Tell whether the polynomial can be factored. If not, change the constant term so that the polynomial is a perfect square trinomial. a) p^2 + 12p + 33 b) x^2 - 16x + 61
Answers
Hello there. To solve this question, we'll have to remember some properties about factorization.
Given the following polynomials, we have to decide whether they are factorable (presumably over the real numbers). If they are not, we have to change the constant term so that it turns into a perfect square trinomial.
a) p² + 12p + 33
As this is a second degree polynomial, we can decide if it is factorable by calculating its discriminant:
For a second degree polynomial with real coefficients ax² + bx + c, it is factorable if and only if:
\(\Delta=b^2-4ac\ge0\)Plugging in a = 1, b = 12 and c = 33, we get:
\(\Delta=12^2-4\cdot1\cdot33=144-132=12\)Therefore this polynomial can be factored.
Its factored form will be something as follows:
\(a\cdot(x-x_1)(x-x_2)_{}\)Where a is the leading coefficient of the polynomial, x1 and x2 are its roots.
We calculate the roots by using the quadratic formula, for a not equal to 0:
\(x_{1,2}=\frac{-b\pm\sqrt[]{\Delta}_{}}{2a}\)Plugging the values, we get:
\(x_{1,2}=\frac{-12\pm\sqrt[]{12}}{2\cdot1}=\frac{-12\pm\sqrt[]{4\cdot3}_{}}{2}=\frac{-12\pm2\sqrt[]{3}_{}}{2}=-6\pm\sqrt[]{3}_{}\)We break the solutions, such that
\(\begin{gathered} x_1=-6-\sqrt[]{3} \\ x_2=-6+\sqrt[]{3} \end{gathered}\)And the factored form is:
\(\begin{gathered} 1\cdot(p-(-6-\sqrt[]{3}))(p-(-6+\sqrt[]{3})) \\ (p+6+\sqrt[]{3})(p+6-\sqrt[]{3}) \end{gathered}\)Rewriting it as a perfect square trinomial, that is, something of the form:
\(x^2+2xy+y^2\)We have to take the coefficient b and divide by two. After this, we square the result and plug as the new constant term:
\(\frac{b}{2}=\frac{12}{2}=6\)Squaring the number
\(6^2=36\)This is the new constant term, such that the perfect square trinomial is then:
\(p^2+12p+36\)b) x² - 16x + 61
Again, we check if it can be factored by plugging the coefficients in the formula for the discriminant:
\(\Delta=(-16)^2-4\cdot1\cdot61=256-244=12\)And this is a factorable polynomial.
Its factored form will be:
\((x-8+\sqrt[]{3})(x-8-\sqrt[]{3})\)But to answer the question, we have to rewrite it as a perfect square polynomial.
In order to do so, we repeat the process, this time using b = -16:
\(\frac{b}{2}=\frac{-16}{2}=-8\)Square the number
\((-8)^2=64\)This is the new constant term, such that:
\(x^2-16x+64\)Is the perfect square trinomial we were looking for.
Related Questions
Molly works at an electronics store. Her last customer purchased a DVD player for $61.29 and five DVDs for $59.31, including taxes. If the customer paid her with two \$100 bills, how much change should she give him?
Answers
Answer:
$79.4
Step-by-step explanation:
Add up both of the prices.
61.29 + 59.31 = 120.6
Subtract that from the 200.
200 - 120.6 = 79.4
A delivery drone approaches a customer's porch, flying 14.1 m above the porch at 22.6 km/h. Calculate the speed the package hits the porch if it's released by the drone. Please report your speed in meters per second to 1 decimal place. Hint: speed is the magnitude of velocity and is thus always positive
Answers
The speed the package hits the porch when it is released by the drone is 16.9 m/s (approx)
Given data: Height of drone, h = 14.1 m
Velocity of drone, v = 22.6 km/h
The speed the package hits the porch when it is released by the drone.
It is given that the drone is flying at a height of 14.1 m above the porch and it releases the package. The package falls freely under gravity.
We can use the formula for the velocity of a freely falling body to calculate the speed the package hits the porch.
v² = u² + 2gh
Here, g is the acceleration due to gravity and u = 0 as the package is initially at rest.
Therefore, the above formula becomes,
v² = 2ghv = √(2gh)
We know that the height of the drone above the porch is h = 14.1 m.
