Answer:
"The Hulk is not green AND the Iron Man is not red"
Step-by-step explanation:
DeMorgan's laws state that the negation of an statement whose structure is "p OR q" is "not p AND not q", and similarly, that the negation of an statement whose structure is "p AND q" is "not p OR not q". The statement we want to negate in our case is "The Hulk is green OR the Iron Man is red". This is an statement whose structure is of the type "p OR q", where p would be "The Hulk is green", and q would be "the Iron Man is red". So according to DeMorgan's laws, its negation should be the statement "not p AND not q". To put them in common english, not p would be "The Hulk is NOT green", and not q would be "The Iron Man is NOT red". So the statement "not p AND not q" is simply "The Hulk is not green AND the Iron Man is not red".
To negate the statement 'the Hulk is green or Iron Man is red' using DeMorgan's Laws, you rephrase it as 'the Hulk is not green and Iron Man is not red'. The formal representation switches the disjunction to a conjunction and applies negation to each individual proposition.
To write the negation of the statement 'the Hulk is green or Iron Man is red' using DeMorgan's Laws, we first need to understand what these laws state. DeMorgan's Laws tell us how to move a negation across a conjunction (and) or a disjunction (or). According to DeMorgan's Laws, the negation of a conjunction is the disjunction of the negations, and the negation of a disjunction is the conjunction of the negations.
In formal logic, the original statement can be represented as (G \/ R), where G represents 'the Hulk is green' and R represents 'Iron Man is red'. Applying DeMorgan's Law to negate this statement would involve negating the entire proposition and then switching the 'or' to 'and'. Thus, the negation of the statement would be \u00AC(G \/ R) which translates to \u00ACG \u2227 \u00ACR using DeMorgan's Laws. This means 'the Hulk is not green and Iron Man is not red'.
Using DeMorgan's Laws is a way to express logical equivalence between propositions. By utilizing these transformation rules, one can simplify complex logical expressions or restate them in a different form without changing their meaning, making them powerful tools in formal logic and mathematics.
The measure of the angles of a triangle are x, 2x, and 3x.What
are the measures of the angles of the triangle ?
Answer: [tex]30^{\circ},\ 60^{\circ},\ 90^{\circ}[/tex]
Step-by-step explanation:
We know that the sum of measure of all the angles of a triangle is 180°.
Given : The measure of the angles of a triangle are x, 2x, and 3x.
Then, the sum of all the angle will be given by :-
[tex]x+2x+3x=180^{\circ}\\\\\Rightarrow\ 6x=180^{\circ}\\\\\Rightarrow\ x=\dfrac{180^{\circ}}{6}=30^{\circ}[/tex]
Then, the measures of angles of the triangle will be : [tex]30^{\circ},\ 2(30^{\circ}),\ 3(30^{\circ})[/tex]
i.e. [tex]30^{\circ},\ 60^{\circ},\ 90^{\circ}[/tex]
Suppose a simple random sample of size nequals=6464 is obtained from a population with mu equals 84μ=84 and sigma equals 16σ=16. (a) Describe the sampling distribution of x overbarx. (b) What is Upper P (x overbar greater than 87.6 )P x>87.6? (c) What is Upper P (x overbar less than or equals 79.2 )P x≤79.2? (d) What is Upper P (81.3 less than x overbar less than 87.6 )P 81.3
Answer:
a. [tex]\bar X[/tex] is distributed [tex]N(84;4)[/tex]
b. [tex]P(\bar X \geq 87.6) = 0.03593[/tex]
c. [tex]P(\bar X \leq 79.2) = 0.00820[/tex]
d. [tex]P(\79.2 \leq \bar X \leq 87.6) = 0.95587[/tex]
Step-by-step explanation:
a.
The central limit theorem states that, for large n, the sampling distribution of the sample mean is approximately normal with mean [tex]\µ[/tex] and variance [tex]\frac{\sigma^2}{n}[/tex], then, the sample mean is distributed as a normal random variable with means [tex]\mu_{\bar X}=\mu=84[/tex] and variance [tex]\sigma^2_{\bar X}=\frac{\sigma^2}{n}=\frac{16^2}{64}=4[/tex].
b.
[tex]P(\bar X \geq 87.6) = 0.03593[/tex]
c.
[tex]P(\bar X \leq 79.2) = 0.00820[/tex]
d.
[tex]P(\79.2 \leq \bar X \leq 87.6) = 0.95587[/tex]
Find the reduced row echelon form of the following matrices and then give the solution to the system that is represented by the augmented matrix. TO 4 7 0 6. a. 2 1 0 0 Lo 3 1 - 4 6. b. 54 30 71 8 6 2 -3 4 3 2 -10]
Answer:
a)
Reduced Row Echelon:
[tex]\left[\begin{array}{cccc}1&1/2&0&0\\0&1&7/4&0\\0&0&1&-4\end{array}\right][/tex]
Solution to the system:
[tex]x_3=-4\\x_2=-\frac{7}{4}x_3=7\\x_1=-\frac{1}{2}x_2=-\frac{7}{2}[/tex]
b)
Reduced Row Echelon:
[tex]\left[\begin{array}{cccc}4&3&0&7\\0&0&2&-17\\0&0&2&-17\end{array}\right][/tex]
Solution to the system:
[tex]x_3=-\frac{17}{2}\\x_1=\frac{7-3x_2}{4}[/tex]
x_2 is a free variable, meaning that it has infinite possibilities and therefore the system has infinite number of solutions.
