The number of belts produced must be positive: X1, X2, X3 >= 0
The shadow price of a constraint can be understand as the change in the objective function value for a unit change in the right-hand side (RHS) value of the constraint, with all other variables held constant.
What are constraints?A constraint is a condition of an optimization problem that should be satisfied the condition.
Here we can formulate this problem as a linear programming model (LPM):
Consider X1 be the number of belt A produced per day
Let X2 be the number of belt B produced per day
Let X3 be the number of belt C produced per day
The objective function is to maximize the profit, that is;
3X1 + 5X2 + 4X3.
The constraints are as follows:
The time spent on machine M1 must be less than or equal to 8 hours per day then;
2X1 + 3X2 <= 8
The time spent on machine M2 must be less than or equal to 10 hours per day then;
3X1 + 2X2 + 5X3 <= 10
The time spent on machine M3 must be less than or equal to 15 hours per day then we have
2X1 + 2X2 + 4X3 <= 15
Thus number of belts produced must be positive:
X1, X2, X3 >= 0
To solve this LPM using the simplex algorithm, you can follow these steps:
w In standard form, the objective function is to minimize the cost, and all constraints are in the form of "less than or equal to" a certain value.
The pivot row is the row that corresponds to the smallest positive ratio between the right-hand side (RHS) value and the corresponding pivot column element the constraint 2X1 + 3X2 ≤ 8.
Now Interpret the shadow prices.
The shadow price of a constraint can be understand as the change in the objective function value for a unit change in the right-hand side (RHS) value of the constraint, with all other variables held constant.
It can help to understand the value of the constraints in the LPM and how they contribute to the optimal solution.
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Neglecting air resistance and the weight of the propellant, determine the work done in propelling a 10-ton satellite to a height of 200 miles above Earth. Assume that the Earth has a radius of 4000 miles. O 4.190.48 mi-ton 1,142.86 mi-ton O 1,904.76 mi-ton 2,857.14 mi-ton 1,523.81 mi-ton
Therefore ,the the the work done in propelling a 10-ton satellite to a height of 200 miles above Earth is 1904.7619 million ton.
Analyze the equation.An expression is composed of a number, a variable, both, or neither, and specific operation symbols. An equation is made up of two expressions, which are separated by an equal sign.
Here,
Use F(x) = C/[tex]x^{2}[/tex]
Where c is a constant and x is the radius of the earth
F(x) is satellite weight
We have to find c.
thus,
=> 10 = c /[tex]4000^{2}[/tex]
=> c =160000000
Therefore,
=> F(x) = 160000000/ [tex]x^{2}[/tex]
Workdone ,
=> W = [tex]\int\limits^{4200}_{4000} {F(x)} \, dx[/tex]
=>W = [tex]\int\limits^{4200}_{4000} {160000000 /x^{2} } \, dx[/tex]
=>W =[tex]\left \{ {{4200} \atop {4000}} \right. \frac{-160000000 }{x}[/tex]
=> W =1904.7619 million ton
Therefore ,the the the work done in propelling a 10-ton satellite to a height of 200 miles above Earth is 1904.7619 million ton.
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It is known that 25 % percent of Americans identify themselves as republican . In a random sample of n=20 Americans , what is the expected number of republicans ? It is known that 25 % percent of Americans identify themselves as republican . In a random sample of n=20 Americans , what is the standard deviation of the number of republicans ? ( Round to 2 decimal places )
Using percentages, we know that out of 20 people, 5 people that is 25% of the random sample consider themselves as republicans.
What is the percentage?A percentage is a figure or ratio stated as a fraction of 100 in mathematics.
We must divide the value by the entire value to find the percentage, and then multiply the resulting number by 100.
As an illustration, 1% of 1,000 chickens is equal to 1/100 of 1,000, or 10 birds, and 20% of the quantity is equal to 20% of 1,000, or 200.
So, we have a random sample of 20 people.
n = 20
And we also know that 25% of Americans identify themselves as republicans.