We can convert the velocity of the drone,
v = 22.6 km/h, into m/s.
1 km/h = 1000 m/3600 s = 0.27778 m/sv = 22.6 km/h = 22.6 × 0.27778 m/s = 6.27708 m/s
Substituting the given values in the above formula, we get,
v = √(2gh)= √(2 × 9.81 m/s² × 14.1 m)≈ 16.91 m/s
Therefore, the speed the package hits the porch when it is released by the drone is 16.9 m/s (approx)
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165% of the value is 3795 kg what is the original value
Answers
Answer:
Answer: 2300kg. formulate:: 3795kg/165%Calculate 3795kg/165%: : 2300kg.
Step-by-step explanation:
I really need help. Can you please show your work too.
Answers
Answer:
Step-by-step explanation:
You just combine common variables.
8a^5 -4
3a^5 + a -2
11a^5 + a -6
The solution to the first is 11a^5+a-6.
The second:
3x^2 -2x + 1
-x^2 + 3x + 1
2x^2 + x + 2
2x^2+x+2
A shipping container will be used to transport several 40-kilogram crates across the country by rail. The greatest weight that can be loaded into the container is 25000 kilograms. Other shipments weighing 10800 kilograms have already been loaded into the container. Write an inequality representing x, the total number of 40-kilogram crates that can be loaded into the shipping container
Answers
So the total number of 40-kilogram crates that can be loaded into the shipping container is x < 355
Let x be the number of 40-kilogram crates that can be loaded into the shipping container. We can write the following inequality:
40x + 10800 < 25000
This inequality states that the total weight of the crates (40x) plus the weight of the other shipments (10800 kilograms) must be less than the maximum weight that can be loaded into the container (25000 kilograms).
Solving for x, we get:
40x < 25000 - 10800
40x < 14200
x < 355
So the total number of 40-kilogram crates that can be loaded into the shipping container is x < 355.
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HELP I GIVE 50 points total if u get it right and get big brain! The perimeter of the triangle shown is 30 inches. What is the length of the shortest side of the triangle? *
Answers
If the perimeter is 30 all around, you need to find the to answers to each equation, then take the answers and add them together and minus the answer from 30 you should get the amount that equals x.
I hope it helps :)
how can you find a 30% increase of some thing which started at 50? 
Answers
increase=(30/100)*50
=15
so,
value after increase
=50+15
=65
Algebra II Question - Tell whether the ordered pair is a solution of the system, if so why?
Answers
Answer:
Oh the answer is yes!
Step-by-step explanation:
If you graph (0,0) it will be in the gray area. Therefore, if the point is inside the gray, then the answer is yes.
introduces a study where researchers studying the relationship between honesty, age, and self-control conducted an experiment on 1G0 children between the ages of 5 and 15. The researchers asked each child to toss a fair coin in private and to record the outcome (white or black) on a paper sheet, and said they would only reward children who report white. Half the students were explicitly told not to cheat and the others were not given any explicit instructions. Differences were observed in the cheating rates in the instruction and no instruction groups, as well as some differences across children's characteristics within each group. (a) Identify the population of interest and the sample in this study. (b) Comment on whether or not the results of the study can be generalized to the population, and if the findings of the study can be used to establish causal relationships.
Answers
Answer:
The solution to this question can be defined as follows:
Step-by-step explanation:
In choice a:
Both children between 5 and 15 years old and 160 years old are also the populations of interest, and Kids from 5 to 15 years of age.
In choice b:
These findings could be compared like we do not know whether the groups were randomly selected, and the allocation is still not random; there could be no casual conclusion because the statement is valid even With sample research, and after randomization, filtering, and testing of 160 specimens (instruction vs no-instruction) Its analysis can be common to the community and replicated. There could be explanatory relations generated by experimental research. This analysis is observational and can therefore be used Causal relationships are established.gggg
Kate and Jalon are scuba diving. Kate is 9
feet below the surface and Jalon is 16 feet
below the surface. If Kate descends 8 more
feet and Jalon rises 4 feet, find the vertical
distance between them.
Answers
Answer:
5 feet
Step-by-step explanation:
Kate would then be 17 feet under
Jalon would then be 12 feet under
Subtract.
What's the circumference of a
circle with a diameter of 31 inches?
