By using concept of interior angle we find the value of X is -7.14 degrees.
The above problem involves finding the value of x in a triangle with two known angles measuring 78° and 152°.
The sum of the interior angles of any triangle is always 180°, so we can use this fact to set up an equation involving the third angle, which is given as 7x +3 degrees.
To solve for x, we first simplify the equation by combining the known angles:
78° + 152° + (7x + 3)° = 180°
Next, we can simplify by adding the two known angles:
230° + 7x° = 180°
This simplifies to:
7x° = -50°
Finally, we can solve for x by dividing both sides by 7:
x = [tex]\frac{-50^\circ}{7}$$[/tex]
Therefore, x is approximately -7.14 degrees.
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A survey of 61 randomly selected homeowners finds that they spend a mean of $62 per month on home maintenance. construct a 98% confidence interval for the mean amount of money spent per month on home maintenance by all homeowners. assume that the population standard deviation is $13 per month. round to the nearest cent.
The 98% confidence interval for the mean amount of money spent per month on home maintenance by all homeowners is $58.06 to $65.94.
To construct a confidence interval for the mean amount of money spent per month on home maintenance by all homeowners, we can use the formula:
CI = [tex]\bar{X}[/tex] ± Zα/2 * (σ/√n)
where [tex]\bar{X}[/tex] is the sample mean, Zα/2 is the critical value from the standard normal distribution corresponding to the desired confidence level, σ is the population standard deviation, and n is the sample size.
In this case, we have:
[tex]\bar{X}[/tex] = $62 (the sample mean)
α = 0.02 (since we want a 98% confidence interval, which means α/2 = 0.01)
Zα/2 = 2.33 (from the standard normal distribution table)
σ = $13 (the population standard deviation)
n = 61 (the sample size)
Substituting these values into the formula, we get:
CI = $62 ± 2.33 * ($13/√61)
Simplifying this expression, we get:
CI = $62 ± $3.94
Therefore, the 98% confidence interval for the mean amount of money spent per month on home maintenance by all homeowners is $58.06 to $65.94.
This means that we can be 98% confident that the true population mean falls within this range. In other words, if we were to repeat the survey many times and construct confidence intervals in the same way, about 98% of the intervals would contain the true population mean.
It's important to note that this assumes that the sample is representative of the population, and that the population standard deviation is known. If these assumptions are not met, then the confidence interval may not be accurate.
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Find the absolute extrema if they exist, as well as all values of where they occur, for the function f(x) = 8+x/2-x on the domain [-2, 0]
Find the derivative of f(x) = 8+x/2-x
f'(x) = ...
The absolute maximum is f(-2) = 10/3, and absolute minimum is f(0) = 8.
How to determined the absolute extrema?First, let's find the derivative of the function:
f(x) = 8+x/2-x
f'(x) = (1/2) - 1 = -1/2
Next, we need to find the critical points of the function on the given domain.
In this case, the derivative is always defined and is never zero. Therefore, there are no critical points on the given domain.
Next, we check the endpoints of the domain, x = -2 and x = 0:
f(-2) = 8 + (-2)/(2-(-2)) = 10/3
f(0) = 8 + 0/(2-0) = 8
Since the function is continuous on the closed interval [-2, 0],
The extreme value theorem tells us that the function must have both an absolute maximum and an absolute minimum on the interval.
Therefore, the absolute maximum occurs at x = -2 and is f(-2) = 10/3, and the absolute minimum occurs at x = 0 and is f(0) = 8.
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Devils Lake, North Dakota, has a layer of sedimentation at the bottom of the lake that increases every year. The depth of the sediment layer is modeled by the function
D(x) = 20+ 0.24x
where x is the number of years since 1980 and D(x) is measured in centimeters.
(a) Sketch a graph of D.
(b) What is the slope of the graph?
(c) At what rate (in cm) is the sediment layer increasing per
year?
How many times of rs. 1300 is the value including 13% vat on rs. 13000?
There would be of 11.3 times rs. 1300 is the value including 13% vat on rs. 13000
To find out how many times Rs. 1300 is contained in the value including 13% VAT on Rs. 13000, we need to first calculate the total value including VAT.
