 15%
Total cost per person after tax and tip = cost per person + (sales tax x cost per person) + (tip x cost per person)
Total cost per person after tax and tip = $20 + (8.875% x $20) + (15% x $20) = $24.18
Total cost for 50 people after tax and tip = $24.18 x 50 = $1209
Therefore, the total cost of the food, including sales tax and tip, for Menu 1 is $1209.
To summarize our calculations:
Budget after discount: $1080
Total cost for Menu 1: $1000
Total cost per person after tax and tip: $24.18
Total cost for 50 people after tax and tip: $1209
So, we can afford Menu 1 within our budget, and the total cost for the food, including sales tax and tip, is $1209.
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200 points giving brainly Which graph shows all real numbers closer to 0 than they are to 4
its has to be an inequality answer
Therefore , the solution of the given problem of inequality comes out to be -4 < x < 0 or 0 < x < 4 .
An inequality is what?An association or group of numbers even without equal sign can represent a disparity in algebra. Equilibrium is always followed by equity. Inequality occurs when norms are still incompatible. Disparity and fairness have different traits. We chose the most prevalent variable because parts are often not similar to one another or close to each other (). It is also possible to evaluate values using disparities of any size.
Here,
"All real numbers closer to 0 than they are to 4" is represented by the inequality:
=> |x - 0| < |x - 4|
When we simplify this inequality, we obtain:
=> |x| < |x - 4|
This inequality is true for x values between x = 0 and x = 4, the two roots of the equation |x| = |x - 4|.
Therefore, the interval (-4, 0) union (0, 4), represented by the following inequality, is the graph that depicts all real numbers as being closer to 0 than they are to 4.
=> -4 < x < 0 or 0 < x < 4
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The option B because it’s the real numbers closer 0 then they are at 4.
What is an inequality?A mathematical comparison stating that one number is larger than, less than, or equal to the other is known as an inequality. Inequalities are written using symbols such as < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to).
It is possible to write the inequality "all real numbers closer to 0 than they are to 4" as follows:
[tex]| x - 0 | < | x - 4 |[/tex]
where | | represents the absolute value function. This inequality holds for all real numbers x that are closer to 0 than they are to 4.
To graph this inequality, we can first graph the two expressions on either side of the inequality sign:
[tex]y = | x - 0 | = | x |[/tex]
and, [tex]y = | x - 4 |[/tex]
These are the graphs of the absolute value functions centered at 0 and 4, respectively. To find the solution to the inequality, we need to shade the region where the inequality is true, which is the region between these two graphs.
So the option B because it’s the real numbers closer 0 then they are at 4.
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Find the length of an edge of a square whose diagonals are of length 10cm
Answer:
Step-by-step explanation:
Diagonals intersect at 90°.
[tex]x^2=5^2+5^2[/tex] (Pythagoras Theorem)
[tex]=50[/tex]
[tex]x=\sqrt{50}[/tex] cm =7.07 cm
3) For a random sample of 100 American cities, it is found that there is a significant linear correlation between the number of robberies last year and the number of schools in the city. Does this suggest that building more schools in a city could lead to more robberies? (1 pt.) Why or why not? (2 pts.)
Building more schools could lead to more robberies.
what is sample space?
In probability theory, the sample space is the set of all possible outcomes of a random experiment or process. It is denoted by the symbol "S".
For example, if we flip a coin, the sample space would consist of the two possible outcomes: heads (H) and tails (T). If we roll a six-sided die, the sample space would consist of the six possible outcomes: 1, 2, 3, 4, 5, and 6.
No, this does not suggest that building more schools in a city could lead to more robberies. Correlation does not necessarily imply causation, and it is important to be cautious in interpreting statistical results.
A significant linear correlation between the number of robberies and the number of schools in a city simply means that there is a strong association between these two variables. It does not provide evidence of causation. In fact, it is more likely that the correlation is due to a third variable that affects both the number of schools and the number of robberies, such as population density or economic inequality.
Therefore, before concluding that building more schools could lead to more robberies, it would be necessary to conduct further research to identify the underlying causes of the correlation. Additionally, it would be important to consider other factors that may affect the relationship between the number of schools and the number of robberies, such as the quality of education, crime prevention strategies, and socioeconomic factors.
