Answer:
Step-by-step explanation:4(3x+y)+2(x-5y)+x²=12x+4y+2x-10y+x²
=14x-6y+x²
=x²+14x-6y
Find the probability of exactly three successes in eight trials of a binomial experiment in which the probability of success is 45%.
The probability of exactly three successes in eight trials is approximately 26.61%
To find the probability of exactly three successes in eight trials of a binomial experiment with a success probability of 45%, use the binomial probability formula:
P(x) = C(n, x) * [tex]p^x[/tex] * [tex](1-p)^{(n-x)}[/tex]
where n is the number of trials, x is the number of successes, p is the probability of success, and C(n, x) is the combination function representing the number of ways to choose x successes from n trials.
In this case, n = 8, x = 3, and p = 0.45.
Calculate the probability:
P(3) = C(8, 3) * [tex]0.45^3[/tex] * [tex](1-0.45)^{(8-3)}[/tex]
P(3) = 56 * 0.091125 * 0.1721865
P(3) ≈ 0.2661
So, the probability of exactly three successes in eight trials is approximately 26.61%.
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A spring has a natural length of 30.0 cm. If a 20.0-N force is required to keep it stretched to a length of 42.0 cm, how much work W is required to stretch it from 30.0 cm to 36.0 cm? (Round your answer to three decimal places.) W = ______ J
The amount of work required to stretch the spring from 30.0 cm to 36.0 cm is 1.411 joules, which can be rounded to three decimal places.
According to Hooke's Law, the amount of work required to stretch or compress a spring by a certain amount is given by the formula:
W = (1/2) k (x2 - x1)²
where W is the work done (in joules), k is the spring constant (in newtons per meter), x1 is the initial displacement (in meters), and x2 is the final displacement (in meters).
In this case, the spring has a natural length of 30.0 cm, which is equivalent to 0.3 meters. To find the spring constant, we can use the fact that a 20.0-N force is required to keep it stretched to a length of 42.0 cm, which is equivalent to 0.42 meters.
Using Hooke's Law, we have
F = k (x2 - x1)20.0 N
= k (0.42 - 0.3) m
=> k = 80.0 N/m
Now we can use Hooke's Law again to find the amount of work required to stretch the spring from 30.0 cm to 36.0 cm, which is equivalent to 0.36 meters.
Using Hooke's Law, we have:
F = k (x2 - x1)W
= (1/2) k (x2 - x1)²W
= (1/2) (80.0 N/m) (0.36 - 0.3) m²W
= 1.411 J
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Find the area of the parallelogram. 8 cm 9 cm 24 cm
Answer:
we know the area of the parallelogram is given by a=b×h
Suppose you purchased a house for $250,000, and three years later it is valued at $280,00. How much equity do you have in the house?
Show your work
According to the given data the equity in the house is $30,000.
What is meant by equity?Equity is the difference between the current market value of the property and the outstanding mortgage balance on the property.
According to the given information:If you purchased the house for $250,000 and it is now valued at $280,000, your equity in the house can be calculated as follows:
Equity = Current market value - Outstanding mortgage balance
Assuming you took out a mortgage for the full purchase price of the house and haven't made any extra payments, your outstanding mortgage balance would be the same as the original mortgage amount, which is $250,000. Therefore, your equity in the house would be:
Equity = $280,000 - $250,000
Equity = $30,000
So, your equity in the house is $30,000.
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Which answer choice below is closest to the length of segment DE?
Answer:
15.71
Step-by-step explanation:
Triangle ABC and triangle ADE are similar triangles so we can find the length of DE using similarity ratio.
[tex] \frac{14}{22} = \frac{10}{de} [/tex]
Cross multiply fractions14×DE = 220
Divide both sides by 14DE = 15.71 approximately.
the size of bass caught in strawberry lake is normally distributed with a mean of 12 inches and a standard deviation of 5 inches. suppose you catch 6 fish. what is the probability the average size of the fish you caught is more than 14 inches? g
The probability that the average size of the fish you caught is more than 14 inches is 0.1635 or 0.164
The size of bass caught in Strawberry Lake is normally distributed with a mean of 12 inches and a standard deviation of 5 inches.
If you catch 6 fish, the probability that the average size of the fish you caught is more than 14 inches is 0.202.
Here,
we need to find the probability that the average size of the fish you caught is more than 14 inches.
Let us denote the size of the bass caught in Strawberry Lake by X.
