Step-by-step explanation:
We will prove by contradiction. Assume that [tex]2^n + 1[/tex] is an odd prime but n is not a power of 2. Then, there exists an odd prime number p such that [tex]p\mid n[/tex]. Then, for some integer [tex]k\geq 1[/tex],
[tex]n=p\times k.[/tex]
Therefore
[tex]2^n + 1=2^{p\times k} + 1=(2^{k})^p + 1^p.[/tex]Here we will use the formula for the sum of odd powers, which states that, for [tex]a,b\in \mathbb{R}[/tex] and an odd positive number [tex]n[/tex],
[tex]a^n+b^n=(a+b)(a^{n-1}-a^{n-2}b+a^{n-3}b^2-...+b^{n-1})[/tex]
Applying this formula in 1) we obtain that
[tex]2^n + 1=2^{p\times k} + 1=(2^{k})^p + 1^p=(2^k+1)(2^{k(p-1)}-2^{k(p-2)}+...-2^{k}+1)[/tex].
Then, as [tex]2^k+1>1[/tex] we have that [tex]2^n+1[/tex] is not a prime number, which is a contradiction.
In conclusion, if [tex]2^n+1[/tex] is an odd prime, then n must be a power of 2.
A nurse is preparing to administer dextrose 5% water (D5W) 250 ml IV to infuse over 2 hr. The nurse should set the IV pump to deliver how many ml/hr?
Answer:
The nurse should set the IV pump to deliver 125 ml/hr.
Step-by-step explanation:
The problem states that a nurse is preparing an IV pump to administer 250ml over 2 hours. So how many ml should be administered each hour?
This problem can be solved by this following rule of three.
250 ml - 2 hours
x ml - 1 hours
[tex]2x = 250[/tex]
[tex]x = \frac{250}{2}[/tex]
[tex]x = 125[/tex]ml.
The nurse should set the IV pump to deliver 125 ml/hr.
To determine the intermittent IV infusion rate, divide the total volume of the fluid (dextrose 5% water) by the total time of infusion. Here, it would be 250 ml divided by 2 hours, which equals 125 ml per hour.
Explanation:To determine how many ml per hour a nurse should set the IV pump to deliver dextrose 5% water (D5W), you should divide the total volume by the total time. In this case, the total dextrose volume is 250 ml and it should be infused over 2 hours. Using the formula:
Total Volume / Total Time = ml per hour250 ml / 2 hr = 125 ml/hrSo, the nurse should set the IV pump to deliver 125 ml per hour of dextrose.
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For a normal distribution with mean equal to 31.5 and standard deviation equal to 11, what is the area under the curve that is between 35 and 45?
Answer:
The area is given by the following integral: [tex]\int\limits^{45}_{35} \frac{1}{\sqrt{242\pi}}e^{-\frac{(x-31.5)^2}{242}} dx[/tex], which can be approximated by: 0.265313
Step-by-step explanation:
A normal distribution is defined as:
[tex]f(x)=\frac{1}{\sqrt{2\pi \sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}[/tex]
, where the greek letter sigma stands for the standard deviation and mu for the mean. Since in our problem we have a mean = 31.5 and a standard deviation = 11, then we can write this function as:
[tex]f(x)=\frac{1}{\sqrt{242\pi}}e^{-\frac{(x-31.5)^2}{242}}[/tex]
Now, we need to find the area below this function, between 35 and 45, and in order to do this, we need to integrate the function. The normal distribution does not has an exact closed form integral, therefore we will have to solve the integral in a software that allows for numerical calculations (I used the online software Wolfram|Alpha).
[tex]\int\limits^{45}_{35} \frac{1}{\sqrt{242\pi}}e^{-\frac{(x-31.5)^2}{242}} dx =0.265313[/tex]
A solution of ibuprofen contains 150g in 800mL What is the percent strength of this solution?
Answer: 18.75%
Step-by-step explanation:
The percent strength of a solution is given by :-
[tex]\% \text{ Strength}=\dfrac{\text{Mass of solute in g}}{\text{Volume of solution in mL}}\times100[/tex]
Given : A solution of ibuprofen contains 150g in 800mL .
Then, the percent strength of this solution will be :-
[tex]\% \text{ Strength}=\dfrac{150}{800}\times100\\\\\Rightarrow\ \% \text{ Strength}=18.75\%[/tex]
Hence, the percent strength of this solution = 18.75%
in june 2012, about 2.4 billion people used the internet - 100 times more than the number of people who used it in June 1995. If the growth in Interne usage is linear (arithmetic), about how many people were added on average each year from 1995 to 2012?
Answer:
About 139,764,705.9 users were added, on average, per year, from 1995 to 2012.
Step-by-step explanation:
The problem states that in june 2012, about 2.4 billion people used the internet - 100 times more than the number of people who used it in June 1995.
So, in June 2012, 2.4 billion people used the internet.
In June 1995, [tex]\frac{2,400,000,000}{100} = 24,000,000[/tex] = 24 million people used the internet.
About how many people were added on average each year from 1995 to 2012?
[tex]2012 - 1995 = 17[/tex]. There were 17 years between 1995 and 2012.