Use 3.14 for pi.
Answers
Answer:
circumference = 97.34
Step-by-step explanation:
circumference = diameter x Pi
31 x 3.14 = 97.34 inches
The circumference of the circle with a diameter of 31 inches is approximately 97.34 inches.
The circumference of a circle can be calculated using the formula C = πd, where C represents the circumference and d represents the diameter of the circle.
Given that the diameter of the circle is 31 inches, we can substitute this value into the formula:
C = 3.14 * 31
C ≈ 97.34 inches
Therefore, the circumference of the circle with a diameter of 31 inches is approximately 97.34 inches.
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NONSENSE=REPORT CORRECT & USEFUL ANSWER=BRAINLIEST. Decimals Round to the Nearest Whole Number Round to the Nearest Tenth 1) 1.56 2) 2.65 3) 3.45 4) 4.49 5) 5. 72
Ty.
Answers
Answer:
1) 1.6
2) 2.7
3) 3.5
4) 4.5
5) 5.7
Step-by-step explanation:
"Five or more, raise the score, Four or less, let it rest"
(basically, if the hundredth decimal is five or greater, round up. If it is four or lower, round down)
Brainliest plz!! :)
The rate of interest that you are paying on a mortgage loan is 6.5 per cent per year. Inflation is running at 3 per cent per year. The real rate of interest that you are paying is therefore: Group of answer choices 6.5 9.5 3 3.5
Answers
The real rate of interest that you are paying is 3.5%.
The rate of interest, often referred to as the interest rate, is the percentage or proportion of the principal amount that is charged or earned over a certain period of time. It represents the cost of borrowing money or the return on investment. The rate of interest is typically expressed as an annual percentage, but it can also be calculated for shorter or longer time periods depending on the context.
The real rate of interest is calculated by subtracting the inflation rate from the nominal interest rate. In this case, the nominal interest rate is 6.5% per year, and the inflation rate is 3% per year.
Real Rate of Interest = Nominal Interest Rate - Inflation Rate
Real Rate of Interest = 6.5% - 3%
Real Rate of Interest = 3.5%
Therefore, the real rate of interest that you are paying is 3.5%.
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compute |u x v| if u and v are unit vectors and the angle between them is π 4.
Answers
Answer:
Step-by-step explanation:
To compute the magnitude of the cross product |u x v|, we need to know the values of u and v. However, you mentioned that u and v are unit vectors, which means their magnitudes are both equal to 1.
The magnitude of the cross product between two vectors u and v is given by |u x v| = |u| * |v| * sin(theta), where theta is the angle between the two vectors.
In this case, since u and v are unit vectors, their magnitudes are both equal to 1. Additionally, you mentioned that the angle between u and v is π/4.
Therefore, |u x v| = 1 * 1 * sin(π/4) = 1 * 1 * (√2/2) = √2/2.
Hence, the magnitude of the cross product |u x v| is equal to √2/2.
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The value of the line integral around the closed path in the figure(Figure 1) is 1.96×10?5 Tm . What is the magnitude of I3?
Answers
As per the integral value of the closed path, the magnitude of I3 is 3.597
The term magnitude in math refers a measure of the size of a mathematical entity.
Here we have the value of the line integral around the closed path 1.96 x 10⁻⁵.
Now, we have to find the value of the magnitude I3,
Here we have also know that the value of I1 and I2 are 18A and 12 A respectively.
Now, as per the line integral of the magnetic field is written as,
=> 1.96 x 10⁻⁵ = 4π⋅10⁻⁷ (I3 + 12)
When we simplify this one then we get,
=> 1.96 x 10²/4π = I3 + 12
=> 15.597 = I3 + 12
Then the value of the magnitude is calculated as
=> I3 = 3.597
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write 10 rational numbers between -1/3 and 1/3
Answers
Step-by-step explanation:
-1/4, -1/5, -1/6, -1/7, -1/8, 1/8, 1/7, 1/6, 1/5, 1/4
The ratio of boys to girls is 4:5 if there are 20 boys in the class find the number of girls. Show workings
Answers
Answer:
25 girls
Step-by-step explanation:
We Know
The ratio of boys to girls is 4:5. For every 4 boys, there are 5 girls.
To get from 4 to 20, we multiply by 5.
We Take
5 x 5 = 25 girls
So, there are 25 girls in class.
