Answer:
A set S is closed under an operation * (we're denoting the operation as asterisk) IF for any two elements a,b in S, the result a*b is also in S.
Step-by-step explanation:
A set being closed under an operation means that whenever you operate elements from the set, the result you get out of it is ALWAYS inside the set. For example, think of the set Z of integer numbers and the operation + (usual addition). If we add ANY two integers, we're going to get another integer. Or said in terms of sets, for any two numbers a,b in Z, a+b is also in Z.
On the other side, not being closed under an operation means you do NOT ALWAYS get results inside the same set. Think of the set of natural numbers N, and the operation - (usual difference). If we do the operation 5-12, we get -7 which is NOT in the set of natural numbers. So N is not closed under subtraction.
The mass of a particular substance is known to grow exponentially at a rate of 17% per week. Its initial mass was 12 grams and, after t weeks, it weighed 56 grams. The equation modelling this growth is 12x1.17 56. Use the method of taking logs to solve this equation for t, giving your answer correct to the nearest week. (Your answer should be a number, without units) Answer:
Answer:
[tex]t = 9.05[/tex] weeks
Step-by-step explanation:
The mass of this particular substance can be modeled by the following exponential function:
[tex]m(t) = m(0)*e^{rt}[/tex]
In which [tex]m(t)[/tex] is the mass in function of time, [tex]m(0)[/tex] is the initial mass and r, in decimal, is the growth rate of the mass.
The problem states that:
The mass of a particular substance is known to grow exponentially at a rate of 17% per week. Its initial mass was 12 grams and, after t weeks, it weighed 56 grams. So:
[tex]r = 17% = 0.17[/tex]
[tex]m(0) = 12[/tex]
[tex]m(t) = 56[/tex]
We have to solve this equation for t. So:
[tex]m(t) = m(0)*e^{rt}[/tex]
[tex]56 = 12*e^{0.17t}[/tex]
[tex]e^{0.17t} = \frac{56}{12}[/tex]
[tex]e^{0.17t} = 4.67[/tex]
To solve for [tex]t[/tex], we put ln in both sides
[tex]ln e^{0.17t} = ln 4.67[/tex]
[tex]0.17t = 1.54[/tex]
[tex]t = \frac{1.54}{0.17}[/tex]
[tex]t = 9.05[/tex] weeks
The total monthly profit for a firm is P(x)=6400x−18x^2− (1/3)x^3−40000 dollars, where x is the number of units sold. A maximum of 100 units can be sold per month.
How many units should be sold to maximize monthly profit? units
What is the maximum monthly profit when this number of units are sold? (Round your answer to the nearest whole number.) dollars
(Hint: When you solve for the critical values in this question, you'll have to solve a quadratic equation. You can do that using the quadratic formula, or you can do it by factoring.)
How to get this answer?
Answer:
Maximum profits are earned when x = 64 that is when 64 units are sold.
Maximum Profit = P(64) = 2,08,490.666667$
Step-by-step explanation:
We are given the following information:[tex]P(x) = 6400x - 18x^2 - \frac{x^3}{3} - 40000[/tex], where P(x) is the profit function.
We will use double derivative test to find maximum profit.
Differentiating P(x) with respect to x and equating to zero, we get,
[tex]\displaystyle\frac{d(P(x))}{dx} = 6400 - 36x - x^2[/tex]
Equating it to zero we get,
[tex]x^2 + 36x - 6400 = 0[/tex]
We use the quadratic formula to find the values of x:
[tex]x = \displaystyle\frac{-b \pm \sqrt{b^2 - 4ac} }{2a}[/tex], where a, b and c are coefficients of [tex]x^2, x^1 , x^0[/tex] respectively.
Putting these value we get x = -100, 64
Now, again differentiating
[tex]\displaystyle\frac{d^2(P(x))}{dx^2} = -36 - 2x[/tex]
At x = 64, [tex]\displaystyle\frac{d^2(P(x))}{dx^2} < 0[/tex]
Hence, maxima occurs at x = 64.
Therefore, maximum profits are earned when x = 64 that is when 64 units are sold.