[tex]2,400,000,000 - 24,000,000 = 2,376,000,000[/tex]. There were 2,376,000,000 internet users added during this 17-year period. To find this number per year, we solve the following rule of three:
1 year - x users
17 years - 2,376,000,000 users
[tex]17x = 2,376,000,000[/tex]
[tex]x = \frac{2,376,000,000}{17}[/tex]
[tex]x = 139,764,705.9[/tex]
About 139,764,705.9 users were added, on average, per year, from 1995 to 2012.
Convert the following pairs of decimal numbers to 5-bit 2's-complement numbers, then add them. State whether or not overflow occurs in each case. (a) 4 and 11 (b) 6 and 14 (c)-13 and 12 (d) -4 and 8 (e) -2 and -9 (f) -9 and -14
Step-by-step explanation:
(a) 4 and 11
binary equivalent of 4 in 5 bit = 00100
binary equivalent of 11 in 5 bit = 01011
decimal number 4 in 2's complement form = 11100
decimal number 11 in 2's complement form = 10101
now,
1 1 1 0 0
+ 1 01 0 1
1 1 000 1
Since, we are doing addition on 5 bit numbers but the result of addition came in 6 digit, so there will be overflow.
(b) 6 and 14
binary equivalent of 6 in 5 bit = 00110
binary equivalent of 14 in 5 bit = 01110
decimal number 6 in 2's complement form = 11010
decimal number 14 in 2's complement form = 10010
now,
1 1 0 1 0
+ 1 00 1 0
1 0 1 1 0 0
Since, we are doing addition on 5 bit numbers but the result of addition came in 6 digit, so there will be overflow.
(c) -13 and 12
binary equivalent of -13 in 5 bit = 10011
binary equivalent of 12 in 5 bit = 01100
decimal number -13 in 2's complement form = 01101
decimal number 12 in 2's complement form = 10100
now,
0 1 1 0 1
+ 1 0 1 0 0
1 0 0 0 0 1
Since, we are doing addition on 5 bit numbers but the result of addition came in 6 digit, so there will be overflow.
(d) -4 and 8
binary equivalent of -4 in 5 bit = 11100
binary equivalent of 8 in 5 bit = 01000
decimal number -4 in 2's complement form = 00100
decimal number 8 in 2's complement form = 11000
now,
0 0 1 0 0
+ 1 1 0 0 0
1 1 1 0 0
Since, we are doing addition on 5 bit numbers and the result of addition also came in 5 digit, so there will not be overflow.
(e) -2 and -9
binary equivalent of -2 in 5 bit = 11110
binary equivalent of -9 in 5 bit = 10111
decimal number -2 in 2's complement form = 00010
decimal number -9 in 2's complement form = 01001
now,
0 0 0 1 0
+ 0 1 0 0 1
0 1 0 1 1
Since, we are doing addition on 5 bit numbers and the result of addition also came in 5 digit, so there will not be overflow.
(f) -9 and -14
binary equivalent of -9 in 5 bit = 10111
binary equivalent of -14 in 5 bit = 10010
decimal number -9 in 2's complement form = 01001
decimal number -10 in 2's complement form = 01110
now,
0 1 0 0 1
+ 0 1 1 1 1
1 1 000
Since, we are doing addition on 5 bit numbers and the result of addition also came in 5 digit, so there will not be overflow.
To convert decimal numbers to 5-bit 2's-complement numbers, convert each number to binary and add them.
Explanation:Conversion to 5-bit 2's-complement numbers:
(a) For 4, convert to binary: 00100. For 11, convert to binary: 01011. Add the binary numbers: 00100 + 01011 = 01111. Since the sum is positive, there is no overflow.
(b) For 6, convert to binary: 00110. For 14, convert to binary: 01110. Add the binary numbers: 00110 + 01110 = 10100. Since the sum is negative, there is overflow.
(c) For -13, convert to binary: 10011. For 12, convert to binary: 01100. Add the binary numbers: 10011 + 01100 = 11111. Since the sum is negative, there is no overflow.
(d) For -4, convert to binary: 11100. For 8, convert to binary: 01000. Add the binary numbers: 11100 + 01000 = 00100. Since the sum is negative, there is overflow.
(e) For -2, convert to binary: 11110. For -9, convert to binary: 10111. Add the binary numbers: 11110 + 10111 = 101001. Since the sum is positive, there is overflow.
(f) For -9, convert to binary: 10111. For -14, convert to binary: 10010. Add the binary numbers: 10111 + 10010 = 110001. Since the sum is negative, there is overflow.
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Solve the system of inequalities by graphing.
Answer:
infinitely many
Step-by-step explanation:
there were too many lines
Answer:
The correct option is D) Infinitely many.
Step-by-step explanation:
Consider the provided graph.
The system of equation has the solution at the point where the line intersects.
Now consider the graph of the equation 2x = 2y-6 and y = x+3
By observing the graph it can be concluded that the graph of 2x = 2y-6 and y = x+3 has the same line.
A system of equation have infinitely many solutions if each equations refers to the same line.
Since 2x = 2y-6 and y = x+3 refer the same line. The system has infinitely many solutions.
