Answer:
a) There is a 61,41% of none of the samples containing high levels of contamination.
b)There is a 32.52% probability that exactly one sample contains high levels of contamination.
c) There is a 38.59% probability that at least one contains high levels of contamination
Step-by-step explanation:
The probabilities are independent from each other. It means that the probability of selecting a lab specimen being contaminated is always 15%, no matter how many contaminated lab specimen have been chosen.
a) There are 3 independent samples. For each sample, the probability of it not being contaminated is 85%. So, the probability that none of the sample are contaminated is
[tex]P = (0.85)^3 = 0.6141 = 61,41%[/tex]
There is a 61,41% of none of the samples containing high levels of contamination.
b) There are 3 independent samples. For each sample, the probability of it being contaminated is 15% and not contaminated 85%.
So the probability the exactly one sample contains high levels of contamination is:
[tex]P = (0.85)^2(0.15) = 0.1084 = 10,84%[/tex]
There can be 3 orderings of the sample in these conditions.(C-NC-NC, NC-C-NC, NC-NC,C), so the probability that exactly one contains high levels of contamination is
P = 3*0.1084 = 0.3252 = 32.52%.
There is a 32.52% probability that exactly one sample contains high levels of contamination.
c) The sum of the probabilities is always 100%.
In relation to the existence of a contaminated sample, either:
-None of the samples are contaminated.
-At least one of the samples are contaminated.
So, the probability of at least one of the samples being contaminated is 100% - the probability that none of the samples are contaminated, that we have already found in a).
So, it is
100% - 61.41% = 38.59%
There is a 38.59% probability that at least one contains high levels of contamination
Ellen said she spent half her money for lunch and half of what was left for a movie. She now has $1.20. How much did she spend for lunch?
Devise a plan
Carry out the plan
Look back (is it reasonable? Did we answer the question?)
Answer: She spend $1.20 for lunch.
Step-by-step explanation:
Let the total amount be 'x'.
Half of her money spend for lunch be [tex]\dfrac{x}{2}[/tex]
Half of her money left for a movie be [tex]\dfrac{x}{2}[/tex]
Amount she has now = $1.20
So, According to question, it becomes ,
[tex]\dfrac{x}{2}=1.20\\\\x=1.20\times 2\\\\x=\$2.40[/tex]
Hence, Amount she spend for lunch is [tex]\dfrac{x}{2}=\dfrac{2.40}{2}=\$1.20[/tex]
Therefore, she spend $1.20 for lunch.
Karen Price has determined that her net worth is $58,000. She has also determined that the face value of her mortgage is $89,000. She has determined that the face value of the rest of her debt is $18,000. What is Karen's debt-to-equity ratio? Multiple Cholce 184 153 3.22 4.94 0.31
Answer:
A. 1.84
Step-by-step explanation:
We have been given that Karen Price's net worth is $58,000. The face value of her mortgage is $89,000. The face value of the rest of her debt is $18,000.
[tex]\text{Debt to equity ratio}=\frac{\text{Total liabilities}}{\text{Total shareholder's equity}}[/tex]
We know that total liabilities include short term debt and long-term debt.
[tex]\text{Debt to equity ratio}=\frac{\$89,000+\$18,000}{\$58,000}[/tex]
[tex]\text{Debt to equity ratio}=\frac{\$107,000}{\$58,000}[/tex]
[tex]\text{Debt to equity ratio}=1.8448[/tex]
[tex]\text{Debt to equity ratio}\approx 1.84[/tex]
Therefore, Karen's debt-to-equity ratio 1.84 and option A is the correct choice.
Describe how an unbounded solution occurs
Answer and explanation :
Unbounded solutions :
Unbounded solution is the case where we can't find the exact solution. In this case there are infinite number of solutions and it is not possible to find exact solution in which these situations occurs.
When we use graphical method to solve the problem then in unbounded solution there is no boundary so that we can determine the maximum possible region in which solution occurs.
The following data describes the magnitude measurements randomly selected from 6 earthquakes recorded in one year from a location in southern Califormia: 6.6 2.2 18.5 7.0 13.7 5.9 The magnitude is measured by MAG on the Richter scale. What type of the data is the magnitude? a) Continuous numeric b) Discrete numeric c) Continuous categorical d) Nominal categorical
Answer: The magnitude is: a) continuous numeric.
Step-by-step explanation:
The magnitude is a numeric variable because it represents quantities. These are variables that you can measure or count. A numeric variable can be classified into discrete or continuous. In the present problem, the magnitude is a continuous variable. It can take any number within a scale, and you can find infinite values between two values on the scale. For example, you could measure earthquakes of magnitude 2.3, 2.4, 2.5, 2.6… and so on, following a continuous scale.
On the other hand, if the variable is numeric and discrete, it can only take certain finite values. For example, when you count the number of trees per acre. The number of trees will be always an integer. You can find 1, 2, or 3 trees, but you’ll never count 2.5 trees.
Categorical variables don’t represent quantities. They represent attributes. For example, apple colors: green and red.