Maximum Profit = P(64) = 2,08,490.666667$
At an effective annual interest rate of i > 0, each of the following two sets of payments has present value K: (i) A payment of 169 immediately and another payment of 169 at the end of two years. (ii) A payment of 225 at the end of two years and another payment of 225 at the end of four years. Calculate K.
Answer:
The present value of K is, [tex]K=251.35[/tex]
Step-by-step explanation:
Hi
First of all, we need to construct an equation system, so
[tex](1)K=\frac{169}{(1+i)} +\frac{169}{(1+i)^{2}}[/tex]
[tex](2)K=\frac{225}{(1+i)^{2}} +\frac{225}{(1+i)^{4}}[/tex]
Then we equalize both of them so we can find [tex]i[/tex]
[tex](3)\frac{169}{(1+i)} +\frac{169}{(1+i)^{2}}=\frac{225}{(1+i)^{2}} +\frac{225}{(1+i)^{4}}[/tex]
To solve it we can multiply [tex](3)*(1+i)^{4}[/tex] to obtain [tex](1+i)^{4}*(\frac{169}{(1+i)} +\frac{169}{(1+i)^{2}}=\frac{225}{(1+i)^{2}} +\frac{225}{(1+i)^{4}})[/tex], then we have [tex]225(1+i)^{2}+225=169(1+i)^{3}+169(1+i)^{2}[/tex].
This leads to a third-grade polynomial [tex]169i^{3}+451i^{2}+395i-112=0[/tex], after computing this expression, we find only one real root [tex]i=0.2224[/tex].
Finally, we replace it in (1) or (2), let's do it in (1) [tex]K=\frac{169}{(1+0.2224)} +\frac{169}{(1+0.2224)^{2}}\\\\K=251.35[/tex]
Calculate the total interest paid on a 30-year, 3.9% fixed-rate $200,000 mortgage loan.
Remember that number of compounding periods in a year n = number of payments expected to be made in a year. If you make monthly mortgage payments, then interest on the loan is compounded monthly.
Give answer in dollars rounded to the nearest cent. Do NOT enter "$" sign in answer.
Answer:
139,600.96
Step-by-step explanation:
We use the payment of a loan formula:
[tex] \displaystyle PMT = \frac{P \left(\displaystyle \frac{r}{n}\right)}{\left[ 1 - \left( 1 + \displaystyle \frac{r}{n}\right)^{-nt} \right]} [/tex]
P is the principal: $200,000. t is the number of years: 30, n is 12 since it is compounded monthly. And r is 0.039 which is 3.9% in decimal form (3.9/100)
So the formula becomes:
[tex] \displaystyle PMT = \frac{200000 \left(\displaystyle \frac{0.039}{12}\right)}{\left[ 1 - \left( 1 + \displaystyle \frac{0.039}{12}\right)^{-12(30)} \right]} [/tex]
And using our calculator we get: PMT = $943.336
Then the total amount of money paid in the mortgage is:
PMT*n*t = $943.336(12)(30) = $339,600.96
Therefore, the interest paid is:
$339,600.96 - $200,000 = $139,600.96
You have to enter it without $ and rounded to the nearest cent so: 139,600.96
Can anyone help me determine the infinite limit of these two problems
lim x--> -2^+(from the right)
[tex]\frac{x-1}{x^{2} (x+2)}[/tex]
lim x-->0 (from both left and right)
[tex]\frac{x-1}{x^{2} (x+2)}[/tex]
lim x-->2π⁻
x csc x
* I used direct substitution to determine the infinite limit of each but I have problems justifying how I got that answer by picking random points to plug into the equation
[tex]\displaystyle\lim_{x\to-2^+}\frac{x-1}{x^2(x+2)}[/tex]
The limit is infinite because the denominator approaches 0 while the numerator does not, since [tex]x+2=0[/tex] when [tex]x=-2[/tex]. Which infinity it approaches (positive or negative) depends on the sign of the other terms for values of [tex]x[/tex] near -2.