Step-by-step explanation:
To find the reduced row echelon form of the matrices, let's use the Gaussian-Jordan elimination process, which consists of taking the matrix and performing a series of row operations. For notation, R_i will be the transformed column, and r_i the unchanged one.
a) [tex]\left[\begin{array}{cccc}0&4&7&0\\2&1&0&0\\0&3&1&-4\end{array}\right][/tex]
Step by step operations:
1. Reorder the rows, interchange Row 1 with Row 2, then apply the next operations on the new rows:
[tex]R_1=\frac{1}{2}r_1\\R_2=\frac{1}{4}r_2[/tex]
Resulting matrix:
[tex]\left[\begin{array}{cccc}1&1/2&0&0\\0&1&7/4&0\\0&3&1&-4\end{array}\right][/tex]
2. Set the first row to 1
[tex]R_3=-3r_2+r_3[/tex]
Resulting matrix:
[tex]\left[\begin{array}{cccc}1&1/2&0&0\\0&1&7/4&0\\0&0&1&-4\end{array}\right][/tex]
3. Write the system of equations:
[tex]x_1+\frac{1}{2}x_2=0\\x_2+\frac{7}{4}x_3=0\\x_3=-4[/tex]
Now you have the reduced row echelon matrix and can solve the equations, bottom to top, x_1 is column 1, x_2 column 2 and x_3 column 3:
[tex]x_3=-4\\x_2=-\frac{7}{4}x_3=7\\x_1=-\frac{1}{2}x_2=-\frac{7}{2}[/tex]
b)
[tex]\left[\begin{array}{cccc}4&3&0&7\\8&6&2&-3\\4&3&2&-10\end{array}\right][/tex]
1. [tex]R_2=-2r_1+r_2\\R_3=-r_1+r_3[/tex]
Resulting matrix:
[tex]\left[\begin{array}{cccc}4&3&0&7\\0&0&2&-17\\0&0&2&-17\end{array}\right][/tex]
2. Write the system of equations:
[tex]4x_1+3x_2=7\\2x_3=-17[/tex]
Now you have the reduced row echelon matrix and can solve the equations, bottom to top, x_1 is column 1, x_2 column 2 and x_3 column 3:
[tex]x_3=-\frac{17}{2}\\x_1=\frac{7-3x_2}{4}[/tex]
x_2 is a free variable, meaning that it has infinite possibilities and therefore the system has infinite number of solutions.
A cell phone company offers two different plans. Plan A costs $99 per month for unlimited talk and text. Plan B costs $0.20 per minute plus $0.10 per text message sent. You need to purchase a plan for your teenage sister. Your sister currently uses 1,750 minutes and sends 1,700 texts each month. What is your sister’s total cost under each of the two plans?
Use De Moivre's Formula to derive the identity sin(40) = 4 cos (0) sin(e) - 4 cos(e) sin (0) as well as a similar "quadruple angle" formula for cos(40). S eeeotoble to leave answer in polar or
By DeMoivre's theorem,
[tex]\cos(4\theta)+i\sin(4\theta)=(\cos\theta+i\sin\theta)^4[/tex]
Expanding the right side gives
[tex]\cos^4\theta+4i\cos^3\theta\sin\theta-6\cos^2\theta\sin^2\theta-4i\cos\theta\sin^3\theta+\sin^4\theta[/tex]
Equating imaginary parts tells us
[tex]\sin(4\theta)=4\cos^3\theta\sin\theta-4\cos\theta\sin^3\theta[/tex]
(Not sure what you mean by sin(e) and cos(e)...)
The brain volumes (cm3) of 20 brains have a mean of 1094.9 cm3 and a standard deviation of 128.9 cm3. Use the given standard deviation and the range rule of thumb to identify the limits separating values that are significantly low or significantly high. For such data, would a brain volume of 1392.7 cm3 be significantly high?
Answer:
Yes, a brain volume of [tex] 1392.7 cm^{3}[/tex] is significantly high.
Step-by-step explanation:
Given
Standard deviation = [tex] 128.9 cm^{3}[/tex]Mean = [tex] 1094.9 cm^{3}[/tex]The range rule of thumb indicates that the usual values are within 2 standard deviations from the mean:
minimum usual value = mean - 2 * standard deviation = [tex]1094.9 - 2*128.9=837.1 cm^{3}[/tex]
maximum usual value = mean + 2 * standard deviation = [tex]1094.9 + 2*128.9=1352.7 cm^{3}[/tex]
We can see that [tex] 1392.7 cm^{3}[/tex] is not between [tex] 837.1 cm^{3}[/tex] and [tex] 1352.7 cm^{3}[/tex], which indicates that this value is unusually high.
Using the range rule of thumb, it is found that:
The limit separating the values that are significantly low is 837.1 cm³.The limit separating the values that are significantly high is 1352.7 cm³.1392.7 cm³ is above 1352.7 cm³, thus, a brain volume of 1392.7 cm³ would be considered significantly high.----------------------------
The range rule of thumb states that:
Measures that are more than 2 standard deviations below the mean are significantly low.Measures that are more than 2 standard deviations above the mean are significantly high.----------------------------
The mean is 1094.9.The standard deviation is 128.9.----------------------------
2 standard deviations below the mean is given by:[tex]1094.9 - 2(128.9) = 837.1[/tex]
The limit separating the values that are significantly low is 837.1 cm³.----------------------------
2 standard deviations above the mean is given by:[tex]1094.9 + 2(128.9) = 1352.7[/tex]
The limit separating the values that are significantly high is 1352.7 cm³.----------------------------
1392.7 cm³ is above 1352.7 cm³, thus, a brain volume of 1392.7 cm³ would be considered significantly high.A similar problem is given at https://brainly.com/question/24126815
You paid $44 to a loan company for the use of $1,153 for 119 days, what annual rate of interest did they charge? (Assume a 360-day year.) If The annual rate of interest is 11.186 %. (Round to three decimal places.)
Answer:
11.70%
Step-by-step explanation:
Given;
Interest paid = $44
Principle amount = $1,153
Time = 119 days = [tex]\frac{\textup{119}}{\textup{365}}\textup{days}[/tex] = 0.326 years
Now,
the interest is calculated as:
interest = Principle × Rate of interest × Time
thus,
$44 = $1,153 × Rate of interest × 0.326
or
Rate of interest = 0.1170
or
in percentage = Rate × 100 = 0.1170 × 100 = 11.70%
A class survey found that 29 students watched television on Monday, 24 on Tuesday, and 25 on Wednesday. Of those who watched TV on only one of these days, 13 choose Monday, 9 chose Tuesday, and 10 chose Wednesday. Every student watched TV on at least one of these days, and 12 students watched TV on all three days. If 14 students watched TV on both Monday and Tuesday, how many students were there in the class
Final answer:
There are 108 students in the class, calculated using the data from the class survey with the principle of inclusion-exclusion.