Hence, the correct option is D) Infinitely many.
What is the volume of the tank in #1 in ft) if the diameter is measured carefully to be 15.00 ft and the height 62.00 ft?
Answer:
The volume of this tank is [tex]V = 10956.30 ft^{3}[/tex], using [tex]\pi = 3.14[/tex]
Step-by-step explanation:
A tank has the format of a cylinder.
The volume of the cylinder is given by:
[tex]V = \pi r^{2}h[/tex]
In which r is the radius and h is the heigth.
The problem states that the diameter is measured carefully to be 15.00 ft. The radius is half the diameter. So, for this tank
[tex]r = \frac{15}{2} = 7.50[/tex] ft
The height of the tank is 62 ft, so [tex]h = 62[/tex].
The volume of this tank is:
[tex]V = \pi r^{2}h[/tex]
[tex]V = pi*(7.5)^2*62[/tex]
[tex]V = 10956.30 ft^{3}[/tex]
The volume of this tank is [tex]V = 10956.30 ft^{3}[/tex], using [tex]\pi = 3.14[/tex]
Compare the values of the underlined digits 506,712 and 324,859.The underlined digit is 5.
Answer:
The first one: It is in the 500,000 place.
The second one: It is in the 50 place.
Step-by-step explanation:
You are evaluating the risks associated with a construction project. Through careful analysis you have developed a list of the following risks, probabilities those risks will happen, and the costs associated with them if they occur.
25% chance of Snowmaggedon which will delay the project at a cost of $35,000
10% chance of cost of construction materials dropping saving the project $70,000
10% probability a labor strike will occur delaying the schedule with a cost of $40,000
80% chance of new regulations mandated calling for higher inspection standards which will cost an additional $15,000 to mitigate
What is the EMV of this project?
Answer:
The EMV of this project is -17,500
Step-by-step explanation:
The EMV of the project is the Expected Money Value of the Project.
This value is given by the sum of each expected earning/cost multiplied by each probability.
So, in our problem
[tex]EMV = P_{1} + P_{2} + P_{3} + P_{4}[/tex]
The problem states that there is a 25% chance of Snowmaggedon which will delay the project at a cost of $35,000. Since this is a cost, [tex]P_{1}[/tex] is negative.
[tex]P_{1} = 0.25*(-35,000) = -8,750[/tex]
There is a 10% chance of cost of construction materials dropping saving the project $70,000. A saving is an earning, so [tex]P_{2}[/tex] is positive
[tex]P_{2} = 0.10*70,000 = 7,000[/tex]
There is a 10% probability a labor strike will occur delaying the schedule with a cost of $40,000.
[tex]P_{3} = 0.10*(-40,000) = -4,000[/tex]
There is a 80% chance of new regulations mandated calling for higher inspection standards which will cost an additional $15,000 to mitigate
[tex]P_{4} = 0.80*(-15,000) = -12,000[/tex]
[tex]EMV = P_{1} + P_{2} + P_{3} + P_{4} = -8,750 + 7,000 - 4,000 - 12,000 = -17,500[/tex]
The EMV of this project is -17,500
If you roll one die two times, what is the probability of getting a 2 on the first roll and a 2 on the second roll? Show work or explain your reasoning. Copyright2016, The Charles A. Dana Center at the University of Texas at Austin
Answer: [tex]\dfrac{1}{36}[/tex]
Step-by-step explanation:
We know that when two events A and B are independent , then the probability of getting A and B will be :-
[tex]P(A\cap B)=P(A)\times P(B)[/tex]
In a fair dice,total outcomes =6
Number of 2's =1
Then, the probability of getting a 2 =[tex]\dfrac{1}{6}[/tex]
If you roll one die two times, then the probability of getting a 2 on the first roll and a 2 on the second roll will be:-
[tex]\dfrac{1}{6}\times\dfrac{1}{6}=\dfrac{1}{36}[/tex]
Hence, the probability of getting a 2 on the first roll and a 2 on the second roll=[tex]\dfrac{1}{36}[/tex]
Evaluate the surface integral ∫∫ F.ds for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation.
F(x,y,z)=xi+yj+5k
S is the coundary of the region enclosed by the cylinder x^2+z^2=1 and the planes y=0 and x+y=2
Use the divergence theorem.
[tex]\vec F(x,y,z)=x\,\vec\imath+y\,\vec\jmath+5\,\vec k\implies\mathrm{div}\vec F(x,y,z)=2[/tex]
By the divergence theorem,
[tex]\displaystyle\iint_S\vec F\cdot\mathrm d\vec S=\iiint_R\mathrm{div}\vec F\,\mathrm dV[/tex]
where [tex]R[/tex] is the region with boundary [tex]S[/tex].
Compute the latter integral in cylindrical coordinates, taking
[tex]\begin{cases}x=r\cos\theta\\y=y\\z=r\sin\theta\end{cases}\implies\mathrm dV=r\,\mathrm dr\,\mathrm d\theta\,\mathrm dy[/tex]
[tex]\displaystyle\iint_R2\,\mathrm dV=2\int_0^{2\pi}\int_0^1\int_0^{2-r\cos\theta}r\,\mathrm dy\,\mathrm dr\,\mathrm d\theta=\boxed{4\pi}[/tex]
In the Holiday Shop the manager wants 20% of the total inventory in the stockroom and the rest displayed on the floor. After meeting these instructions, you placed $35,000 of inventory in the stockroom. What is the dollar amount of the inventory on the selling floor?