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No, Given Sample space does not necessarily suggest that building more schools in a city could lead to more robberies.
What exactly is a sample space?
In probability, a sample space is the set of all possible outcomes of a experiment. It is denoted by the symbol "S" and contains all the distinct outcomes that could occur in a given experiment. For example, if the experiment is to roll a fair six-sided die, the sample space would consist of the numbers 1, 2, 3, 4, 5, and 6. Another example of a sample space could be the possible outcomes of flipping a coin, which would be the set {heads, tails}.
Now,
No, this does not necessarily suggest that building more schools in a city could lead to more robberies.
The presence of a significant linear correlation between the number of robberies and the number of schools indicates that there is a relationship between these two variables. However, correlation does not imply causation. There may be other variables that are responsible for the observed correlation. For example, both the number of schools and the number of robberies may be correlated with the size of the city or the level of economic inequality, which could be the true underlying factors that drive the observed relationship.
Therefore, further analysis and investigation are required to determine the nature of the relationship between the number of schools and the number of robberies, and whether building more schools can indeed lead to more robberies.
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6) what is the value of x? (the figure is not drawn to scale)
7) ABCDE ~ GHJDF. complete the statements
8) find the value of x. the polygons are similar but not drawn to scale.
Based on the similarity theorem, the values are:
6. x = 21; 7. <H = <B, GH/DJ = AB/DC; 8. x = 11
How to Apply the Similarity Theorem?The similarity theorem states that the corresponding pairs of angles of two similar polygons will be congruent to each other, while their corresponding sides will be proportional to each other.
Therefore, we have:
6. x/7 = 12/4
x/7 = 3
x = 21
7. Both polygons are similar, therefore:
<H = <B, while GH/DJ = AB/DC
8. 2x - 4/6 = 9/3
2x - 4/6 = 3
2x - 4 = 18
2x = 18 + 4
2x = 22
x = 11
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alice, bob and carol each think of an expression that is a fraction with 1 as the numerator and a constant integer times some power of x as the denominator. the simplest common denominator of alice's and bob's expressions is 4x2. the simplest common denominator of bob's and carol's expressions is 12x3. the simplest common denominator of alice and carol's expressions is 6x3. find all possible expressions that could be carol's
All possible expressions that could be Carol's are 1/6x3, 1/12x3, 1/18x3, 1/24x3, 1/30x3 .
Alice, Bob and Carol each think of an expression that is a fraction with 1 as the numerator and a constant integer times some power of x as the denominator. The simplest common denominator of Alice's and Bob's expressions is 4x2, while the simplest common denominator of Bob's and Carol's expressions is 12x3. The simplest common denominator of Alice's and Carol's expressions is 6x3. To explain further, Alice's expression can be any fraction with 1 as the numerator and a constant integer times x2 as the denominator, therefore, the denominator of Alice's expression can be any multiple of 4, i.e. 4x2, 8x2, 12x2, 16x2, 20x2 and so on.
Bob's expression can also be any fraction with 1 as the numerator and a constant integer times x2 as the denominator. However, the denominator of Bob's expression must be a multiple of both 4 and 6, i.e. 12x2, 24x2, 36x2, 48x2 and so on. Therefore, Carol's expression must be in the form of a fraction with 1 as the numerator and a constant integer times x3 as the denominator. This means that the denominator of Carol's expression must be a multiple of 6, i.e. 6x3, 12x3, 18x3, 24x3, 30x3 and so on. Hence, all possible expressions that could be Carol's are 1/6x3, 1/12x3, 1/18x3, 1/24x3, 1/30x3 and so on.
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How many solutions does the nonlinear system of equations graphed below
have?
O A Zero
• B. One
O C. Four
O D. Two
Answer:
D. Two.
Step-by-step explanation:
The two graphs touch each other at two different points, meaning they have two solutions.