Explanation:
To find out how many students are in the class, we should analyze the survey data provided. We know that:
29 students watched TV on Monday.
24 students watched TV on Tuesday.
25 students watched TV on Wednesday.
13 students watched TV on only Monday.
9 students watched TV on only Tuesday.
10 students watched TV on only Wednesday.
12 students watched TV on all three days.
14 students watched TV on both Monday and Tuesday.
Since 14 watched on both Monday and Tuesday, and 12 of those also watched on Wednesday, there are 14 - 12 = 2 who watched on Monday and Tuesday but not Wednesday.
We can now calculate the total number of students using the principle of inclusion-exclusion:
Start with the total number of students who watched TV each day: 29 + 24 + 25 = 78.
Subtract the students who were counted twice because they watched on two different days: 78 - 2 (from Monday and Tuesday) = 76.
Subtract the students who watched on all three days, as they have been counted three times: 76 - 12 = 64.
Add back the number of students who watched TV on all three days to account for their previous subtraction: 64 + 12 = 76.
Add the students who watched TV on only one specific day to avoid double-counting those who were included in the two and three days' viewership: 76 + 13 (Monday only) + 9 (Tuesday only) + 10 (Wednesday only) = 108.
Therefore, there are 108 students in the class.
You are visiting a rainforest, but unfortunately,your insect repellent has run out. As a result, at each second, a mosquito lands on your neck with probability 0.5. If a mosquito lands, it will bite you with probability 0.2, and it will never bother you with probability 0.8, independently of other mosquitoes. What is the expected time between successive bites?
Final answer:
The expected time between successive mosquito bites is calculated using the probabilities of landing and biting, resulting in an average time of 10 seconds between bites.
Explanation:
The expected time between successive mosquito bites can be calculated by considering the probabilities of two independent events: a mosquito landing on your neck and a mosquito bite, given that it has landed. As per the data provided, a mosquito lands with a probability of 0.5 per second, and out of those, it will bite with a probability of 0.2. The probability of getting bitten by a mosquito that has landed is then 0.5 (probability of landing) × 0.2 (probability of biting) = 0.1 per second.
To find the expected time between successive bites, we need to consider the inverse of this probability, which tells us that on average, you can expect to get bitten every 1/0.1 or 10 seconds.
The expected time between bites is 10 seconds.
To find the expected time between successive mosquito bites, we need to consider the probability of a mosquito biting you at any given second.
Given the probabilities:
Probability that a mosquito lands on your neck in a second: 0.5 = P(land)Probability that a mosquito bites you if it lands: 0.2 = P(bite | land)Probability that a mosquito doesn't bite you if it lands: 0.8 = [tex]P(not\ bite |\ land)[/tex]The combined probability that a mosquito both lands and bites in any given second is the product of the two probabilities:
[tex]P(bite) = P(land) * P(bite\ |\ land) = 0.5 * 0.2 = 0.1.[/tex]
To find the expected time between successive bites, we take the reciprocal of the probability of being bitten in a given second:
[tex]Expected time = \frac{1}{P(bite)} = \frac{1}{0.1} = 10\ seconds[/tex].
The expected time between successive mosquito bites is approximately 10 seconds.
Suppose you have a gift certificate worth $20 for one long-distance phone call. If the charge is $1.10 for the first minute and $0.42 for each additional minute, what is the longest that you can talk?
Answer:
46 minutes.
Step-by-step explanation:
You have a gift certificate worth $20 for one long-distance phone call.
The charge for the first minute = $1.10
Let the other additional minutes that you can talk = x
The charges for the x minutes = 0.42 per minute
the equation will be : 1.10 + 0.42x = 20
0.42x = 20 - 1.10
0.42x = 18.90
x = [tex]\frac{18.90}{0.42}[/tex]
x = 45
1 minute for $1.10 + 45 minutes for $0.42/min.
you can talk for 46 minutes.
The longest phone call you can make with a $20 gift certificate, given the cost structure of $1.10 for the first minute and $0.42 for each additional minute, is approximately 46 minutes.
Explanation:The subject of this question is Mathematics, specifically related to linear equations and budget constraints in the context of phone call charges. To find the longest time you can talk on the phone using your gift certificate, you'll need to understand the cost structure. The charge is $1.10 for the first minute - that leaves you with $18.90 from the gift certificate for the remaining time ($20 - $1.10). Each additional minute costs $0.42. Now, divide the remaining amount in your gift certificate by the cost per additional minute: $18.90 ÷ $0.42 ≈ 45 minutes. Adding back the first minute, the longest call you can make with the gift certificate is approximately 46 minutes
.
Learn more about Linear Equations here:
https://brainly.com/question/32634451
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What is the area of the region bounded between the curves y=6x^2 and y=4x?
Answer:
The area of the region between the curves y=6x^2 and y=4x is 8/27
Step-by-step explanation:
Use the diagram to visualize the problem, the area colored of blue is the one that needs to be found, let's do it in 3 parts:
Part 1: Find the intersection points of the curves
To do this we put both equations in one and solve it for x:
[tex]6x^2=4x[/tex]
[tex]6x^2-4x=0\\2x(3x-2)=0[/tex]
This equation has 2 possible solutions:
x=0 and x=2/3, so the interval for integration is 0 <= x <= 2/3
Part 2: Find the area below each curve
[tex]A_{blue}=\int\limits^0_{2/3} {6x^2} \, dx \\A_{blue}=2x^3[/tex], evaluate in 0 and 2/3
[tex]A_{blue}=\frac{16}{27}[/tex]
[tex]A_{red}=\int\limits^0_{2/3} {4x} \, dx \\A_{red}=2x^2[/tex], evaluate in 0 and 2/3
[tex]A_{red}=\frac{8}{9}[/tex]
Part 3: Substract the area of the blue curve (y=6x^2) to the area of the red curve (y=4x)
[tex]Area=\frac{8}{9}-\frac{16}{27}\\Area=\frac{8}{27}[/tex]
Solve the equation |x-2|-3=0 first by finding the zeros of
y=|x-2|-3 and then algebraically.