Then the number of republicans in 20 people would be:
20/100 * 25
.2 * 25
5
Therefore, using percentages, we know that out of 20 people, 5 people that is 25% of the random sample consider themselves as republicans.
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NO LINKS!!
Find the average rate of change of the function from x1 to x2.
function f(x) = -9x + 4
x-values x1 = -5, x2 = 0
Answer:
-9
Step-by-step explanation:
Given function:
[tex]f(x)=-9x+4[/tex]
Given x-values:
x₁ = -5x₂ = 0Calculate the value of the function for the two given values of x:
[tex]\begin{aligned}\implies f(x_1)&=-9(-5)+4\\&=45+4\\&=49\end{aligned}[/tex]
[tex]\begin{aligned}\implies f(x_2)&=-9(0)+4\\&=0+4\\&=4\end{aligned}[/tex]
[tex]\boxed{\begin{minipage}{6.3 cm}\underline{Average rate of change of function $f(x)$}\\\\$\dfrac{f(b)-f(a)}{b-a}$\\\\over the interval $a \leq x \leq b$\\\end{minipage}}[/tex]
As -5 < 0:
a = x₁ = -5b = x₂ = 0Therefore:
[tex]\begin{aligned} \implies \textsf{Average rate of change}&=\dfrac{f(x_2)-f(x_1)}{x_2-x_1}\\\\&=\dfrac{4-49}{0-(-5)}\\\\&=\dfrac{-45}{5}\\\\&=-9\end{aligned}[/tex]
*What two numbers have a sum of 352 and a difference of 104?
Answer:
124 and 228
Step-by-step explanation:
Two numbers will be presented as x and y
x + y = 352
x - y = 104
x + y + x - y = 352 + 104
2x = 456
Divide both side by 2
x = 228
x + y = 352
y = 352 - x
y = 352 - 228
y = 124
can I get help with this practice I'm having a lot of trouble with this can I get some help
The formula to find the slope is m = (y₂ - y₁)/(x₂ - x₁), hence option B is correct and the slope of the line will be -1/2.
What is slope?It is possible to determine a line's direction and steepness by looking at its slope. Finding the slope between lines inside a coordinate plane can aid in anticipating if the lines are perpendicular, parallel, or none at all without physically using a compass.
As per the data mentioned in the question,
1.
The formula to find the slope of the line between two points is,
m = (y₂ - y₁)/(x₂ - x₁)
2.
The given points are,
(-4, 3) and (12, -5)
So, the slope of the line that connects given points,
m = (-5 - 3)/(12+4)
m = -8/16
m = -1/2
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2. The width, w, of a rectangular garden is x-2. The area of the garden is ³-2x-4. What is
an expression for the length of the garden?
Ox²-2x-2
Ox²+2x-2
Ox²-2x+2
Ox²+2x+2
Consequently, the rectangular garden's length is [tex]x^{2} +2x+2[/tex]
option D is correct
what is area?Area is the entire amount of space occupied by a flat (2-D) surface or an object's shape. Surface area refers to an open surface or the perimeter of a three-dimensional object, whereas plane region or plane area refers to a shape or planar lamina.
given
the width(w) of the rectangular garden= [tex]x[/tex][tex]-2[/tex]
area of the rectangular garden = [tex]x^{3} -2x-4[/tex]
The area of a rectangle is,
area = length × width(w) ,
If l is the rectangle's length,
w is the width of the rectangle,
By substituting values,
[tex]x^{3}-2x-4[/tex] = ([tex]x-2[/tex]) * length ( l )
[tex]l=\frac{x^{3}-2x-4 }{x-2}[/tex] = [tex]x^{2} +2x+2[/tex] (by long division method)
Consequently, the rectangular garden's length is [tex]x^{2} +2x+2[/tex]
option D is correct
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Sheryl creates a scatter plot to analyze how an increase in the outside temperature above 52°F affects the sale of hot chocolate at her shop. The line shown on the graph is the line of best fit for the data. Which statements are true about the graph? Select all that apply.