Total Inventory
Inventory in the stockroom
Inventory on the selling floor
Answer:
Total inventory: $175,000
Inventory in the stockroom: $35,000.
Inventory on the selling floor: $140,000.
Step-by-step explanation:
Let x be the the total inventory.
We have been given that in the Holiday Shop the manager wants 20% of the total inventory in the stockroom. You placed $35,000 of inventory in the stockroom.
We can set an equation such that 20% of x equals $35,000.
[tex]\frac{20}{100}\cdot x=\$35,000[/tex]
[tex]0.20x=\$35,000[/tex]
[tex]\frac{0.20x}{0.20}=\frac{\$35,000}{0.20}[/tex]
[tex]x=\$175,000[/tex]
Since $35,000 of inventory in the stockroom, so we will subtract $35,000 from $175,000.
[tex]\text{Amount of the inventory on the selling floor}=\$140,000[/tex]
Therefore, $140,000 of the inventory on the selling floor.
we apply 35% of a drug at the morning and 25% of the same drug at the afternoon. if in the evening 28 mL of the drug is left. how many milliliters are we applying during the whole day?
Answer:
we are applying 70 ml during the whole day
Step-by-step explanation:
First, it is necessary to calculate the percentage of the drug that is left in the evening. This is calculated as:
100% - (35% + 25%) = 100% - 60% = 40%
Because, 35% is the percentage of the drug apply at the morning and 25% is percentage of the drug apply at afternoon.
Then, 40% is the percentage of the drug that is left in the evening and it is equivalent to 28 mL. So, the milliliter that we apply during the whole day are the milliliters equivalent to the 100%. We can calculate this by a rule of three as:
40% -------------------- 28 mL
100% ------------------- X
Where X are the milliliters that we apply during the whole day. Solving for X, we get:
[tex]X=\frac{100*28}{40}=70 mL[/tex]
If an injectable solution contains 25μg of a dug substance in each 0.5 mL, how many milliliters would be required to provide a patient with 0.25 mg of the drug substance?
Answer:
5mL would be required to provide a patient with 0.25 mg of the drug substance.
Step-by-step explanation:
The problem states that an injectable solution contains 25μg of a dug substance in each 0.5 mL, and asks how many milliliters would be required to provide a patient with 0.25 mg of the drug substance.
So, the first step is the conversion of 25ug to mg, since the problem asks the answer in mg.
Each mg has 1000ug. So
1mg - 1000ug
xmg - 25ug
1000x = 25
[tex]x = \frac{25}{1000}[/tex]
x = 0.025 mg
It means that each 0.5mL of the solution contains 0.025mg of the drug. How many milliliters would be required to provide a patient with 0.25 mg of the drug substance.
0.5mL - 0.025mg
xmL - 0.25mg
0.025x = 0.5*0.25
[tex]x = \frac{0.5*0.25}{0.025}[/tex]
x = 10*0.5
x = 5mL
5mL would be required to provide a patient with 0.25 mg of the drug substance.
Final answer:
To provide a patient with 0.25 mg of a drug substance, 5 mL of the injectable solution is required when the solution concentration is 25μg per 0.5 mL.
Explanation:
If an injectable solution contains 25μg of a drug substance in each 0.5 mL, the question asks how many milliliters would be required to provide a patient with 0.25 mg of the drug substance. First, it is important to convert 0.25 mg to micrograms (μg) because the concentration of the drug is given in micrograms. Knowing that 1 mg = 1000 μg, we have:
0.25 mg = 0.25 × 1000 μg = 250 μg.
Next, if 25 μg is in 0.5 mL, we find how many times 25 μg goes into 250 μg to determine the volume needed:
250 μg / 25 μg/mL = 10 times
Since 25 μg is contained in 0.5 mL:
10 × 0.5 mL = 5 mL.
Therefore, 5 mL of the injectable solution is required to provide a patient with 0.25 mg of the drug substance.
Two companies have sent representatives to an industry conference. The first company sent 12 representatives and the second company sent 20 representatives. Only 22 will be given the chance to make presentations. What is the probability that exactly 10 representatives from the first company and 12 representatives from the second company will be chosen?
Final answer:
The probability that exactly 10 representatives from the first company and 12 from the second company will be chosen is about 1.288%, calculated using the hyper geometric probability formula.
Explanation:
The question asks for the probability that exactly 10 representatives from the first company and 12 representatives from the second company will be chosen to make presentations at an industry conference. This can be solved using the hypergeometric probability distribution since we are dealing with two groups and selections without replacement. The first group (G1) consists of 12 representatives from the first company, and the second group (G2) consists of 20 representatives from the second company.