Answer:
The solution of given equation are -1 and 5.
Step-by-step explanation:
The given equation is
[tex]|x-2|-3=0[/tex]
We need to solve the above equation by finding the zeros of
[tex]y=|x-2|-3[/tex]
The vertex form of an absolute function is
[tex]y=a|x-h|+k[/tex]
where, a is constant and (h,k) is vertex.
Here, h=2, k=-3. So vertex of the function is (2,-3).
The table of values is
x y
0 -1
2 -3
4 -1
Plot these points on a coordinate plane and draw a V-shaped curve with vertex at (2,-3).
From the given graph it is clear that the graph intersect x-axis at -1 and 5. So, zeroes of the function y=|x-2|-3 are -1 and 5.
Therefore the solution of given equation are -1 and 5.
Now solve the given equation algebraically.
[tex]|x-2|-3=0[/tex]
Add 3 on both sides.
[tex]|x-2|=3[/tex]
[tex]x-2=\pm 3[/tex]
Add 2 on both sides.
[tex]x=\pm 3+2[/tex]
[tex]x=3+2[/tex] and [tex]x=-3+2[/tex]
[tex]x=5[/tex] and [tex]x=-1[/tex]
Therefore the solution of given equation are -1 and 5.
The charge to rent a trailer is $20 for up to 2 hours plus $8 per additional hour or portion of an hour. Find the cost to rent a trailer for 2.9 hours, 3 hours, and 8.5 hours. Then graph all ordered pairs, (hours, cost), for the function.
What is the cost to rent a trailer for 2.9 hours?
What is the cost to rent a trailer for 3 hours?
What is the cost to rent a trailer for 8.5 hours?
What is the cost to rent a trailer for 9 hours?
Graph all ordered pairs, (hours, cost), for the function.
Answer:
The cost to rent a trailer for 2.9 hours is $27.2.
The cost to rent a trailer for 3 hours is $28.
The cost to rent a trailer for 8.5 hours is $72.
The cost to rent a trailer for 9 hours is $76.
Step-by-step explanation:
It is given that the charge to rent a trailer is $20 for up to 2 hours plus $8 per additional hour or portion of an hour.
Let x be the number of hours.
The cost to rent a trailer for x hours is defined as
[tex]C(x)=\begin{cases}20 & \text{ if } x\leq 2 \\ 20+8(x-2) & \text{ if } x>2 \end{cases}[/tex]
For x>2,
[tex]C(x)=20+8(x-2)[/tex]
Substitute x=2.9 in the cost function.
[tex]C(x)=20+8(2.9-2)=27.2[/tex]
The cost to rent a trailer for 2.9 hours is $27.2.
Substitute x=3 in the cost function.
[tex]C(x)=20+8(3-2)=28[/tex]
The cost to rent a trailer for 3 hours is $28.
Substitute x=8.5 in the cost function.
[tex]C(x)=20+8(8.5-2)=72[/tex]
The cost to rent a trailer for 8.5 hours is $72.
Substitute x=9 in the cost function.
[tex]C(x)=20+8(9-2)=76[/tex]
The cost to rent a trailer for 9 hours is $76.
All ordered pairs, in the form of (hours, cost) are (2.9, 27.2), (3,28), (8.5, 72) and (9,76).
The graph of all ordered pairs is shown below.
Final answer:
The costs to rent a trailer for 2.9 and 3 hours are both $28, for 8.5 and 9 hours are both $76. To plot this function, one must mark these costs against the rental hours, showing a flat rate for the first two hours and additional charges thereafter.
Explanation:
The question asks for the cost of renting a trailer for varying numbers of hours and then requires plotting a graph with these costs against the hours. The rental system has a flat rate of $20 for the first 2 hours, and an additional charge of $8 for each hour or part of an hour thereafter.
Cost for 2.9 hours: Since any portion of an hour counts as a full hour, renting for 2.9 hours is essentially renting for 3 hours. The first 2 hours cost $20, and for the additional 0.9 (considered as 1) hour, it's $8. Total cost: $20 + $8 = $28.Cost for 3 hours: The calculation is the same as for 2.9 hours, so the cost is also $28.Cost for 8.5 hours: This includes the first 2 hours plus an additional 6.5 hours (considered as 7 hours). Therefore, the cost is $20 + ($8 x 7) = $20 + $56 = $76.Cost for 9 hours: This is calculated similarly, resulting in a cost of $20 + ($8 x 7) = $76.To graph all ordered pairs, plot points for each of the time periods mentioned with their corresponding costs. Note how the graph demonstrates incremental jumps after the first 2 hours, reflecting the additional $8 charge per hour or part thereof.
Pigeon Hole Principle :
Prove that given any set of n + 1 integers, there must be at least one pair among them whose difference is divisible by n
Step-by-step explanation:
When you divide an integer number by n, you get a remainder of either 0, 1, 2, ..., n-1 (for example 5 divided by 2 leaves a remainder of 1, or 13 divided by 5 leaves a remainder of 3, or 16 divided by 2 leaves a remainder of 0, and so on).
So there are n different remainders we could get when dividing an integer number by n. If we are given n+1 numbers, they each leave a certain remainder when divided by n. Since there are only n possible remainders, and we have n+1 numbers, by the pigeonhole principle we know there must be at least 2 numbers that leave the same remainder when divided by n. Call them numbers a and b, and let's call r the remainder they leave when divided by n. So both a and b are of the form:
[tex] a=kn+r[/tex] (for some integer k)
[tex] b=ln+r[/tex] (for some integer l)
(this is exactly what it means to leave a remainder of r when divided by n)
And so their difference is
[tex] a-b=kn+r-(ln+r)=kn-ln=(k-l)n[/tex]
Which is divisible by n by definition of being divisible (or think of it as a-b being a multiple of n, so it's divisible by n).
