Answer:
(A) 1.52 m²
Step-by-step explanation:
As per the given data of the question,
Height of a person = 5 feet
As we know that 1 feet = 30.48 cm
∴ Height = 152.4 cm
Weight of a person = 120 pounds
And we know that 1 pound = 0.453592 kg
∴ Weight = 54.4311 kg
The Mosteller formula to calculate body surface area (BSA):
[tex]BSA(m^{2})=\sqrt{\frac{Height (cm)\times Weight(kg)}{3600}}[/tex]
Therefore,
[tex]BSA=\sqrt{\frac{Height (cm)\times Weight(kg)}{3600}}[/tex]
[tex]BSA=\sqrt{\frac{152.4\times 54.4311}{3600}}[/tex]
[tex]BSA= 1.517 m^{2} = 1.52 m^{2}[/tex]
Hence, the body surface area of a person = 1.52 m²
Therefore, option (A) is the correct option.
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])
An electricity company charges its customers a fixed base charge of $6 a month, plus 10 cents per kilowatt-hour (kWh) for the first 500 kWh, 11 cents per kWh for the next 500 kWh, and 15 cents for all additional kWh. Express the monthly cost E as a function of the amount x of electricity used.
Answer:
[tex]E(k)= \left[ \begin{array}{c} {6+0.10k\,\,if\,\,0\leq k \leq500} & 6+500\times0.10+0.11(k-500)\,\,if\,\,500<k\leq1000& 6+500\times0.10+500\times0.11+0.15(k-1000)\,\,if\,\,k>1000\end{array}[/tex]
Step-by-step explanation:
The cost function has 3 branches,
Consumptions less than 500Consumptions less than 1000consumptions above a 1000So the first branch the consumer pays 6 dollars plus .10cents for any additional kWh (k)
In the second, they pay the same as the first up to 500kWh, and after that they pay 0.11 for the additional kWh above 500: (k-500) but bellow 1000
In the third branch, for consumptions above 1000, they pay the fix amount, plus .10 for the first 500 ([tex]500\times0.10[/tex]) , .11 for the additional 500 ([tex]500\times0.11[/tex]) , and finally 0.15 for consumptions above 1000: [tex]0.15(k-1000)[/tex]
Final answer:
The monthly cost of electricity, E, as a function of the electricity used, x, is defined using a piecewise function with different rates for usage tiers, including a fixed base charge and variable rates for different consumption brackets.
Explanation:
To express the monthly cost E as a function of the amount x of electricity used by the customer, we consider the following rates:
A fixed base charge of $6 per month.10 cents per kilowatt-hour (kWh) for the first 500 kWh.11 cents per kWh for the next 500 kWh (501 to 1000 kWh).15 cents per kWh for all additional kWh (above 1000 kWh).Therefore, the function is a piecewise function defined as:
E(x) =
6 + 0.10x if 0 ≤ x ≤ 5006 + (0.10 × 500) + 0.11(x - 500) if 500 < x ≤ 10006 + (0.10 × 500) + (0.11 × 500) + 0.15(x - 1000) if x > 1000This function helps calculate the monthly electricity bill based on the usage levels. Remember that the first bracket is for the initial 500 kWh, the second bracket is for 501 to 1000 kWh, and the last bracket applies to usage above 1000 kWh.
Use a direct proof to show that the product of two odd integers is odd.
Step-by-step explanation:
A direct proof is a method that takes an statement p, which we assume to be true, and use it to show directly that another statement q is true. So this method has the following steps:
Assume the statement p is trueUse what we know about p and other facts as necessary to deduce that another statement q is true, that is show p ⇒ q is true.Fact that we need to use:
Every odd integer can be written in the form 2m + 1 for some unique other integer m
Let p be the statement a and b be odd integers and q be the statement that the product of a and b is odd.
Proposition if a and b are odd, then the product of a and b is odd
Proof: Assume that a and b are odd integers, the by definition a = 2m + 1 and b = 2n + 1 for some integers m and n. we will now use this to show that the product of a and b is odd.