The formula for calculating hyper geometric probability is:
[tex]P(X = k) = (C(G1, k) * C(G2, n - k)) / C(G1 + G2, n)[/tex]
Where:
C(G, k) is the combination of k items from a group G.X is the random variable representing the number of successes (in this case, representatives from G1 chosen).k is the number of successes desired (10 representatives from G1).n is the total number of draws (22 representatives in total).To find the probability, we calculate:
[tex]P(X = 10) = (C(12, 10) * C(20, 12)) / C(32, 22)[/tex]
Plugging in the values gives us:
[tex]P(X = 10) = (C(12, 10) * C(20, 12)) / C(32, 22)[/tex]
= [tex](66 * 125,970) / 645,122,40[/tex]
=[tex]8,309,820 / 645,122,40[/tex]
=[tex]0.01288 or 1.288%[/tex]
Therefore, the probability that exactly 10 representatives from the first company and 12 from the second will be chosen is about 1.288%.
The probability that exactly 10 representatives from the first company and 12 from the second company will be chosen is approximately 0.0716%. This is calculated using the combination formula and the hypergeometric distribution.
We will use the concept of combinations and the hypergeometric distribution.
The total number of ways to select 22 representatives out of 32 (12 from the first company and 20 from the second company) is given by the combination formula [tex]\( C(n, k) = \frac{{n!}}{{k!(n-k)!}} \)[/tex] . This reflects the entire sample space.
The number of ways to choose 10 representatives out of 12 from the first company is C(12, 10).The number of ways to choose 12 representatives out of 20 from the second company is C(20, 12).Therefore, the probability P is calculated as:
[tex]\[ P = \frac{{C(12,10) \times C(20,12)}}{{C(32,22)}} \][/tex]
Using a calculator or computing these values manually, we find:
C(12,10) = 66
C(20,12) = 125,970
C(32,22) = 1,166,803,110
Thus, the probability P becomes:
[tex]\[ P = \frac{{66 \times 125,970}}{{1,166,803,110}} \][/tex]
After computation, we get:
[tex]\[ P \approx 0.000716 \][/tex]
Therefore, the probability that exactly 10 representatives from the first company and 12 representatives from the second company will be chosen is approximately 0.000716, or 0.0716%.
Programming (Optimizing an Objective Function) A manufacturer of ski clothing makes ski pants and ski jackets. The profit on a pair of ski pants is $2.00 and the profit on a jacket is $1.50. Both pants and jackets require the work of sewing operators and cutters. There are 60 minutes of sewing operator time and 48 minutes of cutter time available. It takes 8 minutes to sew one pair of ski pants and 4 minutes to sew one jacket. Cutters take 4 minutes on pants and 8 minutes on a jacket. Find the number of pants and jackets the manufacturer should make in order to maximize the profit pairs of pants jackets
Answer:
In order to maximize the profit pairs of pants jackets, the manufactures should make 6 pants and 3 jackets.
Step-by-step explanation:
This problem can be solved by a system of first order equations:
I am going to say that [tex]x[/tex] is the number of pants and [tex]y[/tex] is the number of jackets that the manufacturer should make.
The profit will be maximized when all the avaiable time of the sewing operator and the cutter time has been used.
The problem states that there are 60 minutes of sewing operator time available. It takes 8 minutes to sew one pair of ski pants and 4 minutes to sew one jacket. So:
[tex]8x + 4y = 60[/tex]
We can simplify this equation by 4. So:
[tex]2x + y = 15[/tex]
The problem also states that there are 48 minutes of cutter time available. Cutters take 4 minutes on pants and 8 minutes on a jacket. So:
[tex]4x + 8y = 48[/tex]
Again simplifying by 4,
[tex]x + 2y = 12[/tex].
Now we have to solve the following system:
[tex]2x + y = 15[/tex]
[tex]x + 2y = 12[/tex]
I am going to write y as a function of x in the first equation, and replace it in the second.
[tex]y = 15 - 2x[/tex]
[tex]x + 2y = 12[/tex]
[tex]x + 2(15 - 2x) = 12[/tex]
[tex]x + 30 - 4x = 12[/tex]
[tex]-3x = -18[/tex]
[tex]3x = 18[/tex]
[tex]x = \frac{18}{3}[/tex]
[tex]x = 6[/tex]
Now, replacing
[tex]y = 15 - 2x = 15 - 2(6) = 15 - 12 = 3[/tex]
In order to maximize the profit pairs of pants jackets, the manufactures should make 6 pants and 3 jackets.
Choose all the numbers that are part of Integers.
1
0
-3
5/6
-23
Answer:
all except 5/6
Step-by-step explanation:
All of the numbers listed are in the set of integers, except for the fraction 5/6. It is a rational number, but not an integer.
___
If by "part of integers" you mean that the number can be multiplied by some integer value to make an integer, then 5/6 is "part of 5". It is 1/6 of the integer 5.
1/250 : 2 = 1/150 : x
Answer:
The value of x is [tex]\frac{10}{3}[/tex]
Step-by-step explanation:
Given,
[tex]\frac{1}{250}:2=\frac{1}{150}:x[/tex]
[tex]\frac{1/250}{2}=\frac{1/150}{x}[/tex]
[tex]\frac{1}{500}=\frac{1}{150x}[/tex]
By cross multiplication,
[tex]150x = 500[/tex]
[tex]x=\frac{500}{150}=\frac{500\div 50}{150\div 50}=\frac{10}{3}[/tex]
Show that the sum of two concave functions is concave. Is the product of two concave functions also concave?