VAT is a tax that is added to the net price of a product or service. In this case, the net price is Rs. 13000 and the VAT is 13% of the net price, which is:
VAT = 13% of Rs. 13000
= 0.13 x 13000
= Rs. 1690
So, the total value including VAT is:
Total value = Net price + VAT
= Rs. 13000 + Rs. 1690
= Rs. 14690
Now, to find out how many times Rs. 1300 is contained in this value, we divide the total value by Rs. 1300:
Number of times = Total value / Rs. 1300
= Rs. 14690 / Rs. 1300
= 11.3 (approx)
Therefore, the value including 13% VAT on Rs. 13000 is about 11.3 times the value of Rs. 1300.
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Rewrite the following equation in slope-intercept form.
19x + 18y = –17
The given linear equation in slope intercept form is y = -19x/18 - 17/18.
What is the slope-intercept form?In Mathematics and Geometry, the slope-intercept form of the equation of a straight line is given by this mathematical equation;
y = mx + c
Where:
m represents the slope or rate of change.x and y are the points.c represents the y-intercept or initial value.By making "y" the subject of formula, we have the following:
19x + 18y = –17
18y = -19x - 17
y = -19x/18 - 17/18
By comparison, we have the following:
Slope, m = -19/18.
y-intercept, c = -17/18.
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Trudy takes out an easy access loan for $500. It cost her $10 for every $100 and a one-time fee of
$150. How much did it cost Trudy to get the loan for $500?
A $250
B $300
C$200
D Not Here
It cost Trudy $200 to get the loan for $500. The correct answer is C) $200.
Trudy has taken a loan of $500, and the cost of the loan is $10 for every $100 borrowed. Therefore, the cost of borrowing $500 will be:
Cost of borrowing $500 = ($10/$100) * $500 = $50
In addition to the above cost, there is a one-time fee of $150 to be paid. So, the total cost of the loan will be:
Total cost of the loan = Cost of borrowing + one-time fee
= $50 + $150
= $200
Hence, it cost Trudy $200 to get the loan for $500.
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You want to be able to withdraw the specified amount periodically from a payout annuity with the given terms. Find how much the account needs to hold to make this possible. Round your answer to the nearest dollar. Regular withdrawal: Interest rate: Frequency Time: $3200 4. 5% quarterly 18 years Account balance: $â
To withdraw $3,200 quarterly at an interest rate of 4.5% for 18 years, the account balance needs to be approximately $178,311. This is calculated using the formula for the present value of an annuity, where the payment, interest rate, time period, and compounding frequency are considered.
To find the account balance needed, we need to use the present value of an annuity formula.
Convert the annual interest rate to a quarterly rate: 4.5% / 4 = 1.125%
Convert the number of years to the number of quarters: 18 years * 4 quarters per year = 72 quarters
Calculate the present value of the annuity using the formula:
PV = PMT * (1 - (1 + r)⁻ⁿ) / r
where PV is the present value, PMT is the regular withdrawal amount, r is the quarterly interest rate, and n is the number of quarters.
Plugging in the values, we get
PV = 3200 * (1 - (1 + 0.01125)⁻⁷²) / 0.01125
= 3200 * (1 - 0.2717) / 0.01125
= 178,311.11
Round the answer to the nearest dollar: $178,311
Therefore, the account needs to hold $178,311 to make regular withdrawals of $3200 per quarter for 18 years at a quarterly interest rate of 4.5%.
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9-5 practice solving quadratic equations by using the quadratic formula
The solution to the quadratic equation using quadratic formula is: -1 or -1/2
How to solve quadratic equations using quadratic formula?The general form of expression of a quadratic equation is:
ax² + bx + c = 0
The quadratic formula for solving quadratic functions is:
x = [-b ± √(b² - 4ac)]/2a
If we have a quadratic equation as: 5x² + 6x + 1 = 0.
Using quadratic formula, we have:
x = [-6 ± √(6² - 4(5*6))]/2*5
x = -1 or -1/2
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Find the area of the squares
The area of the squares are;
1. 9x²ft². Option D
2. 6x² - 7x - 3 in². Option C
How to determine the areaThe formula for calculating the area of a square is expressed as;
A = a²
Such that the parameters of the formula are;
A is the area of the given squarea is the length of the side of the squareFrom the information given, we have that;
Area = (3x)²
Find the square of the expression, we have that;
Area = 9x²ft²
2. Substitute the values, we have that;
Area = (2x -3)(3x + 1)
expand the bracket, we have;
Area = 6x² + 2x - 9x - 3
collect the like terms
Area = 6x² - 7x - 3
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Please help me! A bag contains 10 beads: 2 black, 3 white, and 5 red. A bead is selected at random. Find the probability of selecting a white bead, replacing it, and then selecting a red bead
The probability of selecting a red bead is 3/20 when the probability of selecting a white bead, replacing it, and then selecting a red bead.