Answer: 25 girls are in the class
Step-by-step explanation: You can set up the ratio 4:5 as a fraction, \(\frac{4}{5}\) to find your answer. You are given the fact that 20 boys are in the class so now you can solve 2 ways.
Option 1 - Set up the equation algebraically as \(\frac{20}{x}\), where x = number of girls and set that equal to \(\frac{4}{5}\). This way allows you to see that the fraction must have the same ratio as 4:5. You can see that 4 x 5 = 20, so the multiple factor is 5. The variable x must equal 5 x 5, so x = 25.
Option 2 - Multiply the amount of boys given to you by the reciprocal of the ratio. Instead of using \(\frac{4}{5}\), you have to use \(\frac{5}{4}\) because there are more girls than boys in the class. This allows you to finish the problem by multiplying 20 x \(\frac{5}{4}\) to get the result of \(\frac{100}{4}\), which you may know simplifies into 25.
In 2010, the number of participants in a checkers tournament was 2.6 × 10^4 people. In 2013, the number of participants in a checkers tournament was 5.2 × 10^6 people. By what factor did the number of participants increase from 2010 to 2013?
Answers
the data set represents the shoe sizes of 19 students in a fifth grade physical education class 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7.5, 7.5, 8, 8, 8.5, 8.5, 9 a. create a box plot to represent the distribution of the data
Answers
Answer:
The minimum value = 4
The maximum value = 9
The first quartile, Q₁ = 6
The second quartile, (median) Q₂ = 7
The third quartile, Q₃ = 8
The box plot showing the above data is attached
Step-by-step explanation:
The given data set is 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7.5, 7.5, 8, 8, 8.5, 8.5, 9
From the data, we have;
The minimum value = 4
The maximum value = 9
The first quartile, Q₁ = (n + 1)/4th term = (19 + 1)/4 = the 5th term = 6
The second quartile, (median) Q₂ = (n + 1)/2 term = (19 + 1)/2 = the 10th term = 7
The third quartile, Q₃ = 3×(n + 1)/4th term = 3×(19 + 1)/4 = the 15th term = 8
Please find the attached box plot as plotted on Excel.
Pls help me I got it wrongg :((
Answers
Answer:
4 ounces
Step-by-step explanation:
I'm assuming when it says: \(33\frac{1}{3}%\)% it means 33.3333333333 instead of 33/3.
Anyways under that assumption, we currently have 20 ounces in the grape juice, since 4 ounces is grape juice concentrate and 16 ounces is water. So currently let's just say that the total amount of grape juice concentrate is going to be: \((4+x)\) where x is the amount of grape juice concentrate you need to add to make it 33.33%. The next thing to know is how to calculate how much percent a is of b. This can be calculated using the equation: \(\frac{a}{b}*100\). So in this case the a is going to be the grape juice concentrate, and the b is going to be the entire mixture (water and grape juice). Since the concentrate is having x added to it, that means the entire thing is also having x added to it. This means we have the equation: \(\frac{4+x}{20+x} * 100\), since we already have 4 ounces of grape juice concentrate, and we already have 20 ounces in total (16 ounces of water + 4 ounces of grape juice). This is going to be equal to 33.33%, or in other words 33 1/3. So to make this a bit easier, I'm going to convert this mixed fraction into one fraction, this gives you the fraction: \(\frac{100}{3}\). So now that's all left to do is set these two expressions equal to each other and solve for x
Original equation
\(\frac{4+x}{20+x}*100=\frac{100}{3}\)
Divide both sides by 100
\(\frac{4+x}{20+x}=\frac{100}{3}\div 100\)
Keep fraction, change division to multiplication, flip 100
\(\frac{4+x}{20+x}=\frac{100}{3}* \frac{1}{100}\)
The 100 in the numerator will cancel out the one in the denominator
\(\frac{4+x}{20+x}=\frac{1}{3}\)
Multiply both sides by 20+x
\(4+x=\frac{20+x}{3}\)
Multiply both sides by 3
\(3(4+x)=20+x\)
Distribute the 3
\(12+3x=20+x\)
Subtract x from both sides
\(12+2x=20\)
Subtract 12 from both sides
\(2x=8\)
Divide both sides by 2
\(x=4\)
So Kahn needs to pour 4 more ounces of grape juice, to double check this, since the mixture already has 4 ounces of grape juice, this means the final mixture will have a total of 8 ounces of grape juice concentrate, and a total of 24 ounces, since it already had 16 ounces of water, and 4 ounces of grape juice and now there is 4 more ounces of grape juice being added, so a total of 24 ounces. this means that 8/24 will be grape juice concentrate or in other words: \(\frac{1}{3}\) and when you multiply this by 100 (to convert to percent), you get 100/3 or 33 1/3%
Write a quadratic equation in standard form with integral coefficients given
x = 4 is a solution with multiplicity of 2.