1. The y-intercept represents the orders for hot chocolate on a day when the outside temperature was 52°F.
2. Sheryl gets 160 orders for hot chocolate on a day when the outside temperature is 52°F.
3. Sheryl gets 2 fewer orders for each degree increase above 52°F in the outside temperature.
4. Sheryl gets 16 fewer orders for each degree increase above 52°F in the outside temperature.
The true statements about the graph are 1, 2, and 3.
What is a line of fit?A line of the best fit is a straight line that reduces the distance between it and some data. The line of best fit is used to represent a relationship in a scatter plot with numerous data points.
Given:
The temperature above 52° F affects the sale of hot chocolate at her shop,
As you can see from the graph, the y-axis shows the orders for hot chocolate on a day when the outside temperature was 52° F.
From the graph,
at x = 0, y = 160
Thus, Sheryl gets 160 orders for hot chocolate on a day when the outside temperature is 52° F
At x = 0, y = 160 and at x = 16, y = 128
So the difference is 160-128 / 16 - 0 = 2
Thus, Sheryl gets 2 fewer orders for each degree increase above 52° F in the outside temperature.
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Rory is an independent filmmaker and has great ideas for several movies. For each short film or full-length movie she makes, she has to raise money. She figures that it will take 7 months to raise money for each short film and 7 months to raise money for each full-length movie. She has given herself at most 49 months to raise money for the films before she abandons the project.
1
While Rory is raising money, she also has to write the scripts. It will take her of a month to write each short film script and 2 months to write each full-length movie script. She is committed to spending at most 13 months on writing scripts.
a. What is the system of inequalities that models this scenario?
b. What is the graph of the solutions?
continued
The required inequality expressions are 7x + 7y ≤ 49 and x + 2y ≤ 13 an the graph has been plotted with clarity.
How to write an expression for linear inequality?The expression for inequality can be written by taking the suitable sign for the given problem and then relating the variable part to the constant terms.
(a) Suppose the number of short film and full movie are x and y respectively.
Then, the inequality expression for the given case can be written as follows,
The inequality for money raised is,
7x + 7y ≤ 49
And, the inequality for script writing is,
x + 2y ≤ 13
(b) These inequalities can be graphed as follows,
Hence, for the given case the required system of inequalities is 7x + 7y ≤ 49 and x + 2y ≤ 13 and the graph for the solution is drawn properly.
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A quiz team of 5 children is to be selected from a class of 25 children. There are 15 girls and 10 boys in the class. (a) How many teams made up of 3 girls and 2 boys can be selected? (b) How many teams can be selected with at least 3 girls?
Using combination we get 20475 teams can made up of 3 girls and 2 boys and 37128 teams can be selected with at least 3 girls.
Why does combination mean?Combinations are mathematical operations that count the number of potential configurations for a set of elements when the order of the selection is irrelevant. You can choose the components of combos in any order. Permutations and combinations can be mixed up.Since this is just a team of 5 children,
we have no of girls = 15
No of boys = 10
teams made up of 3 girls and 2 boys
girls out of 3 girls can be selected in ¹⁵C₃ ways and
2 boys out of 10 boys can be selected in ¹⁰C₂ ways.
[tex]\implies[/tex] ¹⁵C₃ X ¹⁰C₂
= 455*45 =20475 Ways
Teams can be selected with at least 3 girls
It is the probability of at least 3 girls which means P(3 girls,2 boys) or P(4 girls,1 boy) or P(5 girls, 0 boys)
[tex]\implies[/tex] ¹⁵C₃ X ¹⁰C₂ + ¹⁵C₄ X ¹⁰C₁ + ¹⁵C₅ X ¹⁰C₀
= 455 * 45 +1365*10+3003*1
=20475 +13650 +3003
=37128 ways
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The velocity of a car was read from its speedometer at 10-second intervals and recorded in the table. Use the Midpoint Rule to estimate the distance traveled by the car. (Use the Midpoint Rule with 5 subintervals. (Round your answer to one decimal place.)