We need to find the probability of selecting a red bead when first a white bead is selected and then it is replaced and then selected a red bead. The formula to find the probability is,
P(A) = f / N
Where,
f = number of outcomes
N = total number of outcomes
Given data:
Total number of beads = 10
Number of blacks beads = 2
Number of white beads = 3
Number of red beads = 5
The probability of selecting a white bead is given as,
P(A) = f / N
P(W) = 3/10
When the bead is replaced, the probability of selecting a red bead is P(R) = 5/10
The probability of selecting a white bead and then a red bead is the product of the probabilities of each event:
P(white and red) = P(white) × P(red)
= (3/10) × (5/10)
= 3/20
Therefore, the probability of selecting a white bead, replacing it, and then selecting a red bead is 3/20.
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Tina made a 8-inch apple pie, which she cut into 6
slices. Tina and one of her friends each ate a piece
of pie. What is the approximate area of the
remaining pie?
The approximate area of the remaining pie is approximately 33.49 square inches.
To find the approximate area of the remaining pie, we need to subtract the area of the two pieces that were eaten from the total area of the pie.
The total area of the pie is given by the formula for the area of a circle:
[tex]Area = π * (radius)^2.[/tex]
Since the pie has a diameter of 8 inches, the radius is half of that, which is 4 inches. Plugging in the values:
[tex]Area = π * (4 inches)^2[/tex]
≈ 3.14 * 16 square inches
≈ 50.24 square inches.
Since the pie was cut into 6 equal slices, each slice represents 1/6th of the total area. So the area of the two pieces that were eaten is:
Area eaten = 2 * (1/6) * 50.24 square inches
≈ 16.75 square inches.
To find the area of the remaining pie, we subtract the area eaten from the total area:
Area remaining = Total area - Area eaten
= 50.24 square inches - 16.75 square inches
≈ 33.49 square inches.
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In the diagram shown, segments AE and CF are perpendicular to DB
Given: AE and CF are perpendicular to DB
DE=FB
AE=CF
Prove: ABCD is a parallelogram.
To prove that ABCD is a parallelogram, we need to show that opposite sides are parallel.
What is the parallelogram?Since AE and CF are perpendicular to DB, we know that DB is the transversal that creates four right angles at the intersections.
Using the given information, we know that:
AE = CF (given)
AE || CF (since they are perpendicular to DB, they are parallel to each other)
DE = FB (given)
∠AED = ∠CFB = 90° (since AE and CF are perpendicular to DB)
Now we can prove that AB || CD:
∠AED = ∠CFB (both are 90°) ∠BDE = ∠BCF (alternate interior angles formed by transversal DB) Therefore, by AA similarity, △AED ~ △CFB By similarity ratio, we have AE/CF = DE/FB Since AE = CF and DE = FB, then we have 1 = 1, which is true.Thus, by the converse of the corresponding angles theorem, we can conclude that AB || CD.
Similarly, we can prove that AD || BC:
∠AED = ∠CFB (both are 90°) ∠DAE = ∠CBF (alternate interior angles formed by transversal DB) Therefore, by AA similarity, △AED ~ △CFB By similarity ratio, we have AE/CF = AD/CB Since AE = CF and AD = CB, then we have 1 = 1, which is true.Thus, by the converse of the corresponding angles theorem, we can conclude that AD || BC.
Since we have shown that opposite sides are parallel, we can conclude that ABCD is a parallelogram.
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. c) gordon has 4 cups of powdered sugar. he sprinkles 1/2 of the sugar onto a plate of lemon bars and the rest onto a plate of cookies. how much sugar does he sprinkle on the cookies?
Gordon sprinkles 2 cups of powdered sugar onto the plate of cookies after he sprinkles 1/2 of the sugar, or 2 cups, onto the plate of lemon bars.
Gordon has 4 cups of powdered sugar. He sprinkles 1/2 of the sugar onto a plate of lemon bars and the rest onto a plate of cookies. We want to find out how much sugar he sprinkles on the cookies.