The United States has 435 members of the House of Representatives in Congress. If there are 325.7 million people in the country, what the ratio of members to the people? (Write your answer in scientific notation with 2 digits after the decimal.) рrt sex V8
Answer:
[tex]1.36\times 10^{-6}:1[/tex]
Step-by-step explanation:
We have been given that the United States has 435 members of the House of Representatives in Congress. There are 325.7 million people in the country.
To find the ratio of members to the people, we will find compare both numbers.
1 million equals 1,000,000.
[tex]\text{325.7 million}=325.7\times 1,000,000[/tex]
[tex]\text{325.7 million}=3257,000,000[/tex]
Ratio of members to the people:
435: 3257,000,000
[tex]0.0000013355848941:1[/tex]
[tex]1.3355848941\times 10^{-6}:1[/tex]
[tex]1.36\times 10^{-6}:1[/tex]
Therefore, our required ratio would be [tex]1.36\times 10^{-6}[/tex] members per person.
What is the rate of heat transfer required to melt 1-ton of ice at 32 F in 24 hours?
Answer:
3865.74 J/s
Step-by-step explanation:
mass of ice, m = 1 ton = 1000 kg
time , t = 24 hours
latent heat of fusion of ice, L = 334000 J/kg
Heat required to melt, H = m x L
where, m is the mass of ice and L be the latent heat of fusion
So, H = 1000 x 334000 = 334 xx 10^6 J
Rate of heat transfer = heat / time = [tex]\frac{334\times 10^{6}}{86400}[/tex]
Rate of heat transfer = 3865.74 J/s
thus, the rate of heat transfer is 3865.74 J/s.
One-half liter of solution for intravenous infusion contains 2 g of drug. How many milliliters of the solution would contain 0.5 mg of drug?
Final answer:
To find out how many milliliters of the solution would contain 0.5 mg of the drug, we can set up a proportion using the given information. The solution would contain 0.125 mL of the drug.
Explanation:
To find out how many milliliters of the solution would contain 0.5 mg of the drug, we need to set up a proportion using the given information. We have 2 g of the drug in one-half liter of solution, so the concentration is 4 g/L. We can convert milligrams to grams by dividing by 1000. By setting up the proportion, we have:
4 g/L = 0.5 mg/x mL
Cross-multiplying, we get:
4 g * x mL = 0.5 mg * 1 L
Converting mg to g and mL to L:
4 * x = 0.5 / 1000
x = (0.5 / 1000) / 4
x = 0.000125 L
Since there are 1000 mL in 1 L, we can convert the answer:
x = 0.000125 L * 1000 mL/L
x = 0.125 mL
Can a collection of ideas be called a set?
Answer must be of 1 paragrah(8 lines)
Answer:
Sets
Step-by-step explanation:
1) Set can be defined as a collection of objects that are well defined and distinct.
2) Since each idea has its own unique value or characteristic, they can be considered as objects.
3) Thus, a collection of ideas can be considered as a set.
4) In this case we would define the null set as the set with no ideas.
5) The sets can be represented with the help of curly brackets { }.
6) We can represent it in the set form as:
{Idea 1, Idea 2, Idea 3, Idea 4,...}
7) It can be considered a countable set as we can always count the number of ideas.
8) It is a finite set.
find the solution the each of the following first order linear differential equations:
a) xy' -4y = 2 x^6
b) y' - 5y = 4e^7x
c) dy/dx + 2y = 2/(1+e^4x)
d) 1/2 di/dt + i = 4cos(3t)
Answer:
a. [tex]y=\frac{2}{3}x^7+cx^4[/tex]
b. [tex]y=2e^{7x}-ce^{5x}[/tex]
c. [tex]y=e^{-2x}arctan(e^{2x})+ce^{-2x}[/tex]
d. [tex]i=e^{-2t}\left(\frac{8\left(3e^{2t}\sin \left(3t\right)+2e^{2t}\cos \left(3t\right)\right)}{13}+C\right)[/tex]
Step-by-step explanation:
a) xy' -4y = 2 x^6
[tex]xy'-4y=2x^6\\y'-\frac{4}{x}y=2x^5\\p(x)=\frac{-4}{x}\\Q(x)=2x^5\\\mu(x)=\int P(x)dx=\int \frac{-4}{x}dx=Ln|x|^{-4}\\y=e^{-\mu(x)}\int {e^{\mu(x)}Q(x)dx}\\y=x^4 \int {x^{-4}2x^6}dx\\y=\frac{2}{3}x^7+cx^4[/tex]
b) y' - 5y = 4e^7x
[tex]y'-5y=4e^{7x}\\p(x)=-5\\Q(x)=4e^{7x}\\\mu(x)=\int P(x)dx=\int-5dx=-5x\\y=e^{-\mu(x)}\int {e^{\mu(x)}Q(x)dx}\\y=e^{5x}\int {e^{-5x}4e^{7x}}dx\\y=2e^{7x}-ce^{5x}[/tex]
c) dy/dx + 2y = 2/(1+e^4x)
[tex]\frac{dy}{dx}+2y=\frac{2}{1+e^{4x}}\\p(x)=2\\Q(x)=\frac{2}{1+e^{4x}}\\\mu(x)=\int P(x)dx=\int 2dx=2x\\y=e^{-\mu(x)}\int {e^{\mu(x)}Q(x)dx}\\y=e^{-2x}\int {e^{2x}\frac{2}{1+e^{4x}}}dx\\y=e^{-2x}arctan(e^{2x})+ce^{-2x}[/tex]
d) 1/2 di/dt + i = 4cos(3t)
[tex]\frac{1}{2}\frac{di}{dt}+i=4cos(3t)\\\frac{di}{dt}+2i=8cos(3t)\\p(t)=2\\Q(t)=8cos(3t)\\\mu(t)=\int P(t)dt=\int 2dt=2t\\i=e^{-\mu(t)}\int {e^{\mu(t)}Q(t)dt}\\i=e^{-2t}\int {e^{2t}8cos(3t}dt\\i=e^{-2t}\left(\frac{8\left(3e^{2t}\sin \left(3t\right)+2e^{2t}\cos \left(3t\right)\right)}{13}+C\right)[/tex]
If A, B, and C are mutually exclusive events with P(A) = 0.21, P(B) = 0.32, and P(C) = 0.43, determine the following probabilities. Round your answers to two decimal places.