Then, X ~ N(μ = 12, σ = 5) represents the normal distribution of the size of bass caught in Strawberry Lake.
Let Y be the sample mean of 6 bass caught in the Strawberry Lake.
Then,
We know that Y ~ N(μ = 12, σ = 5/√6) represents the sampling distribution of the sample mean,
where σ = 5/√6
= 2.0412 (approx).
We are given that we have caught 6 fish.
Therefore, the sample size n = 6.
Then,
The probability that the average size of the fish you caught is more than 14 inches can be obtained as follows:
P(Y > 14) = P((Y - μ)/σ > (14 - μ)/σ)
= P(Z > (14 - 12)/2.0412)
= P(Z > 0.977)
= 1 - P(Z < 0.977) (as the standard normal distribution is a continuous distribution)
Using the standard normal distribution table, we get P(Z < 0.977) = 0.8365 (approx) Therefore, P(Y > 14) = 1 - P(Z < 0.977) = 1 - 0.8365 = 0.1635 (approx)
Therefore,
The probability that the average size of the fish you caught is more than 14 inches is 0.1635 or 0.164 (approx).
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preform the indicated operation
(1)/(x^(2)-6x)=(x)/(x^(2)-36)+2
solve for x
[ sorry if im missing anything ]
The solutions to the equation are x = -11/3 and x = 6.
Describe Equation?In mathematics, an equation is a statement that two expressions are equal. It consists of two sides separated by an equal sign (=). The expression on the left side of the equal sign is usually called the left-hand side (LHS), while the expression on the right side is called the right-hand side (RHS).
Equations can take many different forms and can involve various types of functions and operators, such as addition, subtraction, multiplication, division, exponentiation, logarithms, trigonometric functions, and more. They can also involve one or more variables, which can be solved for to obtain a specific value or range of values that make the equation true. Equations are used extensively in mathematics, science, engineering, economics, and many other fields.
To solve the equation for x, we can start by simplifying both sides of the equation and bringing all the terms to one side:
(1)/(x²-6x) - (x)/(x²-36) = 2
We can simplify the left side of the equation by finding a common denominator:
[(1)(x-6) - (x)(x+6)] / [(x-6)(x+6)] = 2
Expanding the numerator and simplifying, we get:
(-x² + 7x - 6) / [(x-6)(x+6)] = 2
Multiplying both sides by the denominator, we get:
-x² + 7x - 6 = 2(x-6)(x+6)
Expanding the right side, we get:
-x² + 7x - 6 = 2(x² - 36)
Simplifying and rearranging, we get:
3x² - 7x - 66 = 0
We can solve this quadratic equation by factoring or using the quadratic formula:
3x² - 7x - 66 = (3x + 11)(x - 6) = 0
So either 3x + 11 = 0 or x - 6 = 0:
3x + 11 = 0 => x = -11/3
x - 6 = 0 => x = 6
Therefore, the solutions to the equation are x = -11/3 and x = 6.
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Maths. Please help me as best as you can.
Answer:
Step-by-step explanation:
She is traveling via Bootle.
Crosby to Bootle is 4 miles.
Bootle to Speke is 12 miles.
That is 16 miles each way.
16 times 2 = 32 miles per day
5 times 32 for the work week.
5 times 32 = 160 miles per week
Give most precise classification for the figure below:
Answer:
parallelogram
Step-by-step explanation:
opposite angles and sides are congruent
a botanist wants to create an srs of size 10 from 60 plants that are arranged in an array of 10 rows of 6 plants each. she numbers the plants in each row from one to six. for each of the 10 rows, she rolls a six-sided number cube and selects the plant corresponding to the number rolled. which statements are true? check all that apply. the sample is a random sample. the sample is an srs. the sample is not a random sample. there are restrictions placed on the sample. each plant has an equal chance of being selected
The statements: the sample is random, the sample is an srs, and restrictions are placed on the sample, each plant has an equal chance of being selected are true.
Each tree has an equal chance of being selected.
This sample is a simple random sample (SRS) because each tree has an equal chance of being selected and the selection of each tree is independent of the others.
Restrictions are imposed on the sample because the botanist selects only one plant from each row, based on a specific criterion (roll a cube).
Hence the following statements are true by probability
The sample is random.
The template is an SRS.
Restrictions are imposed on the sample.
Each tree has an equal chance of being selected.
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Write the general equation for the circle that passes through the points (1, 1), (1, 3), and (9, 2).
You must include the appropriate sign (+ or -) in your answer. Do not use spaces in your answer.