Answer with explanation:
Let us assume that the 2 functions are:
1) f(x)
2) g(x)
Now by definition of concave function we have the first derivative of the function should be strictly decreasing thus for the above 2 function we conclude that
[tex]\frac{d}{dx}\cdot f(x)<0\\\\\frac{d}{dx}\cdot g(x)<0[/tex]
Now the sum of the 2 functions is shown below
[tex]y=f(x)+g(x)[/tex]
Diffrentiating both sides with respect to 'x' we get
[tex]\frac{dy}{dx}=\frac{d}{dx}\cdot f(x)+\frac{d}{dx}\cdot g(x)\\\\[/tex]
Since each term in the right of the above equation is negative thus we conclude that their sum is also negative thus
[tex]\frac{dy}{dx}<0[/tex]
Thus the sum of the 2 functions is also a concave function.
Part 2)
The product of the 2 functions is shown below
[tex]h=f(x)\cdot g(x)[/tex]
Diffrentiating both sides with respect to 'x' we get
[tex]h'=\frac{d}{dx}\cdot (f(x)\cdot g(x))\\\\h'=g(x)f'(x)+f(x)g'(x)[/tex]
Now we can see the sign of the terms on the right hand side depend on the signs of the function's themselves hence we remain inconclusive about the sign of the product as a whole. Thus the product can be concave or convex.
magine that Amy counted 60 numbers per minute and continued to count nonstop until she reached 19,000. Determine a reasonable estimate of the number of hours it would take Amy to complete the counting. It will take Amy approximately (Type a whole number.) hours to count to 19,000.
Answer:
It will take Amy approximately 5.28 hours to count to 19,000.
Step-by-step explanation:
The first step to solve this problem is finding how many minutes it takes for Amy to count to 19,000. In each minute, she counts 60 numbers. So
1 minutes - 60 numbers
x minutes - 19,000 numbers
[tex]60x = 19,000[/tex]
[tex]x = \frac{19,000}{60}[/tex]
[tex]x = 316.7 minutes[/tex]
It is going to take 316.7 minutes for Amy to count to 19,000. How many hours are 316.7 minutes? Each hour has 60 minutes, so:
1 hour - 60 minutes
x hours - 316.7 minutes
[tex]60x = 316.7[/tex]
[tex]x = \frac{316.7}{60}[/tex]
[tex]x = 5.28[/tex]
It will take Amy approximately 5.28 hours to count to 19,000.
Which expression is equivalent to this one:
[tex]\frac{2}{3}[/tex] x 6 + [tex]\frac{2}{3}[/tex] x s
A) 6([tex]\frac{2}{3}[/tex] + s)
B) s(6+[tex]\frac{2}{3}[/tex])
C) [tex]\frac{2}{3}[/tex](6+s)
D) [tex]\frac{2}{3}[/tex] x (6+[tex]\frac{2}{3}[/tex]) x s
Answer:
C) [tex]\dfrac{2}{3}(6+s)[/tex]
Step-by-step explanation:
The distributive property lets you factor out the common factor of 2/3. The result is ...
[tex]\dfrac{2}{3}(6+s)[/tex]
is 0 not an element of an empty set?
Answer:
0 is not an element of an empty set.
Step-by-step explanation:
We are asked to determine whether 0 is not an element of an empty set.
We know that an empty set is an unique set having no elements. The cardinality of an empty set is 0.
Cardinality stands for the count of element is an set. An empty set is denoted by symbols ∅ or { }.
The empty set is like an empty container. The container is there, but nothing is in it.
When 0 is an element of a set, then its cardinality would be 1.
Therefore, 0 is not an element of an empty set.
You wake up with a fever! All you can find at the store is Aspirin, and the bottle says to take 162 mg. The bottle also says each tablet has 81 mg in each tablet. How many tablets should you take? 5. When you don't seem to get any better, you eventually end up in the hospital. The doctor orders 120 mg of medicine to be given twice a day. The nurse comes in with 6 tablets and tells you that there are 30 mg in each tablet. Is this the correct dosage? If not, how many tablets should you get? 6.
Answer:
You should take 2 tablets of Aspirin.
Six tablets is not the correct dosage.
You should get 4 tablets.
Step-by-step explanation:
You wake up with a fever! All you can find at the store is Aspirin, and the bottle says to take 162 mg. The bottle also says each tablet has 81 mg in each tablet. How many tablets should you take?
This can be solved by this following rule of three.
1 tablet - 81mg
x tablets - 162mg
[tex]81x = 162[/tex]
[tex]x = \frac{162}{81}[/tex]
[tex]x = 2[/tex]
You should take 2 tablets of Aspirin.
The doctor orders 120 mg of medicine to be given twice a day. The nurse comes in with 6 tablets and tells you that there are 30 mg in each tablet. Is this the correct dosage?