(a) P(A U B U C)
(b) P(A n B n C)
(c) P(A n B)
(d) P[(A U B) n C]
By their mutual exclusivity,
[tex]P(A\cup B\cup C)=P(A)+P(B)+P(C)=0.96[/tex]
[tex]P(A\cap B\cap C)=0[/tex]
[tex]P(A\cap B)=0[/tex]
For the last probability, first distribute the intersection:
[tex](A\cup B)\cap C=(A\cap C)\cup(B\cap C)[/tex]
Recall that for two event [tex]X,Y[/tex],
[tex]P(X\cup Y)=P(X)+P(Y)-P(X\cap Y)[/tex]
so that
[tex]P((A\cap C)\cup(B\cap C))=P(A\cap C)+P(B\cap C)-P((A\cap C)\cap(B\cap C))[/tex]
[tex]P((A\cap C)\cup(B\cap C))=P(A\cap C)+P(B\cap C)-P(A\cap B\cap C)=0[/tex]
A tank has the shape of an inverted circular cone (point at the bottom) with height 10 feet and radius 4 feet. The tank is full of water. We pump out water (to a pipe at the top of the tank) until the water level is 5 feet from the bottom. The work W required to do this is given by W= ? foot-pounds
The tank as a cone.
As per the question, the tank is given a shape of an inverted circular cone has a point to the bottom with an height of radius of 4 feet. The tank is full of water the pipe can be cued to pump out the water from the top and until which the tank ill have a level of 5 feet from the bottom.
Thus the answer is W equals to 468832 foot-pound
As per the given information the tank consists of the inverted circular cone the Height of cone is equal to 10 feet and radius = 4 feet After water pumped out height = 5 ft. Thus the volume of water pumped out Here we have r/h is constant alwaysHence the Substitute to get volume of water pumped out equals to the Mass of water = density x volume = Work done = force x displacement = mass x accents x displacement. Here acceleration = gravity = 32.2 ft/sec^2. Displacement = height reduced = 5 ft.Hence the W equals to 2912(32.2) that is 468832\pi foot-pound.Learn more about the shape of an inverted.
https://brainly.com/question/23758952.
To calculate the work required to pump water from an inverted circular cone tank, we use a formula that accounts for the weight density of water, volume of water, and height the water is lifted. We integrate from the middle of the tank, where the water level is 5 feet high, up to the top. The work is expressed in foot-pounds and involves an integral that can be solved using calculus.
Explanation:To calculate the work W required to pump water out of an inverted circular cone tank, we must use the concept of work done against gravity. The formula for work is W = γ x V x h, where γ (gamma) represents the weight density of water, V is the volume of water being lifted, and h is the distance the water is lifted.
Since the tank is a cone and water is being lifted from the current water height to the top of the tank, we have to integrate the work done for each infinitesimally small volume δV of water from the water level at 5 feet to the top at 10 feet. The water has a circular cross-section at any height y, with a radius that can be determined by similar triangles.
As the radius of the tank at the top is 4 feet and the height is 10 feet, the radius r at height y is (4/10)*y. The cross-sectional area A at height y is πr^2, which is (π * (4/10)^2 * y^2). The volume element δV is then A δy, and the work element δW is γ * A * (10 - y) δy. The total work is found by integrating δW from 5 to 10 feet.
The weight density of water γ is typically 62.4 lb/ft^3, so the integral becomes: W = ∫ γ * π * (16/100) * y^2 * (10 - y) dy from 5 to 10. This integral can then be evaluated to find the total work W in foot-pounds.