_________ mi
t(s) v(mi/h) t(s) v(mi/h)
0 0 60 56
10 34 70 55
20 52 80 50
30 54 90 49
40 55 100 45
50 51
The estimated distance traveled by the car is 73.5 mi.
To use the Midpoint Rule, we need to divide the time interval into subintervals and evaluate the average velocity over each subinterval.
The time interval is from 0 seconds to 100 seconds, so we will divide this interval into 5 subintervals of length 20 seconds each. The midpoint of each subinterval is the average time, which we can use to estimate the average velocity over that subinterval.
The midpoints of the subintervals are:
Subinterval 1: (0 + 20)/2 = 10 secondsSubinterval 2: (20 + 40)/2 = 30 secondsSubinterval 3: (40 + 60)/2 = 50 secondsSubinterval 4: (60 + 80)/2 = 70 secondsSubinterval 5: (80 + 100)/2 = 90 secondsUsing the data from the table, we can calculate the average velocity over each subinterval:
Subinterval 1: (0 + 34)/2 = 17 mi/hSubinterval 2: (52 + 54)/2 = 53 mi/hSubinterval 3: (55 + 51)/2 = 53 mi/hSubinterval 4: (50 + 49)/2 = 49.5 mi/hSubinterval 5: (49 + 45)/2 = 47 mi/hWe can now use the Midpoint Rule to estimate the distance traveled by the car:
distance = (20/6) * (17 + 53 + 53 + 49.5 + 47)
= (20/6) * 220.5
= <<20/6*220.5=73.5>>73.5 mi
So, the estimated distance traveled by the car is 73.5 mi.
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Emma wrote 8.4x+6.3x+12.6 as an equivalent expression for 4.2(2x+1.5x+3). She said that her equivalent expression is simplified. Do you agree? Explain.
Yes, I agree that this equivalent expression has been simplified.
The solutions to these issues are as follows
Reorder and gather like terms: (8.4x +6.3 x) +12.6
Collect coefficients for the like terms: (8.4 +6.3) × x +12.6
Calculate the sum or difference: 14.7 x +12.6
Answer : 14.7 x + 12.6
What exactly is equivalent expression, and how do we recognize it?
Equivalent expressions are expressions that work the same even though they look different. If two algebraic expressions are equivalent, then the two expressions have the same value when we plug in the same value(s) for the variable(s).
Distribute any coefficients.Combine any like terms on each side of the equation.Arrange the terms in the same order, usually x-term before constants.If all of the terms in the two expressions are identical, then the two expressions are equivalent.To learn more about coefficients refer to:
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An account executive receives a base salary plus a commission. On $50,000 in monthly sales, the account executive receives $6500. On $60,000 in monthly sales, the account executive receives $6800.
(a) Determine a linear function that will yield the compensation y of the sales executive for a given amount of monthly sales x.
(b) Use this model to determine the account executive's compensation for $90,000 in monthly sales.
$
A) A linear function that will yield the compensation y of the sales executive for a given amount of monthly sales x is;
B) The account executive's compensation for $90,000 in monthly sales is; $300
How to model a linear equation?
The general equation of a line in slope intercept form is;
y = mx + b
where;
m is slope
b is y-intercept
x and y are provided in the problem: x = monthly sales and y = compensation
We will have to find m (Slope), which represents the commission percentage, and b (y-intercept), which represents the the base salary.
x₁ y₁ x₂ y₂
(50000, 6500) (60000, 6800)
Slope;
m = (y₂ - y₁) / (x₂ - x₁)
m = (6800 - 6500) / (60000 - 50000)
m = 300/10000
m = 3/100 = 0.03 = 3%
To find b, we have to take one of our ordered pairs [ (50000, 6500) or (60000, 6800) ] and plug it into the equation, y=mx+b.