We need to see how many mg are in 6 tablets. If there are 120mg in 6 tablets, this is the correct dosage. We verify this by the following rule of three:
1 tablet - 30mg
6 tablets - x mg
[tex]x = 30*6[/tex]
[tex]x = 180[/tex]mg.
In 6 tablets, there are 180mg. So, it is not the correct dosage.
If not, how many tablets should you get?
Knowing that each tablet has 30mg, in how many tablets are there 120mg?
1 tablet - 30 mg
x tablets - 120 mg
[tex]30x = 120[/tex]
[tex]x = \frac{120}{30}[/tex]
[tex]x = 4[/tex]
You should get 4 tablets.
Melissa is 29 meters below the surface. After swimming 12 minutes she rises upward 15 meters. What is her final depth?
Answer:
The final depth of Melissa is 14 meters below the surface
Step-by-step explanation:
In this kind of problems, involving directions, etc, usually one position is the positive direction and the other position is the negative direction.
In our problem, we suppose that the surface is the positivo zero, above the surface is the positive position(positive depth) and below the surface is the negative directions(negative depth).
The problem states that Melissa is 29 meters below the surface. So, her initial position is 29 meters in the negative direction, so it is equal to -29.
Then, the problem states that Melissa rises upward 15 meters. Upward is the positive direction, so she moved to the positive direction. It means that we are going to do -29+15 = -14 meters.
It means that the final depth of Melissa is 14 meters below the surface
Determine whether the equation x^3 - 3x + 8 = 0 has any real root in the interval [0, 1]. Justify your answer.
Answer:
The equation does not have a real root in the interval [tex]\rm [0,1][/tex]
Step-by-step explanation:
We can make use of the intermediate value theorem.
The theorem states that if [tex]f[/tex] is a continuous function whose domain is the interval [a, b], then it takes on any value between f(a) and f(b) at some point within the interval. There are two corollaries:
If a continuous function has values of opposite sign inside an interval, then it has a root in that interval. This is also known as Bolzano's theorem.The image of a continuous function over an interval is itself an interval.Of course, in our case, we will make use of the first one.
First, we need to proof that our function is continues in [tex]\rm [0,1][/tex], which it is since every polynomial is a continuous function on the entire line of real numbers. Then, we can apply the first corollary to the interval [tex]\rm [0,1][/tex], which means to evaluate the equation in 0 and 1:
[tex]f(x)=x^3-3x+8\\f(0)=8\\f(1)=6[/tex]
Since both values have the same sign, positive in this case, we can say that by virtue of the first corollary of the intermediate value theorem the equation does not have a real root in the interval [tex]\rm [0,1][/tex]. I attached a plot of the equation in the interval [tex]\rm [-2,2][/tex] where you can clearly observe how the graph does not cross the x-axis in the interval.
A recipe for individual chocolate hazelnut tarts calls for ½ cup of hazelnuts per tart. If 1 cup of hazelnuts weighs 4 ounces, will a 5-kilogram bag of hazelnuts be sufficient to make 75 tarts? 9.
Answer:
The amount is sufficient to make 75 tarts.
Step-by-step explanation:
We have been given that a recipe for individual chocolate hazelnut tarts calls for ½ cup of hazelnuts per tart and 1 cup of hazelnuts weighs 4 ounces.
The half cup of hazelnuts will weigh 2 ounces [tex](\frac{4}{2}=2)[/tex].
1 kg equals 35.274 ounces.
[tex]\text{5 kg}=5\times 35.274\text{ ounces}[/tex]
[tex]\text{5 kg}=176.37\text{ ounces}[/tex]
Since each tart needs ½ cup of hazelnuts and half cup of hazelnuts will weigh 2 ounces, so we will divide 176.37 ounces by 2 to find number of tarts.
[tex]\frac{176.37}{2}=88.185\approx 88[/tex]
Since we can make 88 tarts from 5 kg hazelnuts, therefore, the 5-kilogram bag of hazelnuts be sufficient to make 75 tarts.
PLEASE HELP ASAP PLEASE
Choose the equation below that represents the line passing through the point (2, -4) with a slope of one half.
y = one half x + 5
y = one half x − 3
y = one half x − 5
y = one half x + 3
Answer:
The equation that represents the line passing through the point (2, -4) with a slope of one half is
[tex]f(x) = \frac{1}{2}x - 5[/tex]
Step-by-step explanation:
The equation of a line can be described by a first order equation in the following format:
[tex]f(x) = ax + b[/tex]
In which a is the slope of the line.
Solution:
The line slope is [tex]\frac{1}{2}[/tex], so [tex]a = \frac{1}{2}[/tex].
The equation of the line now is:
[tex]f(x) = \frac{1}{2}x + b[/tex]
The problem states that the line passes through the point(2,-4). This means that when x = 2, f(x) = -4
So:
[tex]f(x) = \frac{1}{2}x + b[/tex]
[tex]-4 = \frac{1}{2}*(2) + b[/tex]
[tex]-4 = 1 + b[/tex]
[tex]b = -5[/tex]
So, the equation that represents the line passing through the point (2, -4) with a slope of one half is
[tex]f(x) = \frac{1}{2}x - 5[/tex]
How much would you invest today to have $9500 in 8 years if the effective annual rate of interest is 4%?