Answers
Answer:
y=(x-4)^2=x^2-8x+16
Step-by-step explanation:
Since we know that x=4 is a solution with multiplicity 2. Therefore x=4 will occur as a factor twice in the quadratic equation.
So
y=(x-4)^2
\(y=(x-4)^2=x^2-8x+16\)
Calculate dy/dx of x= 2a sin²t cos 2t
y= 2a sin²t sin 2t
Answers
Answer:
x=sint
Differentiating w.r.t. t, we get,
dt
dx
=cost
y=cos2t
Differentiating w.r.t. t, we get,
dt
dy
=−2sin2t
Thus,
dx
dy
=−
cost
2sin2t
dx
dy
=
cost
−4sintcost
=−4sint
Step-by-step explanation:
How do I solve x for 2x+8=3x+10
Answers
To solve for x in: \(2x + 8 = 3x + 10\)
Collect like terms
\(→8 - 10 = 3x - 2x \\ - 2 = x\)
Therefore:
\(x = - 2\)
Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. x2-7x 0 74 011 Write the form of the partial fraction decomposition of the rational expression, Do not solve for the constants. 6x+5 (x+ 8) 74.014 Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. 20-3 points LarPCalc10 7.4 023 8 3 4
Answers
To write the form of the partial fraction decomposition of the given rational expressions, we need to express them as a sum of simpler fractions. The general form of a partial fraction decomposition is:
f(x) = A/(x-a) + B/(x-b) + C/(x-c) + ...
where A, B, C, etc., are constants and a, b, c, etc., are distinct linear factors in the denominator.
For the rational expression x^2 - 7x:
The denominator has two distinct linear factors: x and (x - 7). Therefore, the partial fraction decomposition form is:
(x^2 - 7x)/(x(x - 7)) = A/x + B/(x - 7)
For the rational expression 6x + 5 / (x + 8):
The denominator has one linear factor: (x + 8). Therefore, the partial fraction decomposition form is:
(6x + 5)/(x + 8) = A/(x + 8)
For the rational expression 20 - 3 / (4x + 3):
The denominator has one linear factor: (4x + 3). Therefore, the partial fraction decomposition form is:
(20 - 3)/(4x + 3) = A/(4x + 3)
In each case, we write the partial fraction decomposition form by expressing the given rational expression as a sum of fractions with simpler denominators. Note that we have not solved for the constants A, B, C, etc., as requested.
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Arrange the steps in correct order to solve the congruence 2x= (mod 17) using the inverse of 2 modulo 17, which is 9 Rank the options below: 9 is an inverse of 2 modulo 17. The given equation is Zx = 7 (mod 17)_ Multiplying both sides of the equation by 9, we get x= 9 7 (mod 17)_ Since 63 mod 17 = 12,the solutions are all integers congruent to 12 modulo 17, such as 12,29,and-5.
Answers
Answer: Conclude that the solutions to the congruence 2x ≡ 7 (mod 17) are all integers congruent to 12 modulo 17, such as 12, 29, and -5.
Step-by-step explanation:
Verify that 9 is an inverse of 2 modulo 17.
Rewrite the given equation as 2x ≡ 7 (mod 17).
Multiply both sides of the equation by 9 to get 18x ≡ 63 (mod 17).
Simplify the equation using the fact that 18 ≡ 1 (mod 17) to get x ≡ 9*7 (mod 17).
Evaluate 9*7 mod 17 to get x ≡ 12 (mod 17).
Conclude that the solutions to the congruence 2x ≡ 7 (mod 17) are all integers congruent to 12 modulo 17, such as 12, 29, and -5.
Therefore, the correct order of the steps is:
Verify that 9 is an inverse of 2 modulo 17.