Let's use (60000, 6800)
6800 = 0.03(60000) + b
6800 = 1200 + b
b = 6800 - 1200
b = $5600
The base salary = $5600
The equation is y = 0.03x + 5600
b) For $90000 in monthly sales, the executive's compensation is;
y = 0.03(90000) + 5600
y = 2700 + 5600
y = $300
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if the circumference of a circle is 5cm, then the diameter is ____cm.
a. 5 cm
b. 31.4 cm
c. 15.7 cm
d. 1.57 cm
An investor decides to invest some cash in an account paying12%
annual interest, and to put the rest in a stock fund that ends up earning 8%over the course of a year. The investor puts $1500more in the first account than in the stock fund, and at the end of the year finds the total interest from the two investments was $1880. How much money was invested at each of the two rates? Round to the nearest integer.
Let A be the amount of money invested at a 12% annual interest rate and B be the amount of money invested at a 8% annual interest rate.
We know that A = B + 1500, and the total interest earned by the two investments is $1880.
The interest earned by the investment at a 12% annual interest rate is 0.12 * A = 0.12A
The interest earned by the investment at a 8% annual interest rate is 0.08 * B = 0.08B
Therefore, we can write the following equation to represent the situation:
0.12A + 0.08B = 1880
Since A = B + 1500, we can substitute this into the equation to get:
0.12(B + 1500) + 0.08B = 1880
Solving for B, we get:
0.12B + 180 + 0.08B = 1880
Combining like terms, we get:
0.2B + 180 = 1880
Subtracting 180 from both sides, we get:
0.2B = 1700
Dividing both sides by 0.2, we get:
B = 8500
Since A = B + 1500, we can substitute this value into the equation to find the value of A:
A = 8500 + 1500 = 10000
Therefore, the investor invested $8,500 at an 8% annual interest rate and $10,000 at a 12% annual interest rate.
Answer:
Account A (12%)= $10,000
Account B (8% stock fund) = $8,500
Step-by-step explanation:
Annual Interest Formula
[tex]\large \text{$ \sf I=P\left(1+r\right)^{t} -P$}[/tex]
where:
I = Interest.P = Principal amount.r = Interest rate (in decimal form).t = Time (in years).Account A:
P = x + 1500r = 12% = 0.12t = 1 year[tex]\implies \sf Interest=(x+1500) (1+0.12)^1-(x+1500)[/tex]
[tex]\implies \sf Interest=(x+1500) (1+0.12)-(x+1500)[/tex]
[tex]\implies \sf Interest=(x+1500)+0.12(x+1500)-(x+1500)[/tex]
[tex]\implies \sf Interest=0.12(x+1500)[/tex]
[tex]\implies \sf Interest=0.12x+180[/tex]
Account B (stock fund):
P = xr = 8% = 0.08t = 1 year[tex]\implies \sf Interest=x (1+0.08)^1-x[/tex]
[tex]\implies \sf Interest=x (1+0.08)-x[/tex]
[tex]\implies \sf Interest=x+0.08x-x[/tex]
[tex]\implies \sf Interest=0.08x[/tex]
If the total interest from the two investments was $1880:
[tex]\implies \sf 0.12x+180+0.08x=1880[/tex]
[tex]\implies \sf 0.2x+180=1880[/tex]
[tex]\implies \sf 0.2x=1700[/tex]
[tex]\implies \sf x=8500[/tex]
Therefore, the money invested in each of the two accounts is:
Account A = $8,500 + $1,500 = $10,000Account B = $8,500Given the graph of f(x), determine the range of f−1(x).
Rational function with one piece decreasing from the left in quadrant 2 asymptotic to the line y equals 3 and passing through the point 0 comma 2 and asymptotic to the line x equals 2 and another piece decreasing from the left in quadrant 1 asymptotic to the line x equals 2 passing through the point 4 comma 4 asymptotic to the line y equals 3.