The tensile strength of silicone rubber is thought to be a function of curing temperature. A study was carried out in which samples of 12 specimens of the rubber were prepared using curing temperatures of 20◦C and 45◦C. The data below show the tensile strength values in megapascals.(20 C) 2.07 2.14 2.22 2.03 2.21 2.03 2.05 2.18 2.09 2.14 2.11 2.05(45 C) 2.52 2.15 2.49 2.03 2.37 2.05 1.99 2.42 2.08 2.42 2.29 2.01(a) Show the dot plot of the data with both low and high temperature tensile strength value(b) Compute sample mean tensile strength for both samples
Answer:
Sample mean tensile strength for 20°C [tex]\bar X_{20} =2.11[/tex]Mp
Sample mean tensile strength for 45°C [tex]\bar X_{45} =2.235[/tex]Mp
Step-by-step explanation:
A dot plot for combined data allows comparison between the responses of an experiment to two or more independent factors. In this case there are 12 experimental observations of tensile strength on silicone rubber for two levels of the curing temperature factor (30°C and 45°C)
The sample mean can be calculated by:
[tex]\bar X_{20} = \frac{1}{n}\sum{x_i}=2.11[/tex]Mp
[tex]\bar X_{45} = \frac{1}{n}\sum{x_i}=2.235[/tex]Mp
The dot plot can be observed in the attached file.
Show that if A CB, then A = B ( B A ). Show that if A C B, then A U (B \ A) = B. Show, by example, that for sets A, B, and C, AN B = An C does not imply B = C.
Answer: If A ⊂ B, then A = B \ ( B \ A)
ok, when you do B \ A, you are subtracting all the elements in A∩B from B. So the only elements remaining are those who aren't in A.
If we subtract this of B again, we are subtracting of B all the elements that aren't in A, so the only elements remaining are those who belongs in A.
If A ⊂ B then A U (B \ A) = B.
Again, when you do B \ A you are extracting all the elements that belongs to the A∩B from B. So you are extracting al the elements from A. and when you add all the elements of A again, then you recuperate B.
if AnC = AnC does not imply that B = C.
if A = {1,2}, B = {1,2,3,4,5} and C = {1,2,3}
then AnC = {1,2} and AnB = {1,2} but B and C are different.
one interior angle of a polygon is equal to 800 and each of the other interior angles are 128 degrees. Find the number of sides of the polygon.
Answer:
6
Step-by-step explanation:
Given information:
Interior angle of a polygon cannot be more that 180°.
One interior angle = [tex]80^{\circ}[/tex]
Other interior angles are = [tex]128^{\circ}[/tex]
Let n be the number of sides of the polygon.
Sum of interior angles is
[tex]Sum=80+128(n-1)[/tex]
[tex]Sum=80+128n-128[/tex]
Combine like terms.
[tex]Sum=128n-48[/tex] .... (1)
If a polygon have n sides then the sum of interior angles is
[tex]Sum=(n-2)180[/tex]
[tex]Sum=180n-360[/tex] .... (2)
Equating (1) and (2) we get
[tex]180n-360=128n-48[/tex]
Isolate variable terms.
[tex]180n-128n=360-48[/tex]
[tex]52n=312[/tex]
Divide both sides by 52.
[tex]n=\frac{312}{52}[/tex]
[tex]n=6[/tex]
Therefore the number of sides of the polygon is 6.
Solve the following logarithmic equation: In(x +31)-In(4-3x)-5In2 0 x = 2 1 points x= 0 x-0.5 ○ x=0.25 None of the above to save all
Answer:
The solution is [tex]x = 1[/tex]
Step-by-step explanation:
We have the following logarithmic properties:
[tex]ln a + ln b = ln ab[/tex]
[tex]ln a - ln b = ln \frac{a}{b}[/tex]
[tex]n ln a = ln a^{n}[/tex]
We have the following logarithmic equation:
[tex]ln(x + 31) - ln (4-3x) - 5 ln 2 = 0[/tex]
Lets simplify, and try to find properties.
[tex]ln(x + 31) - (ln (4-3x) + 5 ln 2) = 0[/tex]
[tex]ln(x + 31) - (ln (4-3x) + ln 2^{5}) = 0[/tex]
[tex]ln(x + 31) - (ln (4-3x) + ln 32) = 0[/tex]
[tex]ln(x + 31) - ln 32*(4-3x) = 0[/tex]
[tex]ln(x+31) - ln (128 - 96x) = 0[/tex]
[tex]ln \frac{x + 31}{128 - 96x} = 0[/tex]
To eliminate the ln, we apply the exponential to both sides, since e and ln are inverse operations.
[tex]e^{ln \frac{x + 31}{128 - 96x}} = e^{0}[/tex]
[tex]\frac{x + 31}{128 - 96x} = 1[/tex]
[tex]x + 31 = 128 - 96x[/tex]
[tex]97x = 97[/tex]
[tex]x = \frac{97}{97}[/tex]
[tex]x = 1[/tex]
The solution is [tex]x = 1[/tex]
Fill in the blank The retail cost of a TV is 40% more than its wholesale cost. Therefore, the retail cost is _times the wholesale cost The retail cost is 1.4 times the wholesale cost. (Type an integer or a decimal
Answer:
Step-by-step explanation:
let whole sale cost=100
retail cost=100+40% of 100=100+40=140
to find the ratio
(retail cost)/(whole cost)=140/100=14/10=1.4
so retail cost=1.4*whole cost
or retail cost is 1.4 times the whole cost.
The retail cost of a TV, which is 40% more than its wholesale cost, is 1.4 times the wholesale cost. For instance, if the wholesale cost is $500, the retail cost would be $700.