[tex]a\cdot b= (2m+1) \cdot (2n+1)\\a\cdot b = 2m\cdot 2n+2m+2n+1\\a\cdot b =4mn+2m+2n+1\\a\cdot b = 2(2mn+2m+2n) +1\\\:If \:k=2mn+2m+2n\\a\cdot b = 2k+1[/tex]
Hence we have shown that the product of a and b is odd since 2k + 1 is and odd integer. Therefore we have shown that p ⇒ q and so we have completed our proof.
Answer:
Step-by-step explanation:
The proof by the direct method that the product of two odd numbers is odd integer number, is the following:
Let [tex]z_1[/tex] and [tex]z_2[/tex] be two odd integers, then [tex]z_1 = 2a+1[/tex] and [tex]z_2 = 2b +1[/tex], for some integers a and b.
[tex]z_1z_2 = (2a + 1) (2b + 1)\\\\z_1z_2 = 4ab + 2a + 2b + 1\\\\z_1z_2 = 2 (2ab + a + b) +1\\\\z_1z_2 = 2n + 1[/tex]
where [tex]n = 2ab + a + b[/tex], which guarantees that [tex]n[/tex] is an integer number. In this way, [tex]z_1z_2[/tex] is an odd integer.
Determine which of the following sets of three points constitute the vertices of a right triangle: (a) 3 + 5i,2 +2i,5i; (b)2i,3 + 5i,4 + i; (c)6 +4i,7 + 5i, 8 +4i
Answer:
Option (c) is correct
Step-by-step explanation:
Case (a)
A = 3 + 5i = (3, 5)
B = 2 + 2i = (2, 2)
C = 5i = (0, 5)
Use the distance formula to find the distance between two points
[tex]AB = \sqrt{(2-3)^{2}+(2-5)^{2}}=\sqrt{10}[/tex]
[tex]BC = \sqrt{(0-2)^{2}+(5-2)^{2}}=\sqrt{13}[/tex]
[tex]CA = \sqrt{(0-3)^{2}+(5-5)^{2}}=\sqrt{9}[/tex]
For the triangle to be right angles triangle
[tex]BC^{2}=AB^{2}+CA^{2}[/tex]
Here, it is not valid, so these are not the points of a right angled triangle.
Case (b)
A = 2i = (0, 2)
B = 3 + 5i = (3, 5)
C = 4 + i = (4, 1)
Use the distance formula to find the distance between two points
[tex]AB = \sqrt{(3-0)^{2}+(5-2)^{2}}=\sqrt{18}[/tex]
[tex]BC = \sqrt{(4-3)^{2}+(1-5)^{2}}=\sqrt{17}[/tex]
[tex]CA = \sqrt{(4-0)^{2}+(1-2)^{2}}=\sqrt{17}[/tex]
For the triangle to be right angles triangle
[tex]AB^{2}=BC^{2}+CA^{2}[/tex]
Here, it is not valid, so these are not the points of a right angled triangle.
Case (c)
A = 6 + 4i = (6, 4)
B = 7 + 5i = (7, 5)
C = 8 + 4i = (8, 4)
Use the distance formula to find the distance between two points
[tex]AB = \sqrt{(7-6)^{2}+(5-4)^{2}}=\sqrt{2}[/tex]
[tex]BC = \sqrt{(8-7)^{2}+(4-5)^{2}}=\sqrt{2}[/tex]
[tex]CA = \sqrt{(8-6)^{2}+(4-4)^{2}}=\sqrt{4}[/tex]
For the triangle to be right angles triangle
[tex]CA^{2}=BC^{2}+AB^{2}[/tex]
Here, it is valid, so these are the points of a right angled triangle.
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]
One group of scientists (Group A) has measured a time for a particular chemical reaction to be completed to be 7.34 ± 0.05 s. A second group of scientists (Group B) complete a similar experiment and measure the time to be 7.38 ± 0.03s. A third group (Group C) measured the time to be 7.46 ±0.06s. Do results of Group A and B agree with each other within the experimental uncertainty? Do results of Group A and C agree with each other within the experimental uncertainty?