The supply equation for an olive oil press is
S(q) = 5q^2 + 1,000q + 100
where S(q) is the price in dollars at which q units are supplied. Find the quantity supplied in a month when the company sets the price of its olive oil press at $12,820
Answer:
12 units
Step-by-step explanation:
The given function is
[tex]S(q) = 5q^2 + 1000q + 100[/tex]
where, S(q) is the price in dollars at which q units are supplied.
We need to find the quantity supplied in a month when the company sets the price of its olive oil press at $12,820.
Substitute S(q)=12820 in the given function.
[tex]12820 = 5q^2 + 1000q + 100[/tex]
Subtract both sides by 12820.
[tex]0= 5q^2 + 1000q - 12720[/tex]
Taking out GCF.
[tex]0= 5(q^2 + 200q - 2544)[/tex]
Now solve the equation for q by splitting the middle term.
[tex]0= 5(q^2 + 212q-12q - 2544)[/tex]
[tex]0= 5(q(q + 212)-12(q + 212))[/tex]
[tex]0= 5(q + 212)(q-12)[/tex]
Using zero product property we get
[tex]q + 212=0\Rightarrow q=-212[/tex]
[tex]q-12=0\Rightarrow q=12[/tex]
q is number of units. So the value of q can not be negative.
Therefore the quantity supplied in a month when the company sets the price of its olive oil press at $12,820 is 12 units.
sin t sin 3t sin 5t = 1/4(-sin t + sin 3t +sin 7t - sin 9t).
Answer:
Step-by-step explanation:
Sin t . Sin 3t . Sin 5t = 1/4 [ - Sin t + Sin 3t + Sin 7t - Sin 9t ]
Take Right hand side and use the following formula
[tex]sin C - sin D = 2 Cos\left ( \frac{C+D}{2} \right )Sin\left ( \frac{C-D}{2} \right )[/tex]
[tex]Cos C - Cos D = 2 Sin\left ( \frac{C+D}{2} \right )Sin\left ( \frac{D-C}{2} \right )[/tex]
Take right hand side
[tex]\frac{1}{4}\left (Sin 3t - Sin t + Sin 7t - Sin 9t \right )[/tex]
[tex]\frac{1}{4}\left (2 Cos 2t Sin t +2 Sin (-t)Cos 8t \right )[/tex]
[tex]\frac{1}{4}\times 2 Sin t\left (Cos 2t-Cos8t \right )[/tex]
[tex]\frac{1}{4}\times 2 Sin t\ \times 2 \times Sin 5t\times 3t[/tex]
Sin t . Sin 3t . Sin 5t
So, LHS = RHS
A farmer looks out into the barnyard and sees the pigs and the chickens. He says to his daughter, "I count 153 heads and 346 feet. How many pigs and how many chickens are out there?"
Answer: There are 133 chickens and 20 pigs.
Step-by-step explanation:
Let x be the number of chickens and y be the number of pigs.
Given : Number of heads of pigs and chicken = 153
Number of feet = 346
Since one chicken has 2 legs and one pig has 4 legs.
By considering the given information, we have the following system of equations:-
[tex]x+y=153------(1)\\\\ 2x+4y=346---------(2)[/tex]
Multiply 2 on both sides of (1), we get
[tex]2x+2y=306-----(3)[/tex]
Subtract (3) from (2), we get
[tex]2y=40\\\\\Rightarrow\ y=\dfrac{40}{2}=20[/tex]
Put value of y in (1), we get
[tex]x+20=153\\\\\Rightarrow\ x=153-20=133[/tex]
Hence, there are 133 chickens and 20 pigs.
Final answer:
The problem is a system of linear equations in Mathematics, where we find that there are 133 chickens and 20 pigs in the barnyard after setting up and solving the equations based on the given number of heads and feet.
Explanation:
Let's denote the number of chickens as C and the number of pigs as P. Therefore, we have two equations based on the given information:
C + P = 153 (since each animal has one head)
2C + 4P = 346 (since chickens have 2 feet and pigs have 4 feet)
By solving these equations, we can find the values for C and P. Multiplying the first equation by -2 and adding it to the second equation, we eliminate C and get:
-2C - 2P = -306
2C + 4P = 346
-----------------
2P = 40
Dividing both sides by 2, we find that P = 20. Substituting this value back into the first equation, we get C + 20 = 153, which means C = 133.
Therefore, there are 133 chickens and 20 pigs in the barnyard.
Define a basis of a vector space
Answer:
Let V a vector space. And B a subset of elements in V.
B is a basis for V if satisfies the following conditions:
1. V= span(B). It means that every element of V can be written as a finite linear combination of elements of B.
2. B is a linear independent subset.
Joan Messineo borrowed $47,000 at a 5% annual rate of interest to be repaid over 3 years. The loan is amortized into three equal, annual, end-of-year payments.
a. Calculate the annual, end-of-year loan payment.
b. Prepare a loan amortization schedule showing the interest and principal breakdown of each of the three loan payments.
c. Explain why the interest portion of each payment declines with the passage of time.
Answer:
a) The value of the Annual Payment is A=$17,258.80
b) Is the picture in the attachment file
c) As you can see it in the picture with each payment, balance comes down, due it is the interest base, Interest portion comes down too.
Step-by-step explanation:
Hi
a) First of all, we are going to list the Knowns: [tex]VP=47000[/tex], [tex]i=5[/tex]% and [tex]n=3[/tex], Then we can use [tex]A=\frac{VP}{\frac{1-(1+i)^{-n} }{i} } =\frac{47000}{\frac{1-(1+0.05)^{-3} }{0.03} }=17258.80[/tex]. So this is the value of the Annual Payment
Joan Messines's annual payment on her $47,000 loan at 5% interest over 3 years is $17,158.11. The interest portion of each payment declines over time due to the decreasing loan balance, leading to a smaller interest calculation base in each subsequent year.
Joan Messines borrowed $47,000 at a 5% annual rate of interest to be repaid over 3 years. The loan is amortized into three equal, annual, end-of-year payments.
Calculation of the Annual Loan Payment
To calculate the annual payment, we use the formula for an annuity:
PV = PMT [(1 - (1 + r)^-n) / r]
Where:
PV is the present value of the loan (initial loan amount).