8 x 2 + 8 y 2
x
y
= 0
Answer: The general equation for a circle with center (h, k) and radius r is:
(x - h)^2 + (y - k)^2 = r^2
To find the equation of the circle that passes through the points (1, 1), (1, 3), and (9, 2), we can use the fact that the perpendicular bisectors of any two chords of a circle intersect at the center of the circle.
The midpoint and slope of the chord connecting (1, 1) and (1, 3) are:
Midpoint: ((1+1)/2, (1+3)/2) = (1, 2)
Slope: undefined (since x values are the same)
The perpendicular bisector of this chord passes through the midpoint (1, 2) and has a slope of 0. Therefore, its equation is:
y - 2 = 0
Simplifying this equation gives:
y = 2
Similarly, the midpoint and slope of the chord connecting (1, 1) and (9, 2) are:
Midpoint: ((1+9)/2, (1+2)/2) = (5, 1.5)
Slope: (2 - 1)/(9 - 1) = 1/8
The perpendicular bisector of this chord passes through the midpoint (5, 1.5) and has a slope of -8 (the negative reciprocal of the slope of the chord). Therefore, its equation is:
y - 1.5 = -8(x - 5)
Simplifying this equation gives:
y = -8x + 41.5
The intersection of the two perpendicular bisectors (y = 2 and y = -8x + 41.5) is the center of the circle. Solving for x gives:
2 = -8x + 41.5
8x = 39.5
x = 4.9375
Substituting x = 4.9375 into either equation for the perpendicular bisectors gives:
y = 2
Therefore, the center of the circle is (4.9375, 2) and the radius is the distance from the center to any of the three given points. For example, the distance from (4.9375, 2) to (1, 1) is:
r = sqrt((4.9375 - 1)^2 + (2 - 1)^2) = sqrt(24.9219) = 4.992
So the equation of the circle is:
(x - 4.9375)^2 + (y - 2)^2 = (4.992)^2
Expanding and simplifying this equation gives:
x^2 - 9.875x + 24.526 + y^2 - 4y + 0.076 = 24.916
Rearranging terms gives:
x^2 + y^2 - 9.875x - 4y - 0.34 = 0
So the general equation for the circle is:
x^2 + y^2 - 9.875x - 4y - 0.34 = 0
Step-by-step explanation:
from a population of 600 elements, a sample of 100 elements is selected. it is known that the variance of the population is 900. find the approximate standard error of the mean.
The approximate standard error of the mean is 3.
Here is how to find the approximate standard error of the mean from a population of 600 elements, where a sample of 100 elements is selected, and it is known that the variance of the population is 900.
Determine the sample size n, which is 100
Find the population variance, which is 900.
Compute the population standard deviation, which is the square root of the variance, as follows:
σ = √900σ
= 30
Compute the standard error of the mean as follows:
SEM = σ/√nSEM
= 30/√100SEM
= 3
The standard error of the mean (SEM) can be calculated using the formula:
SEM = σ / √n
where σ represents the population standard deviation, and n represents the sample size. In this case, we know that the population variance is 900.
To find the standard deviation, take the square root of the variance:
σ = √900 = 30
Now, we can plug in the values into the formula:
SEM = 30 / √100
SEM = 30 / 10
SEM = 3
The approximate standard error of the mean for this sample is 3.
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HELP ASAP A hockey season ticket holder pays $136.98 for her tickets plus $2.50 for a program each game. A second person pays $17.72 for a ticket to every game, but doesn't buy programs. In how many games will they have paid the same amount?
Therefore, they will have paid the same amount after attending 9 games.
What is equation?An equation is a mathematical statement that expresses the equality between two expressions or values. It consists of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division, as well as other mathematical functions. An equation typically includes an equal sign (=) that indicates that the two sides of the equation are equal in value.
Here,
Let's start by setting up an equation to represent the total cost for each person after attending a certain number of games:
For the season ticket holder: Total Cost = 136.98 + 2.5x, where x is the number of games attended
For the second person: Total Cost = 17.72x
To find the number of games at which they will have paid the same amount, we can set the two equations equal to each other and solve for x:
136.98 + 2.5x = 17.72x
136.98 = 15.22x
x = 9
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Quick for 100 points and 5 stars please help!!
For a certain video game, the number of points awarded to the player is proportional to the amount of time the game is played. For every 1 minute of play, the game awards one half point, and for every 7 minutes of play, the game awards three and one half points.