∠A and
∠
�
∠B are vertical angles. If m
∠
�
=
(
2
�
+
19
)
∘
∠A=(2x+19)
∘
and m
∠
�
=
(
3
�
−
18
)
∘
∠B=(3x−18)
∘
, then find the measure of
∠
�
∠A
If m ∠A=(2x+19)° and m ∠B=(3x−18)° of vertical angles ∠A and ∠B, the the measure of ∠A is 93° (degrees)
Vertical angles are formed when two lines intersect. They are opposite to each other and have the same measure. Therefore, we can set the measure of angle A equal to the measure of angle B and solve for x.
m ∠A = m ∠B
2x + 19 = 3x - 18 (Substitute the given measures of ∠A and ∠B)
19 + 18 = 3x - 2x (Add 2x to both sides and subtract 19 from both sides)
37 = x
Now that we have found the value of x, we can substitute it into the measure of ∠A to find its measure:
m ∠A = 2x + 19
m ∠A = 2(37) + 19 (Substitute x = 37)
m ∠A = 93
Therefore, the measure of ∠A is 93 degrees.
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Complete Question
∠A and ∠B are vertical angles. If m ∠A=(2x+19)° and m ∠B=(3x−18)°, then find the measure of ∠A
A diffused silicon p-n junction has a linearly graded junction on the p-side with a = 3 x 1019/cm4 and uniform doping of 5 x 1014/cm3 on the n-side. If the depletion layer width on the p-side is 0. 6 μm at zero bias, find the total depletion layer width, built-in potential and maximum field at zero bias
The total depletion layer width is 0.982 μm, the built-in potential is 0.84 V, and the maximum field at zero bias is 3.72 x 105 V/cm.
In a diffused silicon p-n junction, the p-side has a linearly graded junction with a doping concentration profile given by a = 3 x 1019/cm4, while the n-side has a uniform doping concentration of 5 x 1014/cm3. The depletion layer width on the p-side at zero bias is given as 0.6 μm.
To find the total depletion layer width, we need to first find the doping concentration at the junction on the p-side. We can do this using the relation:
[tex]Na2 = Ndp2 + 2εφbi/q(a2 - 2/3 a3)[/tex]
Where Na is the acceptor concentration, Ndp is the donor concentration on the p-side, φbi is the built-in potential, q is the electronic charge, a2 and a3 are the second and third terms of the dopant concentration expansion, and ε is the permittivity of the material.
Using the given values, we can find the acceptor concentration on the p-side to be Na = 1.35 x 1017/cm3. Then, we can find the depletion layer width on the n-side using the relation:
[tex]Wn = sqrt((2εφbi/ q)(Na/Nd - 1))[/tex]
where Nd is the doping concentration on the n-side. Since the n-side has a uniform doping concentration, we can use the given value of 5 x 1014/cm3 to find Wn = 0.382 μm.
The total depletion layer width is then the sum of the depletion layer widths on both sides, i.e., W = Wp + Wn = 0.6 + 0.382 = 0.982 μm.
To find the built-in potential, we can use the relation:
[tex]φbi = kT/q ln(NaNd/ni2)[/tex]
where k is the Boltzmann constant, T is the temperature, and ni is the intrinsic carrier concentration. Using the given values and assuming room temperature, we can find φbi to be 0.84 V.
The maximum electric field occurs at the junction and can be found using the relation:
[tex]Emax = sqrt(2qNaεφbi) / W[/tex]
Using the given values, we can find Emax to be 3.72 x 105 V/cm.
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(Please answer quickly!!!!)Simplify the expression.
−9.71 + 2.8
−12.61
−9.73
−6.91
69.10
Answer:
Step-by-step explanation:
−9.71 + 2.8 = -6.91
Answer:
6.91.
9.71
2.8
6.1
if you subtract 9.71 from 2.8 keep everything in point
Orlando skipped rope 135 times in 45 seconds.Assuming a constant rate how many time Orlando skip rope per second
Answer: 3 times per second
Step-by-step explanation:
Orland skipped 135 times in 45 seconds, so you divide 135/45 which is 3, so it is 3 skips per second :)
hope this helped <33
A length is measure as 21cm correct to 2 significant figures. What is the upper bound and lower bound
The length in this instance is accurately measured at 21 cm to two significant figures.
1. Upper Bound:
The upper bound is the highest possible value that the measurement could have while still being correct to 2 significant figures.
In 21 cm, the last significant figure is 1.
The next possible value after 1 is 2.
So, half of the difference between 2 and 1 is 0.5.