PMT is the annual payment.
r is the annual interest rate (expressed as a decimal).
n is the number of years.
Rearranging the formula to solve for PMT yields:
PMT = PV / [(1 - (1 + r)^-n) / r]
Substitute PV = $47,000, r = 0.05 (5%), and n = 3:
PMT = $47,000 / [(1 - (1 + 0.05)^-3) / 0.05]
PMT = $17,158.11 (rounded to the nearest cent).
Loan Amortization Schedule
Year 1: Interest = $47,000 * 5% = $2,350; Principal = $17,158.11 - $2,350 = $14,808.11; Remaining Balance = $47,000 - $14,808.11 = $32,191.89
Year 2: Interest = $32,191.89 * 5% = $1,609.59; Principal = $17,158.11 - $1,609.59 = $15,548.52; Remaining Balance = $32,191.89 - $15,548.52 = $16,643.37
Year 3: Interest = $16,643.37 * 5% = $832.17; Principal = $17,158.11 - $832.17 = $16,325.94; Remaining Balance = $16,643.37 - $16,325.94 = $317.43
Why the Interest Portion Declines Over Time
The interest portion of each payment declines with the passage of time because as the loan principal is paid down, there is a smaller balance on which interest is calculated. This results in a decreasing interest payment and an increasing principal payment with each subsequent payment until the loan is paid off.
The author of a book was told that he would have to cut the number of pages by 17% for the book to sell at a popular price and still make a profit.
Step 1 of 3: If these cuts were made, what percent of the original number of pages was in the final version?
Answer:
83%
Step-by-step explanation:
100% - 17% = 83%
Cutting the number of pages by 17% leaves 83% of the original number of pages.
After reducing the number of pages by 17%, the final version of the book contains 83% of the original number of pages.
Explanation:If the author of a book was told to cut the number of pages by 17%, we want to determine what percent of the original pages remained in the final version. To do this, we subtract the percentage of the pages cut from 100% (which represents the original number of pages).
100% - 17% = 83%.
Therefore, after a 17% cut, 83% percent of the pages are left in the final version of the book.
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Which of these statements are Qualitative, as opposed to Quantitative? Choose all correct answers. A. The flower is red. B. The bug is 5cm long. C. The candy was sour. D. You have three sisters. DQuestion 6 1 pts
Final answer:
The qualitative statements are A. The flower is red. and C. The candy was sour. The quantitative statements are B. The bug is 5cm long. and D. You have three sisters.
Explanation:
The qualitative statement refers to descriptions that do not involve numerical values or measurements. On the other hand, quantitative statements involve numerical values or measurements. Based on these definitions, the qualitative statements in the given options are:
A. The flower is red.
C. The candy was sour.
The quantitative statements in the given options are:
B. The bug is 5cm long.
D. You have three sisters.
Define the following propositions: .p: You drive over 65 miles per hour q You get a speeding ticket Translate the following English sentence into logical expressions using the definitions above (a) You drive over 65 miles per hour, but you do not get a speeding ticket. (b) If you do not drive over 65 miles per hour, then you will not get a speeding ticket (c) You get a speeding ticket, but you did not drive over 65 miles per hour.
Answer:
(a) [tex]p \wedge -q[/tex]
(b) [tex]\neg p \Rightarrow \neg q[/tex]
(c) [tex]q \wedge -p[/tex]
Step-by-step explanation:
(a) You drive over 65 miles per hour, but you do not get a speeding ticket, it can be represented by: [tex]p \wedge -q[/tex]
(b) If you do not drive over 65 miles per hour, then you will not get a speeding ticket, it can be represented by: [tex]\neg p \Rightarrow \neg q[/tex]
(c) You get a speeding ticket, but you did not drive over 65 miles per hour, can be represented by: [tex]q \wedge -p[/tex]
Describe how an infeasible solution occurs
Answer:
Described
Step-by-step explanation:
A solution becomes infeasible when no solution exit and which satisfies all the constraints. We will consider two basic types of infeasibility. The 1st we will call continuous infeasibility and the second one is discrete or integer infeasibility. Continuous infeasibility is the one where a non–MIP problem is infeasible. In this case the feasible region defined by the intersecting constraints is empty. Discrete or integer infeasibility is the one where a MIP problem has a feasible relaxation (note that a relaxation of a MIP is the problem we get when we drop the discreteness required on the variables) but the feasible region of the relaxation contains no solution that satisfies the discreteness requirement.
Final answer:
An infeasible solution occurs when constraints of a problem conflict with each other, making it impossible to find a solution that satisfies all conditions. This is common in linear programming and can happen due to poor design decisions or unreasonable premises. Brainstorming alternative solutions is essential for finding feasible approaches.
Explanation:
An infeasible solution occurs in problem-solving when the constraints of a problem are such that no possible solution satisfies all the conditions. In mathematics and particularly in linear programming, this can happen when the set of constraints creates a situation where no intersection point exists between all the constraint equations - essentially, the requirements for a solution are mutually exclusive and cannot be met simultaneously.
For example, a situation may arise where a design problem contains a premise that produces an unreasonable result, highlighting that some design choices may not align with the physical reality, and thus are infeasible. An extremely large Coulomb force in a physics problem, due to the assumption of an excessively large separated charge, is an instance where the result is obviously incorrect and infeasible based on known physical principles.
When approaching such problems, a common strategy is to brainstorm alternative solutions and explore a wide variety of ideas that could potentially solve the problem within realistic constraints. Some ideas may be discarded as infeasible early on if they conflict with the basic laws of nature or if they stand little chance of practical application. This process is especially useful in engineering and design fields, where practicality and adherence to real-world principles are crucial.
Find ℒ{f(t)} by first using a trigonometric identity. (Write your answer as a function of s.)
f(t) = 16cos(t−π/6)
ℒ{f(t)} = ?