Part A: Find the constant of proportionality. Show every step of your work. (4 points)
Part B: Write an equation that represents the relationship. Show every step of your work. (2 points)
Part C: Describe how you would graph the relationship. Use complete sentences. (4 points)
Part D: How many points are awarded for 18 minutes of play? (2 points)
Step-by-step explanation:
Please find the attached pics for answers.
1) Evaluate. Help me plsss
The logarithm expression is approximately 4.6.
How to evaluate the expression with which logarithmic identity?To evaluate 9㏒₉⁵ - ㏒₃3⁵, we can use the logarithmic property that states:
㏒ₐ(b^c) = c x ㏒ₐ(b)
Using this property, we can simplify the expression as follows:
9㏒₉⁵ - ㏒₃3⁵
= ㏒₉(95^9) - ㏒₃(35)
= ㏒₉(1423892081) - ㏒₃(35)
We can evaluate these logarithmic terms using the change of base formula, which states:
㏒ₐ(b) = log(b) / log(a)
Using this formula, we get:
㏒₉(1423892081) = 8.032
㏒₃(35) = 3.432
Substituting these values back into the expression, we get:
9㏒₉⁵ - ㏒₃3⁵ = 8.032 - 3.432
= 4.6
The above answer is in response to the question below;
Evaluate
9㏒₉⁵ - ㏒₃3⁵
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30 points to whoever solves
Answer:
Step-by-step explanation:
its a 12/14 probabiloty or a 92.5%
Step-by-step explanation:
I'll try:
7 boys choose 5 = 7! / (5!2!) = 21 ways
4 girls choose 2 = 4 ! / (2! 2!) = 6 ways
6 x 21 = 126 ways to choose 5 boys and 2 girls
11 cats total choose 7 = 11! /( 7! 4!) = 330 ways to choose 7 cats
126 of these will be 5 boys and 2 girls
126 out of 330 = 126/330 = .382
square root of 31-4√21
Answer:
-12.762538417
Step-by-step explanation:
Verify the identity so that the left side looks like the right side.
[tex]\frac{1-cos(x)}{sin(x)}=\frac{sin(x)}{1+cos(x)}[/tex]
By trigonometry Taking a look at the equation's left side 1-COSX/SINX is = (SINX/SINX) - (COSX/SINX) = (SINX - COSX)/SINX.
What are a few uses for trigonometry?Several fields, including architecture, physics, astronomy, surveying, oceanography, navigation, electronics, and many more, use trigonometry. These are a few instances:
- In physics, trigonometry is used to calculate forces and analyze waves. - In astronomy, trigonometry is used to calculate distances between stars and planets. - In surveying, trigonometry is used to calculate distances and heights of objects. - In oceanography, trigonometry is used to calculate the heights of waves and tides in oceans.
We must make one side of the equation appear to be the other side in order to prove that 1-COSX/SINX=SINX/1+COSX.
Taking a look at the equation's left side first:
1-COSX/SINX is = (SINX/SINX) - (COSX/SINX) = (SINX - COSX)/SINX.
Let's now adjust the equation's right side:
SINX/1+COSX = (SINX/1+COSX) * (1-COSX/1-COSX)
= (SINX - SINXCOSX)/(1-COSXSINX)
= (SINX - COSXSINX)/(1-COSXSINX)
= (SINX - COSX)/((1-SINXCOSX)/( (1-SINXCOSX)
As the identities on both sides are equal to (SINX - COSX)/SINX, we can conclude that they are correct.
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explain how to take a systematic sample of 100 companies from the 1,803 companies that are members of an industry trade association. state important numerical values used in the process.
To take a systematic sample of 100 companies from a population of 1,803 companies, determine the sampling interval (k) which is N/n, choose a random starting point between 1 and k, and select every kth company until 100 companies are sampled.
To take a systematic sample of 100 companies from the 1,803 companies that are members of an industry trade association, follow these steps
Determine the sampling interval (k), which is the number of companies in the population divided by the desired sample size. In this case, k = 1803/100 = 18.03. Round this number up or down to the nearest whole number based on your sampling preferences.
Choose a random starting point between 1 and k. For example, you could randomly select a number between 1 and 18.
From the starting point, select every kth company in the list of members until you have 100 companies. For example, if the starting point is 4, you would select companies 4, 22, 40, 58, 76, 94, 112, and so on until you reach 100 companies.