Since [tex]x\to-2[/tex] from the right, we're considering values of [tex]x>-2[/tex]. For example, if [tex]x=-1.9[/tex], then [tex]\dfrac{x-1}{x^2}=\dfrac{-2.9}{1.9^2}<0[/tex]; if [tex]x=-1.99[/tex], then [tex]\dfrac{x-1}{x^2}=\dfrac{-2.99}{1.99^2}<0[/tex], and so on. We can keep picking values of [tex]x[/tex] that get closer and closer to -2, and we would see that [tex]\dfrac{x-1}{x^2}[/tex] contributes a negative sign every time. So the limit must be [tex]\boxed{-\infty}[/tex].
[tex]\displaystyle\lim_{x\to0}\frac{x-1}{x^2(x+2)}[/tex]
By similar reasoning above, we see that [tex]\dfrac{x-1}{x+2}[/tex] contributes a negative sign regardless of which side we approach 0 from. [tex]x-1[/tex] is always negative and [tex]x+2[/tex] is always positive, so the net effect is a negative sign and the limit from either side is [tex]\boxed{-\infty}[/tex].
[tex]\displaystyle\lim_{x\to2\pi}x\csc x=\lim_{x\to2\pi}\frac x{\sin x}[/tex]
Direct substitution gives 0 in the denominator. For [tex]x>2\pi[/tex] we have [tex]\sin x>0[/tex], and for [tex]x<2\pi[/tex] we have [tex]\sin x<0[/tex]. Meanwhile, the numerator stays positive, which means the limit is positive or negative infinity depending on the direction in which [tex]x[/tex] approaches [tex]2\pi[/tex], so this limit does not exist.
The rate of sales for a corporation t weeks from now is given by S(t)=59t+5 millions of dollars per week. Find the average sales per week for the first 16 weeks. Set up the integral to solve the problem and express your answer to the nearest dollar.
Answer:
The average sales per week for the first 16 weeks is $477 million dollars per week.
Step-by-step explanation:
Here, you have to find the average value of a continuous function over an interval.
Suppose you have a function [tex]f(x)[/tex] over an interval from a to b. The average of the function in this interval is given by:
[tex]\frac{1}{b-a}\int\limits^b_a {f(x)} \, dx[/tex]
Solution:
In this problem, the function is given by:
[tex]S(t) = 59t + 5[/tex]
The problem asks the average value for the first 16 weeks. It means that our interval goes from 0 to 16. So [tex]a = 0, b = 16[/tex].
The average value is given by the following integral:
[tex]A = \frac{1}{16}\int\limits^{16}_{0} {(59t + 5)} \, dt[/tex]
[tex]A = \frac{59t^{2}}{32} + \frac{5t}{16}, 0 \leq t \leq 16[/tex]
[tex]A =\frac{59*(16)^{2}}{32} + \frac{5*16}{16}[/tex]
[tex]A = $477[/tex]
The average sales per week for the first 16 weeks is $477 million dollars per week.
You have a goal of accumulating $500,000 in an account 30 years from now. If the account earns 9% per year, how much would you have to deposit now to grow to the desired goal?
N= I/Y= PV= PMT= FV= P/Y=
Answer:
$37685.56
Step-by-step explanation:
Given,
Total amount we want to accumulate,A = $500,000
Total time, we have,t = 30 years
Interest rate,r = 9%
We are asked to calculate how much money we should deposit to get the required amount after a certain time period.
So, according to compound interest formula,
[tex]A\ =\ P(1+r)^t[/tex]
Where, P = amount of money we need to deposit
[tex]=>\ 500,000\ =\ P(1+0.09)^{30}[/tex]
[tex]=>\ 500,000\ =\ P(1.09)^{30}[/tex]
[tex]=>\ 500,000\ =\ P\times 13.267[/tex]
[tex]=>\ \dfrac{500,000}{13.267}\ =\ P[/tex]
[tex]=>\ P\ =\ 37685.568[/tex]
So, we need to deposit total amount of $37,685.56.
Suppose you are taking out a loan for $10,000, at an annual percentage rate of 5%, for a term of 5 years. What is the amount of accrued interest?
Principal - $10,000.00
Rate - 5%
Time - 5 years
Answer:
$2500
Step-by-step explanation:
The accrued interest is given by the formula ...
i = Prt
i = interest; P = principal amount of the loan; r = annual interest rate; t = number of years.