Upper Bound = 21 + 0.5
= 21.5 cm
2. Lower Bound:
In 21 cm, the last significant figure is 1, and the next possible value after 1 is 2. So, half of the difference between 1 and 2 is 0.5.
Lower Bound = 21 - 0.5
= 20.5 cm
Therefore, the upper bound is 21.5 cm and the lower bound is 20.5 cm.
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HELP ME!!
In the figure shown below, ab is a tangent to circle “O” at “a”. Also ab=16, and wb=12. Find the length of the radius.
Tell whether the triangle with the given side lengths is a right triangle. 45 in., 26 in., 51 in.
A it is nearly but not a right angled triangle.
Step-by-step explanation:
I used pythagorous theorum.
Answer:
No it is not
Step-by-step explanation:
Hypotenuse is the longest side
A ² = B² + C² (Using Pythagoras theorem)
51² = 45² + 26²
2601 =2025 +676
2601 = 2701
This is not a right angle triangle
Square root of 2701 is 51.57 and that is equivalent to 52
Can someone help me ASAP it’s due tomorrow. I will give brainliest if it’s all done correctly.
The sample space for Ted's experiment is TTT, TTH, THT, HHH, HHT, HTH, THH, HTT ( option C)
What Is sample space ?The sample space of a random experiment is the collection of all possible outcomes.
When we toss a coin, there can be only two outcomes i.e., either head or tail. So, the sample space will be, S = {H, T} where H is the head and T is the tail.
When Ted tossed the coin three times, the sample space include;
TTT, TTH, THT, HHH, HHT, HTH, THH, HTT
This means the total sample space is 8.
Therefore the sample space for Ted's experiment is TTT, TTH, THT, HHH, HHT, HTH, THH, HTT
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What is 268.8 divided by 56 using division ASAP
Answer: 4.8
Step-by-step explanation: 268.8 divided by 56 is 4.8
What is QS and what is c = to
The value of QS is 4.5c + 9.5.
What is parallelogram?A parallelogram is a four-sided plane figure with opposite sides parallel and equal in length. Its opposite angles are also equal.
Since Q is the midpoint of the diagonals RT and SU in parallelogram RSTU, we have:
QS = 1/2 RT ...(1)
QU = 1/2 SU ...(2)
Also, since RU||ST and RS||UT, we know that triangles QRU and QST are similar by AA similarity, so:
QR/QS = QU/QT
Substituting the given values and using (1) and (2), we get:
QR/(6c - 8) = (c + 7)/QT
Cross-multiplying and simplifying, we get:
QR = (6c - 8)(c + 7)/(2c - 1)
QR = (6c^2 + 34c - 56)/(2c - 1)
QR = 3(2c^2 + 11c - 18)/(2c - 1)
QR = 3(2c - 1)(c + 9)/(2c - 1)
QR = 3(c + 9)
Now, using (1), we have:
QS = 1/2 RT
QS = 1/2 (QR + RU)
QS = 1/2 (3(c + 9) + (6c - 8))
QS = 1/2 (9c + 19)
QS = 4.5c + 9.5
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I HAVE 5 MINS PLEASE HELPPPPPPPPP 30point
Never true;
The sum of two irrational numbers is irrational
Always true;
The sum of two rational numbers is rational
The product of two rational numbers is rational
Sometimes true;
The product of two irrational numbers is irrational
What are rational numbers?Every number that can be expressed as a fraction with both integers in the numerator and denominator is said to be rational (whole numbers). Numerator cannot be zero.
Hence every number that can be represented in the form p/q, where p and q are integers and q is not equal to zero, is a rational number.
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Solomon is taking out a $18,320, two-year loan with an APR of 3.29%. What will be the finance charge for this loan to the nearest dollar?
Evaluate : x = 3 , y = -6 then
4y - 6x = xy?
"Substituting x = 3 and y = -6 in the given equation, we get:
4y - 6x = xy
4(-6) - 6(3) = (3)(-6)
-24 - 18 = -18
-42 ≠ -18
Therefore, the given equation is not true for x = 3 and y = -6." (ChatGPT, 2023)
You are enclosing a circluar flower graden with a fence that cost $2. 99 per foot. The radius of a garden is 7 feet. How much will it cost to buy the fence
Answer: 20.93
Step-by-step explanation:
2.99x7
(Please answer quickly!!!)Which expression is equivalent to 6(−3.8y − 11.2)?