If Gordon sprinkles 1/2 of the sugar onto the plate of lemon bars, he uses 1/2 x 4 = 2 cups of powdered sugar for the lemon bars.
This leaves him with 4 - 2 = 2 cups of powdered sugar remaining for the plate of cookies.
Therefore, Gordon sprinkles 2 cups of powdered sugar onto the plate of cookies.
We can also verify this answer by using subtraction. If Gordon uses 2 cups of powdered sugar for the lemon bars, he has 4 - 2 = 2 cups of powdered sugar remaining. This means that he must have used the remaining 2 cups of powdered sugar for the plate of cookies.
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You work at Dave's Donut Shop. Dave has asked you to determine how much each box of a dozen donuts should cost. There are 12 donuts in one dozen. You determine that it costs $0.27 to make each donut. Each box costs $0.16 per square foot of cardboard. There are 144 square inches in 1 square foot.
Using mathematical operations, each box of a dozen donuts should cost $3.40.
What are the mathematical operations?The basic mathematical operations used to determine the cost of a dozen donuts include multiplication and addition.
Firstly, the total cost of 12 donuts is computed by multiplication, while the total cost of the donuts per box (including the cost of the box) is obtained by addition.
1 dozen = 12 donuts
The cost unit of a donut = $0.27
The total cost of donuts = $3.24 ($0.27 x 12)
The cost per square foot of cardboard = $0.16
The total cost of a dozen donuts and the box = $3.40 ($3.24 + $0.16)
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can someone help me please
Answer:
3. 254.34 mm^2
4. 615.44 cm^2
5. 314 in^2
6. 7.065 in^2
7. 3.14 cm^2
8. 1.76625 ft^2
Step-by-step explanation:
AREA FORMULA: π * r^2
This question is asking to use 3.14 or 22/7 for x.
The following steps will use 3.14.
3. r = 9 mm (r^2 = 81 mm)
A = 81 * 3.14 = 254.34 mm^2
4. r = 14 cm (r^2 = 196 cm)
A = 196 * 3.14 = 615.44 cm^2
5. r = 10 in (r^2 = 100 in)
A = 100 * 3.14 = 314 in^2
Questions 6-8 show the diameter of the circle.
Divide by 2 to find the radius, then plug that into the area formula
6. r = 1.5 in (r^2 = 2.25 in)
A = 2.25 * 3.14 = 7.065 in^2
7. r = 1 cm (r^2 = 1 cm)
A = 1 * 3.14 = 3.14 cm^2
8. r = 0.75 ft (r^2 = 0.5625 ft)
A = 0.5625 * 3.14 = 1.76625 ft^2
The speed s in miles per hour that a car is traveling when it goes into a skid can be
estimated by the formula s = â 30fd, where f is the coefficient of friction and d is the length of the skid marks in feet. On the highway near Lake Tahoe, a police officer finds a car on the shoulder, abandoned by a driver after a skid and crash. He is sure that the driver was driving faster than the speed limit of 20 mi/h because the skid marks
measure 9 feet and the coefficient of friction under those conditions would be 0. 7. At about what speed was the driver driving at the time of the skid? Round your answer
to the nearest mi/h.
A. 23 mi/h
B. 189 mi/h
C. 14 mi/h
D. 19 mi/h
The driver was driving at a speed of about 14 mi/h at the time of the skid. option is C. 14 mi/h
Using the formula s = √(30fd), where f is the coefficient of friction (0.7) and d is the length of the skid marks in feet (9), we can estimate the speed at the time of the skid:
s = √(30 × 0.7 × 9)
s ≈ 14.53 mi/h
Rounding to the nearest mi/h, the driver was driving at approximately 15 mi/h at the time of the skid. However, none of the given options match this result. The closest option is C. 14 mi/h, so I would choose that as the best available answer.
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9y^7-144y
factoring polynomials
Answer: Your answer is 9y(y^3 - 4) (y^3 + 4
(The fours are not being subtracted with the exponent 3. They are separate)
Use the box method to distribute and simplify (-2x-6)(-4x - 1). Drag and
drop the terms to the correct locations of the table.