Answer:
1. Group A and B agree with each other.
2. Group A and C do not agree with each other.
Step-by-step explanation:
When we are analizing this problem, we will see what are the ranges of this measured times. Since we are taking into account the error we can see that :
Group A varies from 7.34-0.05 to 7.34+0.05. So the limits are (7.29 ;7.39)Group B varies from 7.38-0.03 to 7.38+0.03. So the limits are (7.35; 7.41)Group C varies from 7.46-0.06 to 7.46+0.06. So the limits are (7.40; 7.52)Question 1 is about the overlapping response in Group A and Group B. And yes, we have an overlap between 7.35 to 7.39. Among this times both group A and B are in agree with each other within the experimental uncertainty.
Question 2 is now referring to Group A and Group C. And no, there isn't any common time where both groups agree with each other.
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?
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
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.
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.
Learn more about Percentage Calculation here:https://brainly.com/question/32197511
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The purpose of this assignment is to make sures linear equations. This skill is required when Chapter 5 is covered. Problem Statement Given is a linear Equation 1 in the format of y mx+b. Equation 1, y = 46.888 +0.78X Plot Equation 1 on a graph paper. Label all axes and markers on the graph. 1. Save your graph in a pdf file zero point is issued for non pdf file format 2. Click on the assignment link. W01 A3a. below to upload your pdf file via attachment by the due date. 3. If you do not see the assignment link it is because the assignment is past due. Wk01 A3a De as
Answer: The plot will be linear as described by the equation of the problem
Step-by-step explanation:
The first thing to do is to determine values of X and place them in one column. The following column should include all values solved by the equation in terms of y. Once you plot the graph, you should have something as the graph attached. To save it as a pdf, simply scan your graph paper to pdf and follow the instructions for assignment submission.
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.
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.
An eight-sided die, which may or may not be a fair die, has four colors on it; you have been tossing the die for an hour and have recorded the color rolled for each toss. What is the probability you will roll a yellow on your next toss of the die? Express your answer as a simplified fraction or a decimal rounded to four decimal places.
Answer:
Case 1
None of the sides is yellow.
Probability of rolling a yellow on your next toss: zero %
Case 2
All the sides are yellow.
Probability of rolling a yellow on your next toss: 100 %
Case 3
At least one of the sides is yellow.
[tex]\frac{q}{N}\times100\%[/tex]
where
N = total number of tosses in one hour.
q = number of tosses you rolled a yellow.
Step-by-step explanation:
Case 1
None of the sides is yellow.
Probability of rolling a yellow on your next toss: zero %
Case 2
All the sides are yellow.
Probability of rolling a yellow on your next toss: 100 %
Case 3
At least one of the sides is yellow.
As you do not know if the die is fair or not, the only way to approximate a probability of rolling a yellow is by making a table of frequencies and record the times you have rolled yellow.
If the number of tosses made in an hour is big enough as to draw a conclusion, then according to the Law of Large Numbers, the probability of rolling a yellow in one toss of the die should be
[tex]\frac{q}{N}\times100\%[/tex]
where
N = total number of tosses
q = number of tosses you rolled a yellow.
Now, suppose you want to roll the dice once more.
As the event of rolling a die is independent of the previous tosses, this means that the probable outcome of the event does not depend on the previous results. So, the probability remains the same.
Probability of tossing a yellow on your next toss of the die
[tex]\frac{q}{N}\times100\%[/tex]
where
N = total number of previous tosses
q = number of tosses you rolled a yellow.
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 ..
The Big Wave Surf Shop sells 8 ounce containers of sunscreen for $8 each. The total daily cost associated with the sale of this item yesterday was $200. What is the shop’s profit if it sold 150 units of sunscreen yesterday?
Answer:
The shop’s profit if it sold 150 units of sunscreen yesterday is $1000
Step-by-step explanation:
The total daily cost associated with the sale of this item yesterday was $200.
Cost of 1 unit = $8
So, Cost of 150 units = [tex]8 \times 150[/tex]
= [tex]1200[/tex]
So, revenue = $1200
Cost = $200
So Profit = Revenue - Cost
Profit = $1200-$200
Profit = $1000
Hence the shop’s profit if it sold 150 units of sunscreen yesterday is $1000
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)...)
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
.