−22.8y − 67.2
−22.8y − 11.2
2.2y − 5.2
2.2y − 11.2
Step-by-step explanation:
6(-3.8y-11.2)
-22.8y-67.2
answer: -22.8y-67.2
Answer: -22.8y - 67.2
Step-by-step explanation: You multiply the number inside of () by whatever number is on the outside, which is 6.
6 x -3.8
6 x 3 = 18
6 x .8 = 4.8
then add on your negative symbol
-22.8
6 x 11 = 66
6 x .2 = 1.2
add them together
67.2
TRIGONOMETRY
Grade 10 Math
Calculate to the nearest 0.1 degree, the size of alpha in the drawing above.
Will give branliest and rate 5 stars for the best answer.
Need it urgently!!!
Thanks.
Check the picture below.
Make sure your calculator is in Degree mode.
[tex]\tan(\alpha )=\cfrac{\stackrel{opposite}{5}}{\underset{adjacent}{4}}\implies \alpha=\tan^{-1}\left( \cfrac{5}{4} \right)\implies \alpha \approx 51.3^o[/tex]
suppose the scores of students on a statistics course are normally distributed with a mean of 278 and a standard deviation of 61. what percentage of the students scored between 278 and 400 on the exam? (give your answer to 3 significant figures.)
Suppose the scores of students on a statistics course are normally distributed with a mean of 278 and a standard deviation of 61. The percentage of the students who scored between 278 and 400 on the exam is 69.8%.
The formula used: To find the percentage of the students who scored between 278 and 400 on the exam, we need to first calculate the z-score of the two values and then find the area between the two z-scores using the standard normal distribution table.
We can use the formula:
z=(x-μ)/σ,
where z is the z-score,
x is the value we want to find the z-score of,
μ is the mean,
and is the standard deviation.
Answer: We are given that the mean is 278 and the standard deviation is 61.
The value of x1 is 278 and the value of x2 is 400.
z1=(x1-μ)/σ
=(278-278)/61=0
z2=(x2-μ)/σ
=(400-278)/61=2.00
Using the standard normal distribution table, the area between the two z-scores is 0.4772.
The percentage of students scoring between 278 and 400 is 0.4772 100% = 47.72%.
Rounding the answer to 3 significant figures, we get 69.8%.
Hence, the percentage of the students who scored between 278 and 400 on the exam is 69.8%.
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PLEASE HELP QUICK!!!!!!
Approximately 4 out of every 6 students in Chloe’s class own pets. Chloe designed a simulation to model data for 25 students.
Answer: 144
Step-by-step explanation:
Amount of numbers in the simulation between 1 & 4 (inclusive): 15 out of a total 25
15 / 25 = 0.6
0.6 * 240 = 144 students
Ava is sewing a quilt. One of the fabric pieces is shaped like a triangle. One of the angles of the triangle measures 106°.
If the other two angles of the triangle have the same measure, what is the measure of each of them?
Enter your answer in the box.
°
find an equation of a line with a slope of -1 and having the point (-2,-6) on the line
Answer: is there a pic?
Step-by-step explanation: is there a pic of what your working on?
Answer:
y=-x-8
Step-by-step explanation:
formular for finding an equation on a line is y-y1=m(x-x1) whiles m is the slope or gradient then x1 is -2 and y1 is -6 then you substitute to the formular you state down
Consider a multinomial distribution with 3 different outcomes, that is,
(Z1, Z2, Z3)∼ Multinomial(n,p1,p2,p3),p1+p2+p3=1. Derive the conditional distribution of Z2 given Z1=z1. That is, work out its pmf and name the distribution.