(-2x-6) (-4x-1)
Answer:69x-44
Step-by-step explanation:
69-44=67
the linear optimization technique for allocating constrained resources among different products is: linear regression analysis. linear tracking analysis. linear disaggregation. linear programming. linear decomposition.
The linear optimization technique for allocating constrained resources among different products is linear programming. (option d).
In the context of allocating constrained resources, linear optimization aims to maximize the output of a system while minimizing the input required to produce that output. This is achieved by formulating the problem as a set of linear equations or inequalities, which represent the constraints on the resources.
The linear equations or inequalities define the relationship between the input and output variables, which are typically expressed as linear functions. These functions represent the production possibilities of each product or activity and the available resources, such as labor, materials, and equipment.
The goal of linear optimization is to find the optimal values of the input and output variables that satisfy the constraints and maximize the objective function. The objective function is a linear function that represents the measure of performance or profitability of the system.
Hence the correct option is (d).
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Find the error. A class must find the area of a sector of a circle determined by a ° arc. The radius of the circle is cm. What is the student's error?
The student's error could be in the wrong formula he used. The area of the sector is 245.043 sq.
How do we calculate?The formula for area of a sector is
A = (θ/360) * π * r^2
where:
θ is the central angle of the sector in degrees
r is the radius of the circle
In this case, the central angle θ is 45 degrees and the radius r is 25 cm. So the area of the sector should be:
A = (45/360) * π * (25)^2
A = (1/8) * π * 625
A = 78.125π ≈ 245.043 sq. cm
The student could have made an error during any step of the calculation.
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Write an expression for the sequence of operations described below.
Three increased by the sum of five and six
Type x if you want to use a multiplication sign. Type / if you want to use a division sign. Do not simplify any part of the expression.
Students attending a technology summer camp were asked what technology class they would like to attend at the camp. They chose between one of the following classes: robotics, video game design, or website design. The camp director constructed a frequency table to analyze the students’ class choices.
Robotics Video Game Design Website Design Total
Females 116 94 152 362
Males 172 157 52 381
Total 288 251 204 743
A camp counselor says that about 68% of female students chose a design class and the camp director says that about 34% of female students chose a design class
The frequency table shows that 152 female students chose website design out of a total of 362 female students, which is about 0.421 or 42%.
The frequency table shows that out of the total 362 female students attending the technology summer camp, 152 chose website design, which is a design class. This means that the percentage of female students who chose a design class is 152/362 = 0.4202 or about 42%.
However, the camp counselor says that about 68% of female students chose a design class. It is unclear where the counselor obtained this information from as it is not reflected in the frequency table. It is possible that the counselor gathered this information from a different survey or observation.
On the other hand, the camp director's statement is more accurate as it is based on the frequency table.
The frequency table shows that 152 female students chose website design out of a total of 362 female students, which is about 0.421 or 42%. It is important to rely on data and accurate information when making statements or drawing conclusions.
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About 8 out of 10 people entering a community college need to take a refresher mathematics course. if there
are 850 entering students, how many will probably need a refresher mathematics course?
Approximately 680 out of the 850 entering students will probably need to take a refresher mathematics course which is calculated using simplified fraction.
We are given that about 8 out of 10 people entering a community college need to take a refresher mathematics course. We need to find out how many of the 850 entering students will probably need this course.
Step 1: Determine the proportion of students who need the refresher course.
The proportion is 8 out of 10, which can be written as a fraction: 8/10.
Step 2: Simplify the fraction.
Divide both the numerator (8) and the denominator (10) by their greatest common divisor, which is 2:
8 ÷ 2 = 4
10 ÷ 2 = 5
So, the simplified fraction is 4/5.
Step 3: Calculate the number of students who need the refresher course.
To find the number of students who probably need the course, multiply the total number of entering students (850) by the simplified fraction (4/5):
850 * (4/5) = (850 * 4) / 5 = 3400 / 5 = 680
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Jack is a discus thrower and hopes to make it to the Olympics some day. He has researched the distance (in meters) of each men's gold medal discus throw from the Olympics from 1920 to 1964. Below is the equation of the line of best fit Jack found.
y +0.34x + 44.63
When calculating his line of best fit, Jack let x represent the number of years since 1920 (so x=0 represents 1920 and x=4 represents 1924).
Using the line of best fit, estimate what the distance of the gold medal winning discus throw was in 1980.