Answer:
[tex]L\{f(t)\}=\frac{8(\sqrt3s+1)}{s^2+1}[/tex]
Step-by-step explanation:
Given : [tex]f(t)=16\cos (t-\frac{\pi}{6})[/tex]
To find : ℒ{f(t)} by first using a trigonometric identity ?
Solution :
First we solve the function,
[tex]f(t)=16\cos (t-\frac{\pi}{6})[/tex]
Applying trigonometric identity, [tex]\cos (A-B)=\cos A\cos B+\sin A\sin B[/tex]
[tex]f(t)=16(\cos t\cos (\frac{\pi}{6})+\sin t\sin(\frac{\pi}{6})[/tex]
[tex]f(t)=16(\frac{\sqrt3}{2}\cos t+\frac{1}{2}\sin t)[/tex]
[tex]f(t)=\frac{16}{2}(\sqrt3\cos t+\sin t)[/tex]
[tex]f(t)=8(\sqrt3\cos t+\sin t)[/tex]
We know, [tex]L(\cos at)=\frac{s}{s^2+a^2}[/tex] and [tex]L(\sin at)=\frac{a}{s^2+a^2}[/tex]
Applying Laplace in function,
[tex]L\{f(t)\}=8\sqrt3L(\cos t)+8L(\sin t)[/tex]
[tex]L\{f(t)\}=8\sqrt3(\frac{s}{s^2+1})+8(\frac{1}{s^2+1})[/tex]
[tex]L\{f(t)\}=\frac{8\sqrt3s+8}{s^2+1}[/tex]
[tex]L\{f(t)\}=\frac{8(\sqrt3s+1)}{s^2+1}[/tex]
Therefore, The Laplace transformation is [tex]L\{f(t)\}=\frac{8(\sqrt3s+1)}{s^2+1}[/tex]
Final answer:
In this college-level mathematics question, the task is to find ℒ{f(t)} by employing a trigonometric identity. By rewriting the function f(t) = 16cos(t−π/6) in terms of sine and utilizing a trigonometric identity, we find ℒ{f(t)} = 16s / (s^2 + 1).
Explanation:
To find ℒ{f(t)} by utilizing a trigonometric identity, first rewrite the function f(t) = 16cos(t−π/6) in terms of sine. Use the trigonometric identity cos(a) = sin(a + π/2) to rewrite cos(t−π/6) as sin(t−π/6 + π/2). This simplifies to sin(t−π/3). Thus, ℒ{f(t)} = 16 * ℒ{cos(t−π/6)}
= 16 * ℒ{sin(t−π/3)} = 16 * (s / (s^2 + 1)).
Therefore, the answer is 16s / (s^2 + 1).
Researchers measured the data speeds for a particular smartphone carrier at 50 airports. The highest speed measured was 75.6 Mbps. The complete list of 50 data speeds has a mean of x overbar equals 15.62 Mbps and a standard deviation of s equal s 20.03 Mbps.
a. What is the difference between carrier's highest data speed and the mean of all 50 data speeds?
b. How many standard deviations is that [the difference found in part (a)]?
c. Convert the carrier's highest data speed to a z score.
d. If we consider data speeds that convert to z scores between minus 2 and 2 to be neither significantly low nor significantly high, is the carrier's highest data speed significant?
Answer:
a) 59.98
b) 2.99
c) 2.99
d) Significantly High
Step-by-step explanation:
Part a)
Highest speed measured = x = 75.6 Mbps
Average/Mean speed = [tex]\overline{x}[/tex] = 15.62 Mbps
Standard Deviation = s = 20.03 Mbps
We need to find the difference between carrier's highest data speed and the mean of all 50 data speeds i.e. x - [tex]\overline{x}[/tex]
x - [tex]\overline{x}[/tex] = 75.6 - 15.62 = 59.98 Mbps
Thus, the difference between carrier's highest data speed and the mean of all 50 data speeds is 59.98 Mbps
Part b)
In order to find how many standard deviations away is the difference found in previous part, we divide the difference by the value of standard deviation i.e.
[tex]\frac{59.98}{20.03}=2.99[/tex]
This means, the difference is 2.99 standard deviations or in other words we can say, the Carrier's highest data speed is 2.99 standard deviations above the mean data speed.
Part c)
A z score tells us that how many standard deviations away is a value from the mean. We calculated the same in the previous part. Performing the same calculation in one step:
The formula for the z score is:
[tex]z=\frac{x-\overline{x}}{s}[/tex]
Using the given values, we get:
[tex]z=\frac{75.6-15.62}{20.03}=2.99[/tex]
Thus, the Carriers highest data is equivalent to a z score of 2.99
Part d)
The range of z scores which are neither significantly low nor significantly high is -2 to + 2. The z scores outside this range will be significant.
Since, the z score for carrier's highest data speed is 2.99 which is well outside the given range, i.e. greater than 2, we can conclude that the carrier's highest data speed is significantly higher.
Final answer:
The highest data speed recorded is 59.98 Mbps above the mean, and this difference equals approximately 2.996 standard deviations. This results in a z-score of 2.996, indicating that the carrier's highest data speed is significantly high.
Explanation:
a. The difference between the carrier's highest data speed and the mean of all 50 data speeds is found by subtracting the mean speed from the highest speed. This is calculated as 75.6 Mbps - 15.62 Mbps = 59.98 Mbps.
b. To find out how many standard deviations this difference is, we divide the difference by the standard deviation of the data speeds: 59.98 Mbps / 20.03 Mbps = 2.996 standard deviations.
c. The z-score for the carrier's highest data speed is calculated by subtracting the mean from the data speed and then dividing by the standard deviation: (75.6 Mbps - 15.62 Mbps) / 20.03 Mbps = 2.996.
d. Since the z-score of 2.996 is greater than 2, it suggests that the carrier's highest data speed is significantly higher than what is considered neither significantly low nor high. In other words, the carrier's highest data speed is significant.
Find the point of intersection of the pair of straight lines.