__
Fill in the values and do the arithmetic.
i = $10000×0.05×5 = $2500
The accrued interest on a $10,000 loan at a 5% annual interest rate for a term of 5 years is $2,500.
The accrued interest on a loan is calculated using the formula: Interest = Principal × rate × time. For a loan amount (Principal) of $10,000 at an annual interest rate (Rate) of 5% over a period (Time) of 5 years, the simple interest calculation would be:
Interest = Principal × rate × time
Interest = $10,000 × 0.05 × 5
Interest = $500 × 5
Interest = $2,500
Therefore, the accrued interest over the term of 5 years on a $10,000 loan at a 5% annual interest rate would be $2,500.
The cost of producing x hundred items is given by the equation C(x) = x2 – 3x + 7 and the revenue generated from sales of x hundred units is given by the equation R(x) = –x2 + 21x – 33. What values of x will the company break even?
Answer:
At x = 2 and 10.
Step-by-step explanation:
Given : The cost of producing x hundred items is given by the equation [tex]C(x) = x^2-3x + 7[/tex]
The revenue generated from sales of x hundred units is given by the equation [tex]R(x) = -x^2 + 21x-33[/tex]
To Find :What values of x will the company break even?
Solution:
Cost function : [tex]C(x) = x^2-3x + 7[/tex]
Revenue function : [tex]R(x) = -x^2 + 21x-33[/tex]
Now to find the company break even :
[tex]-x^2 + 21x-33= x^2-3x + 7[/tex]
[tex]24x= 2x^2+40[/tex]
[tex]12x= x^2+20[/tex]
[tex]x^2-12x+20=0[/tex]
[tex]x^2-10x-2x+20=0[/tex]
[tex]x(x-10)-2(x-10)=0[/tex]
[tex](x-2)(x-10)=0[/tex]
So, x = 2,10
Hence the company break even at x = 2 and 10.
Birth and death rates are often reported as births or
deathsper thousand members of the population. What isthe relative
rate of growth of a population with a birth rate of 30births per
1000 and a death rate of 20 deaths per 1000?
Answer:
10 growth per 1000.
Step-by-step explanation:
Given,
Rate of birth = 30 births per 1000
Rate of death = 20 deaths per 1000
As the growth in population is the difference in the number of the child take birth and the person die.
As we are calculating the rate of birth and rate of growth in per thousands of members, so the growth rate will be also in per thousands.
As we can see on every one thousand people,
total birth = 30
total death = 20
so, total growth = total birth - total growth
= 30 - 20
= 10
As at every 1000 persons, there are 10 persons survive, so the rate of growth will be 10 growth per 1000.
If lim f(x) - 4/(x - 4) = 11, find lim f(x).
Answer:
lim f(x)=(11x-40)/(x-4) // for all x's, except x=4
lim f(x) ≈ ∞ // if x=4
Step-by-step explanation:
lim f(x) - 4/(x - 4) = 11
lim f(x) = 11 + 4/(x - 4)= (11x-40)/(x-4)
at x=4, there is a indetermination point, the function tends to infinity at that point
So:
lim f(x)=(11x-40)/(x-4) // for all x's, except x=4
lim f(x) ≈ ∞ // if x=4
According to the History 202 syllabus, the first paper is worth 25% of the course grade, the second paper is worth 20% of the course grade, and the final exam is worth 55% of the overall course grade. Inara earns a 74% on the first term paper, an 99% on the second term paper, and the final exam has not happened yet. What is the lowest grade that Inara would need on the final exam in order to earn at least an 80% in the course?
Answer:
76%
Step-by-step explanation:
Let f represent the score on the final. Then Inara wants the final grade to be ...
0.25·74 +0.20·99 +0.55·x ≥ 80
38.3 +0.55x ≥ 80
0.55x ≥ 41.7
x ≥ 41.7/0.55 ≈ 75.82
Inara needs a final exam grade of 76% or better to earn at least 80% in the course.
How do I write 8 ten-thousands
Answer:
8 ten-thousands is 80 thousands.
Step-by-step explanation:
To find : How do I write 8 ten-thousands?