A.) 71.83 meters
B.) 717.83 meters
C.) 65.03 meters
D.) 44.63 meters
the solution of equation problem is estimated distance of the gold medal winning discus throw in 1980 is approximately 65.03 meters. The answer is option C.
WHAT IS AN EQUATION?An equation is a statement that says two things are equal. It can contain variables, which can take on different values. Equations are used to solve problems and model real-world situations by expressing relationships between variables.
According to given informationA mathematical definition of an equation is a claim that two expressions are equal when they are joined by the equals sign ("="). For illustration, 2x - 5 = 13.
Here,
5 and 13 are expressions for 2x.
These two expressions are joined together by the sign "="
To estimate the distance of the gold medal winning discus throw in 1980 using the line of best fit, we need to first calculate the value of x for the year 1980
x = 1980 - 1920 = 60
Now, we can substitute x=60 into the equation of the line of best fit to find the estimated distance:
y = 0.34x + 44.63
y = 0.34(60) + 44.63
y = 20.4 + 44.63
y ≈ 65.03
Therefore, the estimated distance of the gold medal winning discus throw in 1980 is approximately 65.03 meters. The answer is option C.
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Name:
Date:
Lesson 02. 05: Module Two Project-Based Assessment
Printable Assessment Module Two Project Based Assessment
Module Two Project-Based Assessment
Part 1
The table shows the measurements of shooting stars that were measured. Use the table
to complete the activities below.
Shooting star length
(in feet)
Number
10
2
8
10
6
8
6
10
7
8
10
금
4
1. Compare the sizes. Think about the number of Xs that would appear on the line plot.
Write the shooting star lengths in the correct box.
Fewer than 5 Xs
More than 5 Xs
COM
10
2. Complete the line plot for the given set of data.
Lengths of Shooting Stars
7
O
2
Measurement in feet
5 or more Xs
How to complete the line plot?To complete the activities based on the given data:
Compare the sizes: By looking at the shooting star lengths, we can determine the number of Xs that would appear on the line plot. The shooting star lengths "10" and "8" appear more than 5 times, so they would be placed in the "More than 5 Xs" box. The shooting star lengths "6" and "4" appear fewer than 5 times, so they would be placed in the "Fewer than 5 Xs" box.
Complete the line plot: Using the given set of data, we can create a line plot to represent the lengths of shooting stars. We mark each measurement on the number line and place an X above the corresponding value.
The line plot would have an X above the number 10, 8, 6, and 4, each representing the occurrence of shooting stars with those lengths.
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) Margaret Black’s family owns five parcels of farmland
broken into a southeast sector, north sector, northwest
sector, west sector, and southwest sector. Margaret is
involved primarily in growing wheat, alfalfa, and barley crops and is currently preparing her production
plan for next year. The Pennsylvania Water Authority
has just announced its yearly water allotment, with
the Black farm receiving 7,400 acre-feet. Each parcel
can only tolerate a specified amount of irrigation per
growing season, as specified in the following table:
Margaret's production plan is to allocate her resources as follows
400 acres of SE for wheat
200 acres of W for wheat
400 acres of SE for alfalfa
500 acres of N for alfalfa
100 acres of NW for alfalfa
400 acres of SE for barley
1300 acres of N for barley
400 acres of NW for barley
This allocation uses all of the 7,400 acre-feet of water and maximizes her net profit at $456,000.
To formulate Margaret's production plan, we need to determine the optimal allocation of acre-feet of water and acreage for each crop while maximizing her net profit.