Important numerical values used in this process are:
Population size (N) = 1,803 companies
Desired sample size (n) = 100 companies
Sampling interval (k) = N/n = 1803/100 = 18.03
Random starting point (any number between 1 and k, depending on sampling preference)
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example: marlon jogs two miles to the park in 25 minutes, turns around, and takes another 55 minutes to walk the same path back to his house. what is the average speed of the round-trip?
The average speed of the round-trip for same path back to his house is given by 3 miles per hour.
The mean value of a body's speed over a period of time is its average speed. As a moving body's speed is not constant over time and fluctuates, the average speed formula is required. The values of total time and total distance travelled may be employed even when the speed varies, and with the aid of the average speed formula, we can identify a single number that sums up the whole motion.
So the average speed is simply: [tex]\frac{distance}{time}[/tex]
In case, the total time = 25 minutes + 55 minutes
which is a total of 80 minutes, or 1.33 hours.
The total distance traveled is two miles + two miles, since he jogged two miles to the park, and then he turns around and walks the same path.
So in total he traveled 4 miles.
Plugging this in to the formula gives you the equation:
[tex]v = \frac{d}{t} \\= \frac{4\ miles}{4/3 \ hour} \\= \frac{4 \ miles}{1} * \frac{3}{4} hours\\= 3mph[/tex]
Therefore, average speed of the round-trip is 3 mph.
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Is the volume of the sphere 2/3 the volume of cylinder now ?
Volume of the sphere is equal to two-third of the volume of a cylinder.
Describe Volume of sphere?Volume is a mathematical concept that refers to the amount of space occupied by a three-dimensional object. It is expressed in cubic units, such as cubic meters or cubic centimeters, and can be calculated using a specific formula for each type of geometric shape.
A sphere is a perfectly round, three-dimensional object that looks like a ball. It has no edges or vertices and is symmetric around its center point. The formula for calculating the volume of a sphere is V = 4/3 πr³, where V is the volume, π is a mathematical constant approximately equal to 3.14, and r is the radius of the sphere.
True.
Assume that both the sphere's and cylinder's radius is r.
Considering that the base's diameter equals the cylinder's height
⇒h = 2r
Sphere radius = cylinder radius = r
Assuming the conditions are met, the sphere's volume:
= [tex]\frac{2}{3}[/tex] × Volume of cylinder
= [tex]\frac{4}{3}[/tex]πr³= [tex]\frac{2}{3}[/tex] × πr² × 2r
=[tex]\frac{4}{3}[/tex]πr³= [tex]\frac{4}{3}[/tex]πr³
Hence, volume of the sphere is equal to two-third of the volume of a cylinder.
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The complete question is:
The volume of a sphere is equal to two-third of the volume of a cylinder whose height and diameter are equal to the diameter of the sphere.
15. Melissa put $900 in her savings account. She spent $25 on gas every week and x dollars on clothes every week. Write TWO expression to show how much money Melissa would have after 10 weeks. Expression 1: Expression 2:
The two expressions to show how much money Melissa would have after 10 weeks are:
1. 900 - 250 - 10x
2. 900 - (10 × 25 + 10x)
Expression 1: To calculate how much money Melissa would have after 10 weeks, first find the total amount she spends
on gas in 10 weeks by multiplying the weekly gas cost by the number of weeks:
25 10 = 250.
Next, find the total amount she spends on clothes in 10 weeks by multiplying the weekly clothes cost (x) by the number of weeks:
x × 10 = 10x.
Now, subtract both of these expenses from her initial savings to find the remaining balance:
900 - 250 - 10x.
Expression 2: You can also find the total expenses in 10 weeks by adding the gas and clothes costs together:
25 + x.
Then, multiply this combined weekly cost by 10 to find the total cost in 10 weeks:
10(25 + x) = 10 × 25 + 10x.
Finally, subtract the total cost from her initial savings:
900 - (10 × 25 + 10x).
So, the two expressions to show how much money Melissa would have after 10 weeks are:
1. 900 - 250 - 10x
2. 900 - (10 × 25 + 10x)
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Work out the equation of the line which has a gradient of 3 and passes through the point (-1,3).
State what each variable may be so that the equation is true. You must have at least one negative number. Explain how you chose the values for a and b. 2^a • 2^b = 2^0
Answer:
[tex] {2}^{a} \times {2}^{b} = {2}^{a + b} = {2}^{0} [/tex]
From this, a + b = 0, meaning a and b are additive inverses of each other.
As an example, let a = 1, so b = -1.