Solution :
We have to write 8 ten thousand,
We know that, according to numeric system
[tex]\text{Thousand is 1000}[/tex]
[tex]\text{Ten-Thousand is 10,000}[/tex]
[tex]\text{8 ten thousand is}\ 8\times 10000[/tex]
[tex]\text{8 ten thousand is 80000}[/tex]
Therefore, 8 ten-thousands is 80 thousands.
How does the number of solutions of a system depend on ranks of the coefficient matrix and the augmented matrix?
Answer:
The number of solutions of a system is given by the number of different variables in the system, this number has to be the same as the number of independent equations. The coefficients and the augmented matrix of the system show these values in a matrix form. A system has a unique solution when the rank of both matrixes and the number o variables in the system are the same.
Step-by-step explanation:
For example, the following system has 2 different variables, x and y.
[tex]x+y=1\\x+2y=5[/tex]
In order to find a unique solution to the system, the number of independent equations and variables in the system must be the same In the previous example, you have 2 independent equations and 2 variables, then the solution of the system is unique.
The rank of a matrix is the dimension of the vector generated by the columns, in other words, the rank is the number of independent columns of the matrix.
According to Rouché-Capelli Theorem, a system of equations is inconsistent if the rank of the augmented matrix is greater than the rank of the coefficient matrix. The inconsistency of the system is because you can't find a combination of the variables that will solve the system.
I need help with these four questions please (68 points)
Answer:
Part 1) The distance is [tex]d=7.3\ units[/tex]
Part 2) The measure of angle 2 is 121°
Part 3) The coordinates of endpoint V are (7,-27)
Part 4) The value of x is 10
Step-by-step explanation:
Part 1) Find the distance between M(6,16) and Z(-1,14)
we know that
the formula to calculate the distance between two points is equal to
[tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex]
substitute the given values in the formula
[tex]d=\sqrt{(14-16)^{2}+(-1-6)^{2}}[/tex]
[tex]d=\sqrt{(-2)^{2}+(-7)^{2}}[/tex]
[tex]d=\sqrt{53}\ units[/tex]
[tex]d=7.3\ units[/tex]
Part 2) Find the measure m∠2
we know that
If two angles are supplementary, then their sum is equal to 180 degrees
In this problem we have
m∠1+m∠2=180°
substitute the given values
[tex](4y+7)\°+(9y+4)\°= 180\°[/tex]
Solve for y
[tex](13y+11)\°= 180\°[/tex]
[tex]13y= 180-11[/tex]
[tex]13y=169[/tex]
[tex]y=13[/tex]
Find the measure of m∠2
[tex](9y+4)\°[/tex]
substitute the value of y
[tex](9(13)+4)=121\°[/tex]
Part 3) The midpoint of UV is (5,-11). The coordinates of one endpoint are U(3,5) Find the coordinates of endpoint V
we know that
The formula to calculate the midpoint between two points is
[tex]M=(\frac{x1+x2}{2},\frac{y1+y2}{2})[/tex]
we have
[tex]M(5,-11)[/tex]
[tex](x1,y1)=(3,5)[/tex]
substitute and solve for (x2,y2)
[tex](5,-11)=(\frac{3+x2}{2},\frac{5+y2}{2})[/tex]
so
Equation 1
[tex]5=(3+x2)/2[/tex]
[tex]10=3+x2[/tex]
[tex]x2=7[/tex]
Equation 2
[tex]-11=(5+y2)/2[/tex]
[tex]-22=(5+y2)[/tex]
[tex]y2=-27[/tex]
therefore
The coordinates of endpoint V are (7,-27)
Part 4) GI bisects ∠DGH so that ∠DGI is (x-3) and ∠IGH is (2x-13) Find the value of x
we know that
If GI bisects ∠DGH
then
∠DGI=∠IGH
Remember that bisects means, divide into two equal parts
substitute the given values
[tex]x-3=2x-13[/tex]
solve for x
[tex]2x-x=-3+13[/tex]
[tex]x=10[/tex]
If an intravenous fluid is adjusted to deliver 15 mg of medication to a patient per hour, how many milligrams of medication are delivered per half minute?