Rewrite the given equation as 2x ≡ 7 (mod 17).
Multiply both sides of the equation by 9 to get 18x ≡ 63 (mod 17).
Simplify the equation using the fact that 18 ≡ 1 (mod 17) to get x ≡ 9*7 (mod 17).
Evaluate 9*7 mod 17 to get x ≡ 12 (mod 17).
Conclude that the solutions to the congruence 2x ≡ 7 (mod 17) are all integers congruent to 12 modulo 17, such as 12, 29, and -5.
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You play dots against an opponent who is your equal. Which of the following is more likely: (a) winning three games out of four or winning five out of eight? (b) winning at least three out of four or at least five out of eight?
Answers
Winning three games out of four or winning five out of eight Winning three games out of four is more likely than winning five out of eight.
To determine which outcome is more likely, we compare the probabilities of winning three out of four games and winning five out of eight games.
(a) Winning three games out of four:
Assuming you and your opponent have an equal chance of winning each game (1/2 probability for each game), we can calculate the probability using the binomial probability formula:
P(X=k) = (n choose k) * p^k * (1-p)^(n-k)
For winning three out of four games:
P(X=3) = (4 choose 3) * (1/2)^3 * (1/2)^(4-3)
= 4 * (1/8) * (1/2)
= 1/4
(b) Winning five games out of eight:
Using the same approach, we can calculate the probability of winning five out of eight games:
P(X=5) = (8 choose 5) * (1/2)^5 * (1/2)^(8-5)
= 56 * (1/32) * (1/2)
= 7/32
Comparing the probabilities, we see that the probability of winning three out of four games is 1/4, while the probability of winning five out of eight games is 7/32. Since 1/4 is greater than 7/32, winning three games out of four is more likely than winning five games out of eight.
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The electricity accounts of residents in a very small town are calculated as follows: If 500 units or fewer are used, the cost is 2 cents per unit. If more than 500 but not more than 1000 units are used, the cost is $10 for the first 500 units and 5 cents for every unit in excess of 500 . If more than 1000 units are used, the cost is $35 for the first 1000 units plus 10 cents for every unit in excess of 1000 . ■ A basic service fee of $5 is charged, no matter how much electricity is used. Write a program that enters the following five consumptions and use an if statement to calculate and display the total charge for each one: 200,500,700,1000,1500. (Answers: $9,$15,$25,$40,$90 )
Answers
The output is
The total charge for 200 units is $9.00
The total charge for 500 units is $15.00
The total charge for 700 units is $25.00
The total charge for 1000 units is $40.00
The total charge for 1500 units is $90.00
Here's a Python program that calculates and displays the total charge for each consumption using the given conditions:
```python
# Function to calculate the total charge for a given consumption
def calculate_total_charge(consumption):
basic_service_fee = 5 # Basic service fee of $5
total_charge = basic_service_fee # Start with the basic service fee
if consumption <= 500:
# If 500 units or fewer are used
total_charge += consumption * 0.02
elif consumption <= 1000:
# If more than 500 but not more than 1000 units are used
total_charge += 10 + (consumption - 500) * 0.05
else:
# If more than 1000 units are used
total_charge += 35 + (consumption - 1000) * 0.1
return total_charge
# List of consumptions
consumptions = [200, 500, 700, 1000, 1500]
# Calculate and display the total charge for each consumption
for consumption in consumptions:
total_charge = calculate_total_charge(consumption)
print(f"The total charge for {consumption} units is ${total_charge:.2f}")
```
When you run this program, it will output the following results:
```
The total charge for 200 units is $9.00
The total charge for 500 units is $15.00
The total charge for 700 units is $25.00
The total charge for 1000 units is $40.00
The total charge for 1500 units is $90.00
```
The program defines a function `calculate_total_charge` that takes the consumption as an input and calculates the total charge based on the given conditions. It uses an if statement to check the consumption range and applies the corresponding cost calculation. The basic service fee is added to the total charge in each case. The program then iterates over the list of consumptions and calls the `calculate_total_charge` function for each consumption, displaying the results accordingly.
Keywords: Python program, electricity accounts, total charge, consumption, if statement, basic service fee, cost calculation.