ℝ
(2, ∞)
(−∞, 2) ∪ (2, ∞)
(−∞, 3) ∪ (3, ∞)
Given the graph of f(x), the range of f⁻¹(x) is; (−∞, 2) ∪ (2, ∞)
How to find the range of the function?The range of a function is defined as the set that is composed by all the output values on a function. Thus, on the graph, the range of a function is composed by the values of y of the function.
For the inverse function, the input and the output are exchanged, and as such given the graph of a function, we can say that the range of the inverse function is given by the domain of the initial function, which is, the values of x of the graph of the original function.
From the description of the rational function, we are told that the asymptote is given as: x = 2
Thus, the domain of the graphed function, would be the same as the range of the inverse function, will be given by the interval:
(−∞, 2) ∪ (2, ∞).
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Complete the following.
(a) Consider the following statement.
The water temperature is 70.
Check all the statements below that are negations of this statement.
The water temperature is less than 80.
It is not the case that the water temperature is 70.
The water temperature is not 70.
(b) Consider the following conditional statement.
If the dress is red, then Keisha likes the dress.
What is the conclusion of this statement?
It is not true that Keisha owns a dress.
Keisha likes the dress.
The dress is red.
Keisha loves the dress.
The dress likes Keisha.
a) The option that is a negation of the given statement is; The water temperature is not 70.
b) The conclusion of the given conditional statement is; Keisha likes the dress.
How to Interpret Conditional Statements?In mathematics, a conditional statement is defined as a statement that can be written in the form “If P then Q,” where P and Q are sentences.
a) We are given the statement as;
The water temperature is 70.
Now, negation is a false/opposite form of the given statement and as such among the options, the only one that is a negation is that
"The water temperature is not 70."
b) We are given the conditional statement as;
If the dress is red, then Keisha likes the dress.
Thus, the conclusion of this is that likes the dress.
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What values of v and w make Δ P Q R ≅ Δ K J I ?
v=_____
w=_____
The values of v and w that will make ΔPQR congruent to ΔKJI are:
v = 12; w = 20.
What are Congruent Triangles?Based on the CPCTC, every corresponding sides and corresponding angles of two congruent triangles are equal to each other.
Therefore, given that ΔPQR ≅ ΔKJI
PR = KI, and QR = JI
Substitute the values:
5v = 6v - 12
5v - 6v = -12
-v = -12
v = 12
w - 1 = 3w - 41
w - 3w = 1 - 41
-2w = -40
w = -40/-2
w = 20
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Question 1-28
What is the range of possible values for x?
B
A
8.2
11.7
(11x-4)
D
84°
C
The range of possible values for x is 4/11 < x < 8
How to determine the range of possible values for x?From the question, we have the following parameters that can be used in our computation:
Triangles = 2
On these triangles, we have the following angles
Angle 1 = 11x - 4
Angle 2 = 84
Angle 1 cannot exceed angle 2
So, we have
11x - 4 < 84
Add 4 to both sides
11x < 88
Divide
x < 8
Angle 1 cannot exceed be negative or 0
So, we have
11x - 4 > 0
Add 4 to both sides
11x > 4
Divide
x > 4/11
Hence, the range is 4/11 < x < 8
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Solve:(6x^2+5x+1)÷(x+2)
Answer:
6x+6x2+3
Step-by-step explanation:
6x2+5x+1+x+2
Combine 5x and x to get 6x.
6x2+6x+1+2
Add 1 and 2 to get 3.
6x2+6x+3
-16(d + 1) = -21 solve for d
Step-by-step explanation:
To solve the equation -16(d + 1) = -21 for d, we can first use the distributive property to expand the expression -16(d + 1):
-16(d + 1) = -16d - 16
Then, we can set the two sides of the equation equal to each other and solve for d:
-16d - 16 = -21
-16d = -5
d = 5/16
Therefore, the value of d that satisfies the equation -16(d + 1) = -21 is d = 5/16.
Write an equation for the absolute value function below:
The absolute value equation represented in the graph is
|x + 3|
What is absolute value?Without taking direction into account, absolute value describes how far away from zero a certain number is on the number line.