z3 = n − k − z1 − z2
A multinomial distribution with 3 different outcomes is given by (Z1, Z2, Z3)∼ Multinomial(n,p1,p2,p3), where p1 + p2 + p3 = 1. The conditional distribution of Z2 given Z1=z1 is to be derived. The pmf and distribution will be found.Let k be the number of trials with Z1 = z1. Therefore, the number of trials with Z2 = z2 is n − k. Then, the conditional distribution of Z2 given Z1=z1 is(Z1,Z2=n−k,Z3)∼Multinomial(k,p1,p2,p3)×Multinomial(n−k−1,p1′,p2′,p3′),where p1' = p1 / (1 − p1), p2' = p2 / (1 − p1), and p3' = p3 / (1 − p1).As a result, the pmf of the conditional distribution of Z2 given Z1=z1 is given byP(Z2 = z2 | Z1 = z1) = (k!/(z1!z2!(n−k−z1−z2)!))×(p1z1p2z2p3n−k−z1−z2)×(n−k−1)!/(z3!(n−k−z3−1)!(p1'z1+p2'z2+p3'z3)n−k−1), where z3 = n − k − z1 − z2. The distribution obtained is named a "conditional multinomial distribution".
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find the equation of a straight line passing through a point (-2, 1) and perpendicular to the line -3x +5y +2=0
Answer:
y= -5/3x - 7/3
Step-by-step explanation:
1. solve the line
-3x +5y + 2 = 0
+3x +3x
5y + 2 = +3x
-2 -2
5y = +3x - 2
/5 /5
y= 3/5x -2/5
2. find slope
perpendicular lines have an opposite reciprocal slope to original line so your slope is now -5/3
3. find b ( y - intercept )
point ( -2 , 1 )
y - y1 = m ( x - x1 )
y - 1 = -5/3 ( x + 2 )
y -1 = -5/3x - 10/3
+1 + 1
y= -5/3x - 7/3
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Find the directions in which the function increases and decreases most rapidly at P_0. Then find the derivatives of the function in these directions. f(x, y) = x^2 + xy + y^2, P_0(-1, 4) The direction in which the given function f(x, y) = x^2 + xy + y^2 increases most rapidly at P_0 (-1, 4) is u = i + j. (Type exact answers, using radicals as needed.) The direction in which the given function f(x, y) = x^2 + xy + y^2 decreases most rapidly at P_0 (- 1, 4) is - u = i + j. (Type exact answers, using radicals as needed.) The derivative of the given function f(x, y) = x^2 + xy + y^2 in the direction in which the function increases most rapidly at P_0 (-1, 4) is D_u f = (Type an exact answer, using radicals as needed.) The derivative of the given function f(x, y) = x^2 + xy + y^2 in the direction in which the function decreases most rapidly at P_0 (-1, 4) is D_-u f = (Type an exact answer, using radicals as needed.)
The direction in which the given function[tex]f(x, y) = x^2 + xy + y^2[/tex]increases most rapidly at P_0 (-1, 4) is [tex]u = < 2/√53, 7/√53 >[/tex], the direction in which the function decreases most rapidly at P_0 is [tex]-u = < -2/√53, -7/√53 >[/tex], the derivative of the function in the direction of the gradient is[tex]D_u f = √53,[/tex]and the derivative of the function in the opposite direction of the gradient is [tex]D_-u f = -√53.[/tex]
The direction in which a function increases most rapidly at a given point is the direction of the gradient of the function at that point. The gradient of the function [tex]f(x, y) = x^2 + xy + y^2[/tex] is given by:
[tex]∇f = < 2x + y, x + 2y >[/tex]
At the point P_0 (-1, 4), the gradient is:[tex]∇f(P_0) = < 2(-1) + 4, -1 + 2(4) > = < 2, 7 >[/tex]
The direction in which the function increases most rapidly at P_0 is the unit vector in the direction of the gradient:
[tex]u = ∇f(P_0)/||∇f(P_0)|| = < 2/√53, 7/√53 >[/tex]
The direction in which the function decreases most rapidly at P_0 is the opposite of the direction of the gradient:[tex]-u = < -2/√53, -7/√53 >[/tex]
The derivative of the function in the direction of the gradient is the dot product of the gradient and unit vector in the direction of the gradient:
[tex]D_u f = ∇f(P_0) · u = < 2, 7 > · < 2/√53, 7/√53 > = (4 + 49)/√53 = 53/√53 = √53[/tex]
The derivative of the function in the opposite direction of the gradient is the dot product of the gradient and the unit vector in the opposite direction of the gradient:
[tex]D_-u f = ∇f(P_0) · -u = < 2, 7 > · < -2/√53, -7/√53 > = (-4 - 49)/√53 = -53/√53 = -√53[/tex]
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