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To make a small vase, Elisa uses no more than 4.5 ounces of clay. To make a large vase, she uses at least 12 ounces of clay. Which compound inequality represents the number of ounces of clay, c, that Elisa uses to make one vase of either size? 4.5 < c < 12 4.5 ≤ c ≤ 12 c < 4.5 or c > 12 c ≤ 4.5 or c ≥ 12
Answers
The compound inequality that represents the number of ounces of clay, "c", Elisa uses to make one vase of either size is [(c ≤ 4.5) or (c ≥ 12)].
As per the question statement, Elisa uses no more than 4.5 ounces of clay to make a small vase and at least 12 ounces of clay to make a large vase.
We are required to find out the compound inequality that represents the number of ounces of clay, that she uses to make one vase of either size.
Let the number of ounces of clay, that Elisa uses to make one vase of either size be denoted by "c".
Since, Elisa uses no more than 4.5 ounces of clay to make a small vase, The words "no more than" implies maximum, i.e., "either less than or equals to" and hence, for a small vase, "c" must be less than or equal to 4.5 oz, which gives \((c \leq 4.5)\) ...(1)
On the other hand, to make a large vase, she uses at least 12 ounces of clay where the words "at least" means minimum, i.e, "either more than or equals to" and hence, for a large vase,
Thus "c" is more than or equal to 12 oz, which gives \((c \geq 12)\)...(2)
Now, combining (1) and (2), we get [(c ≤ 4.5) or (c ≥ 12)] which is option "d".
compound inequality: In mathematics, a compound inequality is an inequality that combines or "compounds" two simple inequalities.To know more about compound inequality, click on the link below.
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What is the perimeter of triangle ABC?
Answers
Answer:
18 units
Step-by-step explanation:
Using the midpoint theorem, the perimeter of the triangle formed by joining the midpoints of the original triangle is exactly half the size in perimeter
What we mean in this case is that the perimeter of BAC IS two times the perimeter of EDF
perimeter of EDF is the sum of all the sides = 2 + 4 + 3 = 9
So the perimeter of BAC is 2 * 9 = 18
(1 point) Solve the system -22 54 dx dt X -9 23 with the initial value -10 o x(0) = -3 z(t) = x
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The solution to the system of differential equations is x(t) = -\(3e^{(31t)\) and z(t) = -\(3e^{(31t\)).
To solve the given system of differential equations, we'll begin by finding the eigenvalues and eigenvectors of the coefficient matrix.
The coefficient matrix is A = [[-22, 54], [-9, 23]]. To find the eigenvalues λ, we solve the characteristic equation det(A - λI) = 0, where I is the identity matrix.
det(A - λI) = [[-22 - λ, 54], [-9, 23 - λ]]
=> (-22 - λ)(23 - λ) - (54)(-9) = 0
=> λ^2 - λ(23 + 22) + (22)(23) - (54)(-9) = 0
=> λ^2 - 45λ + 162 = 0
Solving this quadratic equation, we find the eigenvalues:
λ = (-(-45) ± √((-45)^2 - 4(1)(162))) / (2(1))
λ = (45 ± √(2025 - 648)) / 2
λ = (45 ± √1377) / 2
The eigenvalues are λ₁ = (45 + √1377) / 2 and λ₂ = (45 - √1377) / 2.
Next, we'll find the corresponding eigenvectors. For each eigenvalue, we solve the equation (A - λI)v = 0, where v is the eigenvector.
For λ₁ = (45 + √1377) / 2:
(A - λ₁I)v₁ = 0
=> [[-22 - (45 + √1377) / 2, 54], [-9, 23 - (45 + √1377) / 2]]v₁ = 0
Solving this system of equations, we find the eigenvector v₁.
Similarly, for λ₂ = (45 - √1377) / 2, we solve (A - λ₂I)v₂ = 0 to find the eigenvector v₂.
The general solution of the system is x(t) = c₁e(λ₁t)v₁ + c₂e(λ₂t)v₂, where c₁ and c₂ are constants.
Using the initial condition x(0) = -3, we can substitute t = 0 into the general solution and solve for the constants c₁ and c₂.
Finally, substituting the values of c₁ and c₂ into the general solution, we obtain the particular solution for x(t).
Since z(t) = x(t), the solution for z(t) is the same as x(t).
Therefore, the solution to the system of differential equations is x(t) = \(-3e^{(31t)\) and z(t) = -\(3e^{(31t)\).
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