The retail cost of a TV is 40% more than its wholesale cost. To find out how many times more the retail cost is compared to the wholesale cost, we need to understand percentage increase calculations.
Let the wholesale cost of the TV be represented by 1 (or 100%). An increase of 40% on this cost means the TV now costs:
1 + 0.40 = 1.40.
Therefore, the retail cost of the TV is 1.4 times the wholesale cost.
For example, if the wholesale cost of a TV is $500, then the retail cost would be:
$500 × 1.4 = $700.
Find all real values of ‘t‘ so that angle between the vectors u = (t − 2, 6 − t, −4) and v = (−4, t − 2, 6 − t) is 120◦ .
Answer:
for all values
Step-by-step explanation:
u = (t - 2, 6 - t, - 4)
v = ( - 4, t - 2, 6 - t)
Angle between them, θ = 120°
Use the concept of dot product of two vectors
[tex]\overrightarrow{A}.\overrightarrow{B}=A B Cos\theta[/tex]
Magnitude of u = [tex]\sqrt{(t-2)^{2}+(6-t)^{2}+(-4)^{2}}[/tex]
= [tex]\sqrt{2t^{2}-16t+56}[/tex]
Magnitude of v = [tex]\sqrt{(t-2)^{2}+(6-t)^{2}+(-4)^{2}}[/tex]
= [tex]\sqrt{2t^{2}-16t+56}[/tex]
[tex]\overrightarrow{u}.\overrightarrow{v}=-4(t-2)+(6-t)(t-2)-4(6-t)=-t^{2}+8t-28[/tex]
By the formula of dot product of two vectors
[tex]Cos120 = \frac{-t^{2}+8t-28}{\sqrt{2t^{2}-16t+56}\times \sqrt{2t^{2}-16t+56}}[/tex]
[tex]-0.5\times {2t^{2}-16t+56} = {-t^{2}+8t-28}}[/tex]
[tex]{-t^{2}+8t-28}} = {-t^{2}+8t-28}}[/tex]
So, for all values of t the angle between these two vectors be 120.
An infusion pump is infsing heparin at a rate of 11.3 mL/hr. The infusion bag hanging has 25,000 units of heparin in 500 mL solution. Calculate the flow rate (units/hr) for these heparin infusions. Round the answer to the nearest whole number.
Answer:
565 units/hour
Step-by-step explanation:
As given in question,
rate of infusion of pump = 11.3 mL/hr
amount of heparin infusion bag contains = 25,000 units
amount of solution in infusion bag = 500 mL
Since, 500 mL of solution contains 25000 units of heparin
[tex]\textrm{So, 1 mL of solution will contain heparin of amount}=\dfrac{25000}{500}units[/tex]
= 50 units
Since, 11.3 mL of solution can flow in 1 hour
So, the heparin contains in 11.3 mL of solution = 11.3 x 50
= 565 units
As, 565 units of heparin can flow in 1 hour so the rate of flow of heparin will be 565 units/hour.
When an object falls through air, there is a drag force that depends on the product of the cross sectional area of the object and the square of its velocity, that is, Fair = CAv2, where C is a constant. Determine the dimensions of C. (Use the following as necessary: M for mass, T for time, and L for length.)
Answer:
[tex]\textrm{Dimension of C }=\ [ML^{-3}T^{0}][/tex]
Step-by-step explanation:
As given in question drag force depends upon the product of the cross sectional area of the object and the square of its velocity
and drag force can be given by
[tex]F=CAv^2[/tex] (1)
It is given that
Dimension of mass = [M]
Dimension of length = [L]
Dimension of time = [T]
So, by using above dimension we can write
the dimension of force,
[tex]F=[MLT^{-2}][/tex]
dimension of cross-section area,
[tex]A=[L^2][/tex]
and dimension of velocity
[tex]v=[LT^{-1}][/tex]
now, by putting these values in equation (1), we will get
[tex]F=CAv^2[/tex]
[tex]=>[MLT^{-2}]=C[L^2][LT^{-1}]^2[/tex]
[tex]=>C=[ML^{-3}T^0][/tex]
Hence, the dimension of constant C will be,
[tex]C=[ML^{-3}T^0][/tex]
The dimensions of C from the expression above is ML^-3
Units and DimensionGiven the function that relates drag force with the cross-sectional area of the object and the square of its velocity expressed as:
[tex]F_{air} = CAv^2[/tex]
Make C the subject of the formula to have:
[tex]C =\frac{F}{Av^2}[/tex]
Given the following dimensions
[tex]M =MLT^{-2}[/tex]
A = L²
v = [tex]LT^{-1}[/tex]
Substitute into the formula. the dimension of C will be given as:
[tex]C=\frac{MLT^{-2}}{L^2L^2T^{ -2}}\\C= ML^{-3}[/tex]
Henc the dimensions of C from the expression above is ML^-3
Learn more on Units and Dimension here: https://brainly.com/question/28464
(a) How many prime numbers are (b) How many prime numbers are also abundant numbers?