Pls 50 point for each person will mark brainiest for the best answer
Please show me the three common methods for solving quadratic equations
factoring, using the square roots, completing the square and the quadratic formula.
thanks
Answer:
The three most common methods for solving quadratic equations are factoring, completing the square, and the quadratic formula.
Here's a quadratic equation you can solve by factoring:
x^2 - 4x + 3 = 0
(x - 1)(x - 3) =0
x = 1, 3
What is the sign of
−
9
⋅
(
0
−
3
)
−9⋅(
−3
0
)minus, 9, dot, left parenthesis, start fraction, 0, divided by, minus, 3, end fraction, right parenthesis?
The expression is undefined, so the sign cannot be determined.
How to determine sign for the given problem?
The following given expression can be more simplified as follows:
-9 * (0 - 3) - (9 * (-3/0))
= -9 * (-3) - 9 * undefined [Note: Division by zero is undefined]
= 27 - undefined
As anything subtracted from undefined remains undefined, the overall result is undefined.
Therefore, the sign of the expression cannot be determined.
Undefined values represent the absence of a meaningful result or outcome. In this case, the expression involves division by zero, which is undefined. As a result, any operation involving an undefined value will also be undefined, including subtraction. Thus, the overall result is undefined.
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Answer:
ITS ZEROOOO
Step-by-step explanation:
i guesses and ended up getting it right.
m∠CBD=6x+27 ∘ so you have to find C B D with this
For the given triangle, m∠CBD = 63 degrees.
What is triangle?
A triangle is a two-dimensional geometric shape with three sides and three angles. It is one of the basic shapes in geometry and is formed by connecting three non-collinear points with line segments.
In a triangle, the sum of the angles is always 180 degrees.
So, we can write:
m∠ABD = m∠ABC + m∠CBD
Substituting the given values, we get:
96 = (7x - 9) + (6x + 27)
Simplifying the equation:
96 = 13x + 18
Subtracting 18 from both sides:
78 = 13x
Dividing by 13:
x = 6
Therefore:
m∠ABD = 96 degrees
m∠CBD = 6x + 27 = 6(6) + 27 = 63 degrees
m∠ABC = 7x - 9 = 7(6) - 9 = 33 degrees
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algebra 1 mid term help!!
Answer:
Yes,
Step-by-step explanation:
Because no x- Values repeat.
the product of twice a number and four
The product of twice a number and four is 8 times the number.
What is product of a number?The product means a number that you get by multiplying two or more other numbers together.
Equation:The product of twice a number and four can be represented algebraically as:
4(2x)
where x is the number we are referring to.
Simplifying the expression, we get:
4(2x) = 8x
Therefore, the product of twice a number and four is 8 times the number.
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Jay is planting a maximum of 60 bulbs of lilies and tulips in her garden. She wants to plant
at least twice as many tulips (x) as lilies (y). Tulip bulbs cost 1.60 each and lily bulbs cost
$1.25 each. How many bulbs of each should Jay purchase to minimize her costs? (Solution
should contain object function, constraints, and graph).
The minimum cost occurs when Jay plants 30 tulip bulbs and 15 lily bulbs, and the minimum cost is $67.50.
Let x be the number of tulip bulbs that Jay plants, and let y be the number of lily bulbs that Jay plants.
Since Jay wants to plant at least twice as many tulips as lilies, we have the constraint:
x ≥ 2y
Also, the total number of bulbs that Jay plants cannot exceed 60, so we have the constraint:
x + y ≤ 60
We want to minimize the total cost of the bulbs that Jay purchases, which is given by the object function:
C = 1.6x + 1.25y
To solve this problem, we can use linear programming. First, we graph the two constraints on a coordinate plane:
The shaded region represents the feasible region, which is the region that satisfies both constraints. The vertices of the feasible region are (0, 0), (40, 20), (60, 0), and (30, 15).
Next, we evaluate the object function at each vertex to find the minimum value. We have:
C(0, 0) = 0
C(40, 20) = 1.6(40) + 1.25(20) = 92
C(60, 0) = 1.6(60) + 1.25(0) = 96
C(30, 15) = 1.6(30) + 1.25(15) = 67.5
Therefore, the minimum cost occurs when Jay plants 30 tulip bulbs and 15 lily bulbs, and the minimum cost is $67.50.
Graphically, we can see that the optimal solution occurs at the intersection of the two constraints, which is the point (30, 15). This is where the slope of the object function is equal to the slope of the feasible region, which indicates that the object function is optimized at this point.
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