A number can never have a negative absolute value.
How to write the absolute value equationThe equation is written in the form below
|x|
The a transformation which involves a translation of 3 units to the left took place. the transformation rule for 3 units to the left is addition of 3, hence we have the equation
|x + 3|
We can therefore conclude that the absolute value equation of the function is |x + 3|
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what is a acute angle
Answer: Measure less 90 degrees
Step-by-step explanation: Less than 90 degrees measure acute angle. 90 degrees in right corner. Obtuse corners are larger than 90 degrees
Jen drew a scale drawing of a summer camp
Answer:
This is a statement. Not a question.
Step-by-step explanation:
A dance teacher divides 5 dance classes into 6 equal groups. Each dance class has 18 students. How many dance students are in each group
First, multiply [tex]18*5 = 90[/tex]
Then, divide [tex]90 / 6 = 15[/tex]
Therefore, each group has 15 students.
if s is the part of the sphere that lies above the cone find the following: 1. s can be parametrized by a vector equation
If surface , S is the part of the sphere, x² + y²+ z² =1 ,lies above the cone, x² + y² = z
a)| ⃗rᵩ × ⃗rθ| = ⟨sin²φ cosθ,sin²φsinθ, sinφcosφ⟩
b)∫∫ z² ds = ₀∫ˣ₀∫ʸcos²φ dφdθ,
S
where x = 2π and y = π/4
What is Parametrizing Surfaces?
A surface in space given in Cartesian coordinates as f(x,y,z) = 0, can be parametrized as a vector function with two parameters,
r(u,v)= ⟨r₁(u,v) , r₂(u,v) , r₃(u,v)⟩ , (u,v)∈R²
We have, S is a part of Sphere , x² + y²+ z² =1 , lies above the cone, x² + y² = z and surface S parametrized by following vector equation ,
r(θ, φ) = ⟨sinφ cosθ , sinφsinθ,cosφ⟩ --(1)
⃗rᵩ = ∂r/∂φ =⟨cos φ cosθ,cos φ sinθ,-sin φ⟩
⃗rθ= ∂r/∂θ=⟨- sinφ sinθ , sinφ cosθ ,0⟩
| ⃗rᵩ × ⃗rθ| =| i j k |
|cosφcosθ cosφ sinθ -sinφ |
|- sinφsinθ sinφ cosθ 0 |
= i( 0 + sinφ cosθsinφ) -j(0- (- sinφsinθ)(-sinφ)) + k(sinφ cosθ cosφcosθ - (-sinφ sinθ ) cosφ sinθ)
= (sin²φcosθ )i + (sin²φsinθ)j + (sinφcosφ(sin²θ+cos²θ))k
= sin²φcosθ )i + (sin²φsinθ)j +(sinφcosφ)k
| ⃗rᵩ× ⃗rθ|=⟨sin²φcosθ ,sin²φsinθ,sinφcosφ⟩
b) from part (a) we get,
x = sinφ cosθ
y = sinφsinθ
z = cosφ
are spherical coordinates of S where, 0≤φ≤π/4 and 0≤ θ≤ 2π.
so, ∫∫ z² ds = ₀∫ˣ₀∫ʸcos²φ dφ dθ ,
S
where x = 2π and y = π/4
So,Surface integeral is equals to ₀∫ˣ₀∫ʸcos²φ dφ dθ.
To learn more about Parametrizing Surfaces, refer:
https://brainly.com/question/29772771
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Complete question:
If Sis the part of the sphere x2 + y2 + x2 = 1 that lies above the cone z= x2 + y2, find the following: a)S can be parametrized by a vector equation
r(θ, φ) = ⟨sinφ cosθ , sinφsinθ , cosφ ⟩ then a)| ⃗rᵩ × ⃗rθ| = ?
b) ∫∫ z² ds = ?