Answer:
a) There are infinite prime numbers, b) All prime numbers are also abundant numbers
Step-by-step explanation:
To prove a) let's first prove that if n divides both integers A and B then also divides the difference A-B
If n divides A and B, there are integers j, k such that
A = nj and B= nk,
So
A-B= nj - nk = n(j-k)
But j-k is also an integer, which means that n divides also A-B
Now, to prove that there are infinite prime numbers , we will proceed with Reductio ad absurdum.
We will suppose that there are only a finite number of primes and then arrive to a contradiction.
Suppose there are only n prime numbers,
{p1,p2,... pn}
then take P=p1.p2...pn the product of all of them
and consider P+1
If P+1 is prime the proof is complete for P+1 is not in the list.
if P+1 is not prime then by the Fundamental Theorem of Arithmetic there is a prime in the list that must divide P+1, let's say pk
Then pk also divides P+1-P=1 which is a contradiction because no prime divides 1.
b) To prove this, recall that an abundant number is a number for which the sum of its proper divisors is greater than the number itself.
Given that a prime number P is only divided by P and 1, the sum of its divisors is P+1 which is greater than P. So P is abundant
How many possible ways are there to fill in answers to a
quizwith five multiple choice questions when the choices are a, b,
andc?
Answer: There are 15 possible ways to fill in answers .
Step-by-step explanation:
Given : The number of multiple choice questions = 5
The total number of choices for each question {a, b,
and c} = 3
Now by using the fundamental principle of counting , the number of possible ways to fill in answer is given by :-
[tex]5\times3=15[/tex]
Therefore, there are 15 possible ways to fill in answers .
An die (six faces) has the number 1 painted on three of its faces, the number 2 painted on two of its faces, and the number 3 painted on one face. Assume that each face is equally likely to come up. Find a sample space for this experimen
Answer: {1 ,2 ,3 }
Step-by-step explanation:
We know that a sample space is a set of possible occurring oin an experiment.
Given : An die (six faces) has the number 1 painted on three of its faces, the number 2 painted on two of its faces, and the number 3 painted on one face.
We assume that each face is equally likely to come up.
When we toss a dice , then the possible occurring = 1 , 2, 3
Then, the sample space for this experiment will be {1 ,2 ,3 }
Final answer:
Explaining the sample space for an experiment with a die displaying numbers in different frequencies.
Explanation:
An die (six faces) has the number 1 painted on three of its faces, the number 2 painted on two of its faces, and the number 3 painted on one face. The sample space for this experiment would be: {1, 1, 1, 2, 2, 3}.
This means that when you roll the die, the possible outcomes are: 1, 1, 1, 2, 2, 3.
A survey of 250 adults found that during the last year, 70 traveled by plane but not by train, 70 traveled by train but not by plane, 20 traveled by bus but not by plane or by train, 45 traveled by bus and plane, 20 traveled by all three, and 185 traveled by plane or train. How many did not travel by any of these modes of transportation? and plane, 20 traveled by all three, and 15 traveled by blanc eled by plane but not by train, 95
People who didn't travel by any mode = 45
Step-by-step explanation:In the question,
Total number of people included in survey = 250
People who traveled by plane but not by train = 70
i.e.
a + e = 70
People who traveled by train but not by plane = 70
i.e.
c + d = 70
People who traveled by Bus but not by Plane or Train = 20
i.e.
f = 20 ..........(1)
People who traveled by bus and plane both = 45
i.e.
e + g = 45
People who traveled by all three = 20
i.e.
g = 20
People who traveled by Plane or Train = 185
i.e.
a + b + c + d + e + g = 185 ........(2)
So,
e = 45 - g = 45 - 20 = 25
e = 25
Now, on putting in eqn. (2) we get,
a + b + c + d + 25 + 20 = 185
a + b + c + d = 140 .......(3)
Now,
We need to find out,
Number of people travelling with any of these three is,
a + b + c + d + e + f + g
So,
On putting from eqn. (3) and (1), we get,
a + b + c + d + e + f + g = 140 + 25 + 20 + 20 = 205
So,
Number of people who didn't travel by any mode =Total people - Number of people travelling by any three
People who didn't travel by any mode = 250 - 205 = 45
Final answer:
By analyzing the given data and using set theory, we determined that 90 adults did not travel by plane, train, or bus in the last year.
Explanation:
To solve this question, we need to find out how many adults did not travel by plane, train, or bus. We are given several subsets of people who use various combinations of these modes of transportation, and we can use a principle in set theory to determine the answer.
We know that 185 adults traveled by plane or train. This number includes those who traveled by both modes. There is an overlap of people who used all three modes, which is 20. So, to find the sole plane and train travelers, we subtract the people who used all three modes from those who traveled by plane but not by train and vice versa.
We calculate the number of plane-only and train-only travelers: 70 (plane only) + 70 (train only) - 20 (all three) = 120.
Since 185 traveled by either plane or train, the number that traveled by either without the bus is 185 - 20 (all three) = 165. The number that traveled by plane or train only is 165 - 45 (bus and plane) = 120.
Adding those who traveled by bus but not by plane or train (20) to those who traveled by all three (20) gives us 40. Therefore, 120 (plane or train only) + 40 = 160. To find out the number of adults who did not travel by any of these three modes, we subtract 160 from the total number of surveyed adults (250).