10x - 4y = 43
-3x - 3y = -15
(x, y) = ( , )
Answer:
(x,y) = ([tex]\frac{9}{2}[/tex],[tex]\frac{1}{2}[/tex])
Step-by-step explanation:
We have to find point of intersection of two lines.
the given equations of line are:
10x - 4y = 43 - (1)
-3x - 3y = -15 - (2)
Multiplying the first equation by 3 we have:
(10x - 4y = 43)×3 = 30x - 12 y = 129 - (3)
Multiplying second equation by 10 we have :
(-3x - 3y = -15)×10 = -30x -30y = -150 - (4)
Now, adding equation (3) and (4) we have:
-42y = -21
⇒ y = [tex]\frac{1}{2}[/tex]
Now, putting this value of y in equation (1), we have
10x - 2 = 43
⇒ 10x = 45
⇒x = [tex]\frac{9}{2}[/tex]
Hence, the intersection of given two lines is (x,y) = ([tex]\frac{9}{2}[/tex],[tex]\frac{1}{2}[/tex])
Drag a statement or reason to each box to complete this proof.
If −2(x+1)=8, then x=−5.
(Look at the photo)
Answer:
The proof will be as follows
Statement Reason
1. -2(x+1) = 8 Given
2. -2(x+1)/ -2 = 8/-2 Division Property of equality
3. x+1 = -4 Simplifying
4. x+1-1 = -4-1 Subtraction property of Equality
5. x = -5 Simplifying
The option of Distributive property of equality will not be used ..
Consider a business model in which two products, X and Y are produced. There are 160 pounds of material and 120 hours of labor available. It requires 4 pounds of material and 2 hours of labor to produce one unit of X. It requires 4 pounds of material and 6 units of labor to produce one unit of Y.
The profit for X is $35 and the profit for Y is $55. The business needs to know how many units of each product to produce to maximize resources and maximize profits. Express your answer in (x = , y = )
Maximize Z = 35x + 55y
subject to:
4x + 4y = 160
2x + 6y = 120
Answer:
The maximum profit is $1600 at x=30 and y=10.
Step-by-step explanation:
Let x be the number of units of product X.
y be the number of units of product Y.
The profit for X is $35 and the profit for Y is $55.
Maximize [tex]Z = 35x + 55y[/tex] ..... (1)
It requires 4 pounds of material and 2 hours of labor to produce one unit of X. It requires 4 pounds of material and 6 units of labor to produce one unit of Y.
Total material = 4x+4y
Total labor = 2x+6y
There are 160 pounds of material and 120 hours of labor available.
[tex]4x+4y\leq 160[/tex] .... (2)
[tex]2x+6y\leq 120[/tex] ..... (3)
[tex]x\geq 0,y\geq 0[/tex]
The related line of inequality (2) and (3) are solid line because the sign of equality "≤" contains all the point on line in the solution set.
Check the inequalities by (0,0).
[tex]4(0)+4(0)\leq 160[/tex]
[tex]0\leq 160[/tex]
This statement is true.
[tex]2x+6y\leq 120[/tex]
[tex]2(0)+6(0)\leq 120[/tex]
[tex]0\leq 120[/tex]
It means shaded region of both inequalities contain (0,0).
The extreme points of common shaded region are (0,0), (0,20), (40,0) and (30,10).
At (0,0),
[tex]Z = 35(0) + 55(0)=0[/tex]
At (0,20),
[tex]Z = 35(0) + 55(20)=110[/tex]
At (40,0),
[tex]Z = 35(40) + 55(0)=140[/tex]
At (30,10),
[tex]Z = 35(30) + 55(10)=1600[/tex]
Therefore the maximum profit is $1600 at x=30 and y=10.
In the following pair of polynomials, the second polynomial is a factor of the first. Factor completely x^3+3x^2-18x , x-4
Answer:
The required factors are: x, (x + 6) and (x - 3).
Step-by-step explanation:
As per the question,
The given polynomial is:
[tex]x^{3}+3x^{2}-18x[/tex]
Now,
BY factorization, we get
[tex]x^{3}+3x^{2}-18x[/tex]
[tex]=x(x^{2}+3x-18)[/tex]
By splitting the mid-term, that is split 3x like:
3x = 6x - 3x
Therefore,
[tex]x(x^{2}+6x-3x-18)[/tex]
Now on further solving by taking common factor out, we get
[tex]=x[x(x+6)-3(x+6)][/tex]
[tex]=x(x+6)(x-3)[/tex]
Therefore, the given second polynomial (x - 4), is not a factor of given polynomial [tex]x^{3}+3x^{2}-18x[/tex].
Hence, the given polynomial has three factor x, (x + 6) and (x - 3).
A patient is instructed to take three 50-mcg tablets of pergolide mesylate (Permax) daily. How many milligrams of the drug would the patient receive weekly?
Answer:
The patient would receive 1.05mg of the drug weekly.
Step-by-step explanation:
First step: How many mcg of the drug would the patient receive daily?
The problem states that he takes three doses of 50-mcg a day. So
1 dose - 50mcg
3 doses - x mcg
x = 50*3
x = 150 mcg.
He takes 150mcg of the drug a day.
Second step: How many mcg of the drug would the patient receive weekly?
A week has 7 days. He takes 150mcg of the drug a day. So:
1 day - 150mcg
7 days - x mcg
x = 150*7
x = 1050mcg
He takes 1050mcg of the drug a week.
Final step: Conversion of 1050 mcg to mg
Each mg has 1000 mcg. How many mg are there in 1050 mcg? So
1mg - 1000 mcg
xmg - 1050mcg
1000x = 1050
[tex]x = \frac{1050}{1000}[/tex]
x = 1.05mg
The patient would receive 1.05mg of the drug weekly.
The patient receives 0.15 milligrams of pergolide mesylate daily and thus 1.05 milligrams weekly.
Explanation:To calculate the weekly dosage of pergolide mesylate in milligrams, we first need to understand the daily dosage. The patient takes three 50-mcg tablets daily, which is 150 mcg daily. Knowing that 1 mg (milligram) is equal to 1000 mcg (micrograms), we can convert the daily dose to milligrams by dividing by 1000 which is 0.15 mg daily. To find the weekly dosage, we multiply this daily total by 7 (since there are 7 days in a week), which gives us 1.05 milligrams weekly.
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