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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 fraction of defective integrated circuits produced in a photolithography process is being studied. A random sample of 300 circuits is tested, revealing 17 defectives. (a) Calculate a 95% two-sided confidence interval on the fraction of defective circuits produced by this particular tool. Round the answers to 4 decimal places. less-than-or-equal-to p less-than-or-equal-to (b) Calculate a 95% upper confidence bound on the fraction of defective circuits. Round the answer to 4 decimal places. p less-than-or-equal-to
Answer:
(a) The confidence interval is: 0.0304 ≤ π ≤ 0.0830.
(b) Upper confidence bound = 0.0787
Step-by-step explanation:
(a) The confidence interval for p (proportion) can be calculated as
[tex]p \pm z*\sigma_{p}[/tex]
[tex]\sigma=\sqrt{\frac{\pi*(1-\pi)}{N} }\approx\sqrt{\frac{p(1-p)}{N} }[/tex]
NOTE: π is the proportion ot the population, but it is unknown. It can be estimated as p.
[tex]p=17/300=0.0567\\\\\sigma=\sqrt{\frac{p(1-p)}{N} }=\sqrt{\frac{0.0567(1-0.0567)}{300} }=0.0134[/tex]
For a 95% two-sided confidence interval, z=±1.96, so
[tex]\\LL = p-z*\sigma=0.0567 - (1.96)(0.0134) = 0.0304\\UL =p+z*\sigma= 0.0567 + (1.96)(0.0134) = 0.0830\\\\[/tex]
The confidence interval is: 0.0304 ≤ π ≤ 0.0830.
(b) The confidence interval now has only an upper limit, so z is now 1.64.
[tex]UL =p+z*\sigma= 0.0567 + (1.64)(0.0134) = 0.0787[/tex]
The confidence interval is: -∞ ≤ π ≤ 0.0787.
Final answer:
To calculate a confidence interval for the fraction of defective circuits, use the formula for the confidence interval of a proportion. The 95% two-sided confidence interval for the fraction of defective circuits is 0.0182 ≤ p ≤ 0.0951. The 95% upper confidence bound on the fraction of defective circuits is 0.0951.
Explanation:
To calculate a confidence interval for the fraction of defective circuits, we can use the formula for the confidence interval of a proportion. Let p-hat be the proportion of defectives in the sample, which is equal to 17/300 = 0.0567. We can calculate the standard error as [tex]\sqrt{((p-hat*(1-p-hat))/n)[/tex], where n is the sample size.
Using a 95% confidence level, we can find the critical value from the standard normal distribution, which is approximately 1.96. The lower bound of the confidence interval is given by p-hat - z*(standard error), and the upper bound is given by p-hat + z*(standard error).
Lower bound: 0.0567 - [tex]1.96*(\sqrt{((0.0567*(1-0.0567))/300))[/tex] = 0.0182Upper bound: 0.0567 + [tex]1.96*(\sqrt{((0.0567*(1-0.0567))/300))[/tex]= 0.0951Therefore, the 95% two-sided confidence interval for the fraction of defective circuits is 0.0182 ≤ p ≤ 0.0951. The 95% upper confidence bound on the fraction of defective circuits is 0.0951.