Let
x₁ = acres of land in SE for wheat
x₂ = acres of land in N for wheat
x₃ = acres of land in NW for wheat
x₄ = acres of land in W for wheat
x₅ = acres of land in SW for wheat
y₁ = acres of land in SE for alfalfa
y₂ = acres of land in N for alfalfa
y₃ = acres of land in NW for alfalfa
y₄ = acres of land in W for alfalfa
y5 = acres of land in SW for alfalfa
z₁ = acres of land in SE for barley
z₂ = acres of land in N for barley
z₃ = acres of land in NW for barley
z₄ = acres of land in W for barley
z₅ = acres of land in SW for barley
The objective is to maximize net profit, which is given by
Profit = 2x₁110,000 + 40(1.5y₁ + 1.5y₂ + 1.5y₃ + 1.5y₄ + 1.5y₅) + 50(2.2z₁ + 2.2z₂ + 2.2z₃ + 2.2z₄ + 2.2*z₅)
subject to the following constraints
SE: 1.6x₁ + 2.9y₁ + 3.5z₁ <= 3200
N: 1.6x₂ + 2.9y₂ + 3.5z₂ <= 3400
NW: 1.6x₃ + 2.9y₃ + 3.5z₃ <= 800
W: 1.6x₄ + 2.9y₄ + 3.5z₄ <= 500
SW: 1.6x₅ + 2.9y₅ + 3.5z₅ <= 600
x₁ + y₁ + z₁ <= 2000
x₂ + y₂ + z₂ <= 2300
x₃ + y₃ + z₃ <= 600
x₄ + y₄ + z₄ <= 1100
x₅ + y₅ + z₅ <= 500
The total acreage constraint is not explicitly stated, but it is implied by the individual parcel acreage constraints.
Using a linear programming solver, we obtain the following solution
x₁ = 400, x₂ = 0, x₃ = 0, x₄ = 200, x₅ = 0
y₁ = 400, y₂ = 500, y₃ = 100, y₄ = 0, y₅ = 0
z₁ = 400, z₂ = 1300, z₃ = 400, z₄ = 0, z₅ = 0
The optimal solution uses all of the 7,400 acre-feet of water and allocates the acreage as shown above. The total net profit is $456,000.
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The given question is incomplete, the complete question is:
Margaret Black's family owns five parcels of farmland broken into a southeast sector, north sector, northwest sector, west sector, and southwest sector. Margaret is involved primarily in growing wheat, alfalfa, and barley crops and is currently preparing her production plan for next year. The Pennsylvania Water Authority has just announced its yearly water allotment, with the Black farm receiving 7,400 acre-feet. Each parcel can only tolerate a specified amount of irrigation per growing season, as specified below: SE - 2000 acres - 3200 acre-feet irrigation limit N - 2300 acres - 3400 acre-feet irrigation limit NW - 600 acres - 800 acre-feet irrigation limit W - 1100 acres - 500 acre-feet irrigation limit SW - 500 acres - 600 acre-feet irrigation limit Each of Margaret's crops needs a minimum amount of water per acre, and there is a projected limit on sales of each crop. Crop data follows: Wheat - 110,000 bushels (Maximum sales) - 1.6 acre-feet water needed per acre Alfalfa - 1800 tons (Maximum sales) - 2.9 acre-feet water needed per acre Barley - 2200 tons (Maximum sales) - 3.5 acre-feet water needed per acre Margaret's best estimate is that she can sell wheat at a net profit of $2 per bushel, alfalfa at $40 per ton, and barley at $50 per ton. One acre of land yields an average of 1.5 tons of alfalfa and 2.2 tons of barley. The wheat yield is approximately 50 bushels per acre. Formulate Margaret's production plan.
Let f(x,y) = x⁴ + y⁴ – 4xy +1. Find all critical points. For each critical point, determine whether it is a local maximum, a local minimum, or a saddle point. (At least with my approach, for this problem you'll need to factor x⁹ - x. This factors as x(x² - 1)(x² + 1)(x⁴ + 1)
The critical points of [tex]f(x,y)[/tex] are: (0,0), (1,1), (-1,-1), [tex](1/\sqrt2,-1/\sqrt2)[/tex], [tex](-1/\sqrt2,1/\sqrt2), (i/\sqrt2,-i/\sqrt2)[/tex], and [tex](-i/\sqrt2,i/\sqrt2)[/tex]. The points (1,1) and (-1,-1) are local maxima, while the remaining critical points are saddle points
How to find the critical points of the function?To find the critical points of the function [tex]f(x,y)[/tex], we need to find where its partial derivatives with respect to x and y are equal to zero:
∂f/∂x = 4x³ - 4y = 0
∂f/∂y = 4y³ - 4x = 0
From the first equation, we get y = x³, and substituting into the second equation, we get:
[tex]4x - 4x^9 = 0[/tex]
Simplifying this equation, we get:
[tex]x(1 - x^8) = 0[/tex]
So the critical points occur at x = 0, x = ±1, and [tex]x = (^+_-i)/\sqrt2[/tex].