S
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Use the properties to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume the variable is positive)
1. ln((xy)^5)
2. ln (seventh root(t))
3. ln (√x²/y^5)
Properties of log:
[tex]log\ ab=log\ a + log\ b[/tex][tex]log\ a/b =log\ a-log\ b[/tex][tex]log\ a^b=b\ log\ a[/tex]Evaluate given using the properties above:
Q1
[tex]ln((xy)^5) = 5ln(xy) = 5(ln x + ln y) = 5\ ln\ x + 5\ ln\ y[/tex]Q2
[tex]ln(\sqrt[7]{t} ) = ln(t^{1/7})=1/7\ ln\ t[/tex]Q3
[tex]ln(\sqrt{x^2}/y^5)=ln(x/y^5)= ln\ x - ln\ y^5 = ln\ x - 5\ ln\ y[/tex]Answer:
[tex]\textsf{1.} \quad 5 \ln x + 5 \ln y[/tex]
[tex]\textsf{2.} \quad \dfrac{1}{7} \ln t[/tex]
[tex]\textsf{3.} \quad \ln x - \dfrac{5}{2}\ln y \;\; \;\;\textsf{or} \;\;\;\; \ln x - 5\ln y[/tex]
Step-by-step explanation:
[tex]\boxed{\begin{minipage}{6 cm}\underline{Natural log laws}\\\\Product law: \;$\ln xy=\ln x + \ln y$\\\\Quotient law: $\ln \left(\dfrac{x}{y}\right) = \ln x - \ln y$\\\\Power law: \;\;\;\;$\ln x^n=n \ln x$\\\end{minipage}}[/tex]
Question 1Apply the power law followed by the product law:
[tex]\begin{aligned}\ln (xy)^5 & = 5 \ln (xy)\\ & = 5\left( \ln x + \ln y \right) \\ & = 5 \ln x + 5 \ln y\end{aligned}[/tex]
Question 2[tex]\textsf{Apply the exponent rule} \quad \sqrt[n]{a}=a^{\frac{1}{n}}[/tex]
then apply the power law:
[tex]\begin{aligned}\ln \sqrt[7]{t} & = \ln t^{\frac{1}{7}} \\ & = \dfrac{1}{7} \ln t\end{aligned}[/tex]
Question 3It is not completely clear where the square root sign begins and ends, so I have provided answers for both permutations:
[tex]\begin{aligned}\ln \left(\sqrt{\dfrac{x^2}{y^5}}\right) & = \ln \left(\dfrac{\sqrt{x^2}}{\sqrt{y^5}}\right) \\\\& = \ln \left(\dfrac{x}{y^{\frac{5}{2}}}\right) \\\\&=\ln x - \ln y^{\frac{5}{2}}\\\\&=\ln x - \dfrac{5}{2}\ln y\end{aligned}[/tex]
[tex]\begin{aligned}\ln \left(\dfrac{\sqrt{x^2}}{y^5}\right)& = \ln \left(\dfrac{x}{y^5}\right) \\\\&=\ln x - \ln y^5\\\\&=\ln x - 5\ln y\end{aligned}[/tex]
Solve the proportion using cross products. Round to the nearest hundredth if necessary.
12Miles/27hours=4miles/Xhours
A. 28
B. 12
C. 9
D. 18
Answer:
C) x = 9
Step-by-step explanation:
[tex]\frac{12}{27}[/tex] = [tex]\frac{4}{x}[/tex]
[tex]\frac{12}{27}[/tex] ÷ [tex]\frac{3}{3}[/tex] = [tex]\frac{4}{9}[/tex] This is the fraction simplified
[tex]\frac{4}{9}[/tex] = [tex]\frac{4}{x}[/tex] Simplified, I think that it is easier to see that x = 9
Convergence or Divergence? Prove using a test.
when conducting a study comparing more than two groups of parametric (normally distributed) data, which of the following statistical tests should be used? g
Answer:
.
Step-by-step explanation:
sorry i hit my daily limit and i need more answers
The y-intercept can be described all except for one of them. Identify the statement that does not always represent the y-intercept.