Solve the differential equation x^2 y"-xy' +y 0
Answer:
[tex]y(x)\ =\ \sqrt{x}[C_1cos\dfrac{\sqrt{3}}{2}logx+C_2sin\dfrac{\sqrt{3}}{2}logx][/tex]
Step-by-step explanation:
Given differential equation is
[tex]x^2y"-xy'+y=0[/tex] (1)
Let's assume that
[tex]x=e^t[/tex]
[tex]=>\ t\ =\ logx[/tex]
then,
[tex]\dfrac{dx}{dt}=e^t[/tex]
[tex]and\ \dfrac{d^2x}{dt^2}=e^t[/tex]
We can write,
[tex]\dfrac{dy}{dx}=\dfrac{dy}{dt}.\dfrac{dt}{dx}[/tex]
[tex]=e^{-t}\dfrac{dy}{dt}[/tex]
Similarly,
[tex]\dfrac{d^2y}{dt^2}=\dfrac{d^2y}{dt^2}.\dfrac{dt^2}{dx^2}[/tex]
[tex]=e^{-2t}.\dfrac{d^2y}{dt^2}[/tex]
Putting these values in equation (1), we will get
[tex]e^{2t}.e^{-2t}.\dfrac{d^y}{dt^2}-e^t.e^{-t}\dfrac{dy}{dt}+y=0[/tex]
[tex]=>\dfrac{d^2y}{dt^2}-\dfrac{dy}{dt}+y=0[/tex]
So, the characteristics equation can be given as
[tex]D^2-D+1=0[/tex]
[tex]=>D\ =\ \dfrac{1+\sqrt{1-4}}{2}\ or\ \dfrac{1-1\sqrt{1-4}}{2}[/tex]
[tex]=>D=\ \dfrac{1}{2}+i\dfrac{\sqrt{3}}{2}\ or\ \dfrac{1}{2}-i\dfrac{\sqrt{3}}{2}[/tex]
Hence, the general solution of the equation can be give by
[tex]y(t)\ =\ e^{\dfrac{t}{2}}[C_1cos\dfrac{\sqrt{3}}{2}t+C_2sin\dfrac{\sqrt{3}}{2}t][/tex]
Now, by putting the value of t in above solution, we will have
[tex]y(x)\ =\ e^{\dfrac{1}{2}logx}[C_1cos\dfrac{\sqrt{3}}{2}logx+C_2sin\dfrac{\sqrt{3}}{2}logx][/tex]
[tex]y(x)=\ \sqrt{x}[C_1cos\dfrac{\sqrt{3}}{2}logx+C_2sin\dfrac{\sqrt{3}}{2}logx][/tex]
Hence, the solution of above given differential equation can be given by
[tex]y(x)=\ \sqrt{x}[C_1cos\dfrac{\sqrt{3}}{2}logx+C_2sin\dfrac{\sqrt{3}}{2}logx][/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.
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
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.
Assume that the readings on the thermometers are normally distributed with a mean of 0 C and a standard deviation of 1 C. A thermometer is randomly selected and tested. If 1.7% of the thermometers are rejected because they have readings that are too low, but all other thermometers are acceptable, find the reading that separates the rejected thermometers from the others.
Answer: -2.12°C
Step-by-step explanation:
Let x denotes the reading of the thermometers .
We assume that the readings on the thermometers are normally distributed.
Let a be the reading that separates the rejected thermometers from the others.
Given: Population mean : [tex]\mu=0[/tex]
Standard deviation: [tex]\sigma= 1[/tex]
Also, [tex]P(x<a)=0.017[/tex]
By using the z-table , the z-value corresponds to the p-value (one -tailed)0.017 is [tex]\pm2.12[/tex].
Now, [tex]z=\dfrac{a-\mu}{\sigma}[/tex]
i.e. [tex]\pm2.12=\dfrac{a-0}{1}[/tex]
i.e. [tex]\pm2.12=a[/tex]
For left tailed , [tex]a=-2.12[/tex]
It means the reading that separates the rejected thermometers from the others = -2.12°C.
Compare the values of the underlined digits 631,485 and 682 the underlined is 6.
Answer:
First one: Its place value is 600,000.
Second one: Its place value is 600.
Step-by-step explanation:
A normal curve with mean = 25 has an area of .3531 between 25 and 34. What is the standard deviation for this normal curve?
Answer:
The desviation is 8 4/7 or 8.571
Step-by-step explanation:
The conversion for any variable X to a standard z is
[tex]Z=\frac{X-\[Mu]}\\{\[Sigma]}[/tex]
where mu is the mean and sigma de desviation
You can find the value of Z whith the tables of the accumulated probability function. The accumulated probability for the mean is 0.5 .Remenber that the accumlated probability function represent the area at the left of an abscissa. Then
0.5+0.3531=0.8531
Acording to the accumulated probability function table, a Z=1.05 has an area of 0.8531 at its left.
Now it is only solving the equation
[tex]\[Sigma]=\frac{X-\[Mu]}{Z}[/tex]
σ=[tex]\frac{34-25}{1.05}[/tex]
σ=8.571
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.
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]