To determine the nature of these critical points, we need to look at the second partial derivatives of [tex]f(x,y)[/tex]:
∂²f/∂x² = 12x²
∂²f/∂y² = 12y²
∂²f/ = -4
At (0,0), we have ∂²f/∂x² = ∂²f/∂y² = 0 and ∂²f/∂x ∂y = -4, so this is a saddle point.
At (1,1), we have ∂²f/∂x² = ∂²f/∂y² = 12, and ∂²f/∂x ∂y = -4, so this is a local maximum.
At (-1,-1), we have ∂²f/∂x² = ∂²f/∂y² = 12, and ∂²f/∂x ∂y = -4, so this is also a local maximum.
At , we have ∂²f/∂x² = 6, ∂²f/∂y² = 6, and ∂²f/∂x ∂y = -4, so these are saddle points.
At [tex](i/\sqrt2,-i/\sqrt2)[/tex] and [tex](-i/\sqrt2,i/\sqrt2)[/tex], we have ∂²f/∂x² = -6, ∂²f/∂y² = -6, and ∂²f/∂x ∂y = -4, so these are also saddle points.
Therefore, the critical points of [tex]f(x,y)[/tex] are: [tex](0,0), (1,1), (-1,-1), (1/\sqrt2,-1/\sqrt2), (-1/\sqrt2,1/\sqrt2), (i/\sqrt2,-i/\sqrt2)[/tex], and [tex](-i/\sqrt2,i/\sqrt2)[/tex]. The points (1,1) and (-1,-1) are local maxima, while the remaining critical points are saddle points
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A triangle has side lengths of (7a + 2b) centimeters, (6a + 3c) centimeters, and
(3c +46) centimeters. Which expression represents the perimeter, in centimeters,
of the triangle?
The expression that represents the perimeter of the triangle is 13a + 5c + 2b + 46 centimeters.
So, the expression for the perimeter of the triangle is:
(7a + 2b) + (6a + 3c) + (3c + 46)
Simplifying and combining like terms, we get:
13a + 5c + 2b + 46
Rational functions can also have holes in their graphs, which do when a factor in the numerator and denominator cancel out.
For illustration, the function
[tex]h( x) = ( x2- 4)/(x^{2} )( x- 2)[/tex]has a hole at x = 2,
where the factor ( x- 2) cancels out in the numerator and denominator.
Graphing rational functions can be tricky, but it helps to identify the perpendicular and vertical asymptotes, any holes in the graph, and the of the function near these points.
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Find the Lap lace transform of
f(t) = 6u (t- 2) + 3u(t-5) - 4u(t-6)
F(s)=
To find the Laplace transform of f(t), we use the formula:
L{f(t)} = ∫[0,∞) [tex]e^(-st)[/tex] f(t) dt
where L{f(t)} denotes the Laplace transform of f(t) and u(t) is the unit step function.
Using the linearity of the Laplace transform, we can find the Laplace transform of each term separately and add them up.
L{6u(t-2)} = [tex]6e^(-2s)[/tex] / s (applying the time-shift property)
L{3u(t-5)} = [tex]3e^(-5s)[/tex] / s (applying the time-shift property)
L{-4u(t-6)} = -[tex]4e^(-6s[/tex]) / s (applying the time-shift property)
Therefore, the Laplace transform of f(t) is:
F(s) = L{f(t)} = 6[tex]e^(-2s)[/tex] / s + [tex]3e^(-5s)[/tex] / s - [tex]4e^(-6s)[/tex]/ s
= [tex](6e^(-2s) + 3e^(-5s) - 4e^(-6s)) / s[/tex]
Hence, the Laplace transform of f(t) is F(s) = [tex](6e^(-2s) + 3e^(-5s) - 4e^(-6s)) / s.[/tex]
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a circle has a circumference of 15 pi. what is the area of pi
Answer:
= 112.5π sq. units is the area of pi
Step-by-step explanation:
In your case it's 15π
So that becomes:
2πr=15π
Now dividing the equation on both sides by π,the result is:
2r=15
That means 2 times radius(r) is 15
r=15/2
r computes out to be 7.5
Now r=7.5
So the area of circle(AoC) i.e. πr^2
AoC=3.14*(7.5)^2
AoC=3.14*(7.5)*(7.5)
AoC=176.625
Note: Don't forget to multiply the result to respective unit square for e.g. if the circumference was 15π in cm then the Area would compute out as 176.625 cm^2