Finally, the number of adults who did not travel by any mode is 250 - 160 = 90.
Therefore, 90 adults did not travel by plane, train, or bus.
Convert 500 cubic feet to liters
Answer:
500 cubic feet is equal to 14158.4 liters.
Step-by-step explanation:
Since, we know that,
1 square feet = 28.3168 liters,
Thus, the number of liters in 500 cubic feet = 500 × number of liters in 1 square feet
[tex]=500\times 28.3168[/tex]
[tex]=14158.4[/tex]
Therefore, 500 cubic feet is equal to 14158.4 liters.
500 cubic feet is 14158.4 liters.
To convert 500 cubic feet to liters, follow these steps:
Using the conversion factor:
1 ft³ = 28.3168 L
So, to convert 500 cubic feet to liters:
500 ft³ × 28.3168 L/ft³ = 14158.4 L
Food mix A contains 2% fat, and food mix B contains 7% fat. A 20-kilogram diet mix of foods A and B is formed. If x kilograms of food A are used, write an algebraic expression that represents the total number of kilograms of fat in the final food mix. Simplify the expression.
Answer:
The required expression is y = 1.4 - 0.05x
Step-by-step explanation:
Consider the provided information.
Food mix A contains 2% fat and food mix B contains 7% fat.
Let x kilograms of food A are used, in a 20-kilogram mixture.
Thus, 20 - x kilograms of food B are used, in a 20-kilogram mixture.
Now It is given that A contains 2% fat and food mix B contains 7% fat.
2% and 7% can be written as 0.02 and 0.07 respectively. Let represent the total fat with y.
Thus, the required expression is:
y = 2% (x) + 7% (20 - x)
y = 0.02 (x) + 0.07 (20 - x)
y = 0.02x + 1.4 - 0.07x
y = 1.4 - 0.05x
Hence, the required expression is y = 1.4 - 0.05x
Answer the questions about the following function.
f left parenthesis x right parenthesis equals StartFraction 16 x squared Over x Superscript 4 Baseline plus 64 EndFractionf(x)=16x2 x4+64
(a) Is the point
left parenthesis negative 2 StartRoot 2 EndRoot comma 1 right parenthesis−22,1
on the graph of f?
(b) If
x equals 2 commax=2,
what is f(x)? What point is on the graph of f?
(c) If
f left parenthesis x right parenthesis equals 1 commaf(x)=1,
what is x? What point(s) is (are) on the graph of f?
(d) What is the domain of f?
(e) List the x-intercepts, if any, of the graph of f.
(f) List the y-intercept, if there is one, of the graph of f.
Answer:
(a) yes, (-2√2, 1) is on the graph
(b) f(2) = 4/5, the point is (2, 4/5)
(c) (-2√2, 1), and (2√2, 1)
(d) all real numbers
(e) (0, 0)
(f) (0, 0)
Step-by-step explanation:
You want various points on the graph of the function ...
[tex]\displaystyle f(x)=\frac{16x^2}{x^4+64}[/tex]
(a) (-2√2, 1)Yes, this point is on the graph. The value of f(x) can be found easily by realizing -2√2 = -√8, so ...
x² = 8x⁴ = 8² = 64and the function value is ...
[tex]f(-2\sqrt{2})=\dfrac{16\cdot8}{64+64}=\dfrac{128}{128}=1[/tex]
(b) f(2)Substituting 2 for x, we have ...
[tex]f(2) = \dfrac{16\cdot 2^2}{2^4+64}=\dfrac{64}{16+64}=\dfrac{64}{80}\\\\\boxed{f(2)=\dfrac{4}{5}}[/tex]
The point on the graph is (2, 4/5).
(c) f(x) = 1The answer to part (a) tells you that one of the points where f(x) = 1 is ...
(-2√2, 1)
Since the sign of x is irrelevant, another point where x=1 is ...
(2√2, 1)
(d) DomainThere are no values of x that make the denominator of this rational function zero, so its domain is all real numbers.
(e) X-interceptThe only x-value where f(x) = 0 is x = 0.
The x-intercept is (0, 0).
(f) Y-interceptThe function crosses the y-axis at the origin.
The y-intercept is (0, 0).
Solve each of the following equations for x. (a) 5x-7=28 (b) 12-5x= x+30 (c) 5(x+2)= 1-3x
Зx-y=-5 X+2y=3
Answer:
(a) 7
(b) -3
(c) [tex]-\frac{9}{8}[/tex]
(d) -1
Step-by-step explanation:
(a) 5x - 7 = 28
5x = 28 + 7
5x = 35
⇒ x = 7
(b) 12 - 5x = x + 30
-5x = x + 30 - 12
-5x = x + 18
-5x - x = 18
-6x = 18
⇒ x = -3
(c) 5(x+2) = 1 - 3x
5x + 10 = 1 - 3x
5x = 1 - 3x - 10
5x + 3x = -9
8x = -9
⇒ x = [tex]-\frac{9}{8}[/tex],
(d) Given system of equations,
Зx-y = -5 ------(1),
x + 2y = 3 ----(2),
Equation (2) + 2 equation (1),
x + 6x = 3 - 10⇒ 7x = -7 ⇒ x = -1
A TV costs $125.67 with a discount of 15% and 8.5% tax. What is the total that you will have to pay?