Suppose that an investment of $5750 accumulates to $11533.20 at the end of 13 years, then the effective annual interest rate is i= ?
At an effective annual rate of interest of 5.3%, the present value of $7425.70 due in t years is $3250. Determine t
Answer:
a) You should invest $6941.90 today.
b) The effective annual interest rate is 11%.
c) t is approximately 6.
Step-by-step explanation:
These are compound interest problems. The compound interest formula is given by:
[tex]A = P(1 + \frac{r}{n})^{nt}[/tex]
Where A is the amount of money, P is the principal(the initial sum of money), r is the interest rate(as a decimal value), n is the number of times that interest is compounded per unit t and t is the time the money is invested or borrowed for.
a) How much would you invest today to have $9500 in 8 years if the effective annual rate of interest is 4%?
Here, we want to find the value of P when [tex]A = 9500, t = 8, n = 1, r = 0.04[/tex].
[tex]9500 = P(1 + \frac{0.04}{1})^{8}[/tex]
[tex]P = \frac{9500}{1.3685}[/tex]
[tex]P = 6941.90[/tex].
You should invest $6941.90 today.
b) Suppose that an investment of $5750 accumulates to $11533.20 at the end of 13 years, then the effective annual interest rate is i= ?
Here, we have that [tex]A = 11533.20, P = 5750, t = 13, n = 1[/tex], and we want to find the value of i, that is r on the formula above the solutions.
[tex]11533.20 = 5750(1 + r)^{13}[/tex]
[tex]\sqrt[13]{11533.20} = \sqrt[13]{5750(1 + r)^{13}}[/tex]
[tex]2.05 = 1.94(1 + r)[/tex]
[tex]r = 0.11[/tex]
The effective annual interest rate is 11%.
c) At an effective annual rate of interest of 5.3%, the present value of $7425.70 due in t years is $3250. Determine t.
Here, we have that [tex]A = 7425.70 + 3250 = 10675.7, P = 7425.7, r = 0.053, n = 1[/tex] and we have to find t. So
[tex]10675.7 = 7425.7(1 + \frac{0.053}{1})^{t}[/tex]
[tex](1 + 0.053)^{t} = \frac{10675.7}{7425.7}[/tex]
[tex](1.0553)^{t} = 1.4377[/tex]
We have that:
log_{a}a^{n} = n
So
[tex]log_{1.0553} (1.0553)^{t} = log_{1.0553} 1.4377[/tex]
[tex]t = 5.95[/tex]
t is approximately 6.
the simple interest on the loan. (Round your answer to the nearest cent.)
$875 at 6.85% for 5 years 6 months.
the total amount due for the simple interest loan. (Round your answer to the nearest cent.)
$6400 at 5.3% for 4 years 9 months.
the interest rate on a loan charging $960 simple interest on a principal of $3750 after 8 years.
the term of a loan of $350 at 4.5% if the simple interest is $63.
Answer:
1) $329.66
2) $8011.20
3) 3.2%
4) 4
Step-by-step explanation:
Simple interest formula: I = P*r*t
the simple interest on the loan: $875 at 6.85% for 5 years 6 months
5 years 6 months = 5,5 years
6.85% = 0.0685
I = 875*0.0685*5.5 = 329.66
the total amount due for the simple interest loan: $6400 at 5.3% for 4 years 9 months.
4 years 9 months = 4 + 9/12 = 4 + 0.75 = 4.75
5.3% = 0.053
I = 6400*0.053*4.75 = 1611.20
Total amount due: 6400+1611.20 = 8011.20
the interest rate on a loan charging $960 simple interest on a principal of $3750 after 8 years.
I = 960
P = 3750
t = 8
960 = 3750*r*8
960 = 30000*r
r = 0.032
r = 3.2%
the term of a loan of $350 at 4.5% if the simple interest is $63.
P = 350
r = 4.5% = 0.045
I = 63
t = ?
63 = 350*0.045*t
63 = 15.75*t
t = 4
Suppose you go shopping for a new futon bed for your room. The model you really like happens to be on sale for $1200. It's original price is $1400. What percent of the original price will you save if you purchase it?
Answer:
If you purchase it, you will save 16.67% of the original price.
Step-by-step explanation:
Percentage problems can be solved as a simple rule of three problem:
In a rule of three problem, the first step is identifying the measures and how they are related, if their relationship is direct of inverse.
When the relationship between the measures is direct, as the value of one measure increases, the value of the other measure is going to increase too. In this case, the rule of three is a cross multiplication.
When the relationship between the measures is inverse, as the value of one measure increases, the value of the other measure will decrease. In this case, the rule of three is a line multiplication.
A percentage problem is an example where the relationship between the measures is direct.
The problem states that the model you really like happens to be on sale for $1200 and it's original price is $1400. It means the you saved $1400-$1200 = $200. Thus, the problem wants to know how much $200 is of $1200. So, we have the following rule of three
$1200 - 100%
200 - x%
1200x = 200*100
[tex]x = \frac{20000}{1200}[/tex]
x = 16.67%.
If you purchase it, you will save 16.67% of the original price.