Answer:
The total you have to pay for the TV is $115.9
Step-by-step explanation:
When we have a discount we have to make a substraction, when we have tax we have to sum so:
$125.67 * 15 % = 18.85
125.67 - 18.85 = 106.82
Now we have to add the tax
106.82 * 8.5% = 9.08
106.82 + 9.08 = 115.9
What is the total resistance of a parallel circuit that has two loads? Load one has a resistance of 10 ohms. Load two has a resistance of 24 ohms. (YOU MUST SHOW YOUR WORK)!!
Answer:
The total resistance is [tex]7.0588\Omega[/tex]
Step-by-step explanation:
Attached please find the circuit diagram. The circuit is composed by a voltage source and two resistors connected in parallel: [tex]R_1=10\Omega [/tex] and [tex]R_2=24\Omega [/tex].
First step: find the total current
For finding the current that the voltage source can provide, you must find the current consumed by each load and then add both. To do that, take first into account that the voltage is the same for both resistors ([tex]R_1[/tex] and [tex]R_2[/tex]).
[tex]I_{R_1}=\frac{V_S}{R_1}[/tex][tex]I_{R_2}=\frac{V_S}{R_2}[/tex]The total current is:
[tex]I_{TOTAL}=I_{R_1}+I_{R_2}=\frac{V_S}{R_1}+\frac{V_S}{R_2}=\frac{R_2\cdot V_S+R_1\cdot V_S}{R_1\cdot R_2}[/tex]
[tex]I_{TOTAL}=V_S\cdot \frac{R_1+R_2}{R_1\cdot R_2}[/tex]
Now, the total resistance ([tex]R_{TOTAL}[/tex]) would be the voltage divided by the total current:
[tex]R_{TOTAL}=\frac{V_S}{I_{TOTAL}}[/tex]
If you replace [tex]I_{TOTAL}[/tex] by the expression obtained previously, the total resistance would be:
[tex]R_{TOTAL}=\frac{V_S}{V_S\cdot \frac{R_1+R_2}{R_1\cdot R_2}}[/tex]
After simplifying the terms you should get:
[tex]R_{TOTAL}=\frac{R_1\cdot R_2}{R_1 + R_2}}[/tex]
Now, you must replace the values of the resistors:
[tex]R_{TOTAL}=\frac{(10\Omega )\cdot (24\Omega)}{10\Omega + 24\Omega}}=\frac{120}{17}\Omega=7.0588\Omega [/tex]
Thus, the total resistance is [tex]7.0588\Omega[/tex]
Northwest Molded molds plastic handles with a variable cost of $1.00 per handle. The fixed cost to run the molding machine is $2560 per week. If the company sells the handles for $3.00 each, how many handles must be molded weekly to break even? What is the profit if 1500 handles are produced and sold?
Answer:
To break even it must be molded 1280 handles weekly.
The profit if 1500 handles are produced and sold is $440
Step-by-step explanation:
To break even, the amount of total cost must be the same as the amount of revenues.
Total Cost is Fixed cost plus unitary variable cost multiplied by the produce quantity.
Total cost= FC + vc*Q
Where
FC=Fixed cost
vc=unitary variable cos
Q=produce quantity
Revenue= Price * Q
Break even FC + vc*Q=Price * Q
Isolating Q
FC=(Price * Q)-(vc*Q)
FC=(Price-vc) * Q
Q= FC/(Price-vc)
Q= $2560/($3.00-$1.00)=1280
If we sold 1500 handles
Profit = Revenue- Total cost =(Price * Q)-(FC + vc*Q)
P=$3.00 *1500-$2560 - $1.00*1500=
P=$4500-$2560-$1500=440
Final answer:
Northwest Molded must sell 1,280 handles to break even, based on their fixed weekly costs of $2,560 and a variable cost of $1.00 per handle with a selling price of $3.00 each. If they produce and sell 1,500 handles, they will make a profit of $440.
Explanation:
To calculate the break-even point for Northwest Molded, we need to determine the number of handles that must be sold to cover the total costs, which include both fixed and variable costs.
The fixed cost to operate the molding machine is $2,560 per week, and the variable cost per handle is $1.00.
Each handle is sold for $3.00. The break-even point is reached when total cost equals total revenue, which can be found using the break-even formula:
Break-even point in units = Fixed costs / (Selling price per unit - Variable cost per unit)
Thus, for Northwest Molded:
Break-even point in units = $2,560 / ($3.00 - $1.00) = $2,560 / $2.00 = 1,280 handles
To calculate the profit for producing and selling 1,500 handles, we need to compute the total revenue and subtract the total costs:
Total Revenue = Selling price per unit × Number of units sold = $3.00 × 1,500 = $4,500
Total Costs = Fixed costs + (Variable cost per unit × Number of units sold) = $2,560 + ($1.00 × 1,500) = $4,060
Profit = Total Revenue - Total Costs = $4,500 - $4,060 = $440
Therefore, in order to break even, Northwest Molded must mold and sell 1,280 handles weekly, and the company would make a profit of $440 if they produced and sold 1,500 handles.