If the intravenous fluid is adjusted to deliver 15 mg of medication per hour, the dosage rate would be 0.5 mg of medication delivered per half minute.
First, let's convert the 1 hour dosage rate of 15 mg to minutes. Since there are 60 minutes in an hour, divide 15 mg by 60 to find the dosage rate per minute.
15 mg / 60 min = 0.25 mg/min
Next, convert the dosage rate from minutes to half minutes. Since there are 2 half minutes in a minute, multiply the dosage rate per minute by 2 to find the dosage rate per half minute.
0.25 mg/min * 2 = 0.5 mg/half min
Therefore, if the intravenous fluid is adjusted to deliver 15 mg of medication per hour, the dosage rate would be 0.5 mg of medication delivered per half minute.
Know more about unit conversion,
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If 15 mg of medication is delivered per hour, the rate per half-minute would be 0.125 mg. This is calculated by converting 15 mg/hour to 0.25 mg/minute, and then halving to find the dose per half-minute.
Explanation:The student's question pertains to the calculation of dosage rates in medical treatment, specifically how much medication is delivered per half-minute given a rate of 15 mg of medication per hour. To figure this out, it's a simple conversion from hours to minutes and then to half-minutes.
Firstly, convert the hourly rate into a minute rate. There are 60 minutes in an hour. So, 15 mg per hour is the same as 15/60 = 0.25 mg per minute.Then, to convert to half minutes, you need to divide the minute rate by 2, because there are two half-minutes in one minute. Hence, 0.25 mg per minute is the same as 0.25/2 = 0.125 mg per half-minute.Learn more about Dosage Calculation here:https://brainly.com/question/31339825
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Use the row operations tool to solve the following system of equations, obtaining the solutions in fraction form.
15x + 13y + 4z = 8
11x + 13y + 9z = 1
3x + 5y + 7z = -5
8x + 8y + 2z = 6
7x + 5y + 2z = 2
Give the values for x, y, and z with the fractions reduced to lowest terms (for example 1/2 rather than 3/6).
x =
y =
z =
Answer:
[tex]x=-\frac{7}{26}, y=\frac{37}{26}, z=-\frac{21}{13}[/tex]
Step-by-step explanation:
The matrix representation of the system of linear equations is:
[tex]\left(\begin{array}{ccccc}15&13&4&\vdots&8\\11&13&9&\vdots&1\\3&5&7&\vdots&-5\\8&8&2&\vdots&6\\7&5&2&\vdots&2\end{array}\right)[/tex]
First apply the following row operations:
[tex]\begin{array}{c}R_{2}\to R_{2}+(-\frac{11}{15})R_{1}\\R_{3}\to R_{3}+(-\frac{1}{5})R_{1}\\R_{4}\to R_{4}+(-\frac{8}{15})R_{1}\\R_{5}\to R_{5}+(-\frac{7}{15})R_{1}\end{array}[/tex]
The resulting matrix is:
[tex]\left(\begin{array}{ccccc}15&13&4&\vdots&8\\0&\frac{52}{15}&\frac{91}{15}&\vdots&-\frac{73}{15}\\0&\frac{12}{5}&\frac{31}{5}&\vdots&-\frac{33}{5}\\0&\frac{16}{15}&-\frac{2}{15}&\vdots&\frac{26}{15}\\0&-\frac{16}{15}&\frac{2}{15}&\vdots&-\frac{26}{15}\end{array}\right)[/tex]
Then apply the row operations:
[tex]\begin{array}{c}R_{3}\to R_{3}+(-\frac{12}{5}\cdot \frac{15}{52})R_{2}\\R_{4}\to R_{4}+(-\frac{-16}{15}\cdot \frac{15}{52})R_{2}\\R_{5}\to R_{5}+(\frac{16}{15}\cdot \frac{15}{52})R_{2}\end{array}[/tex]
The resulting matrix is:
[tex]\left(\begin{array}{ccccc}15&13&4&\vdots&8\\0&\frac{52}{15}&\frac{91}{15}&\vdots&-\frac{73}{15}\\0&0&2&\vdots&-\frac{42}{13}\\0&0&-2&\vdots&\frac{42}{13}\\0&0&2&\vdots&-\frac{42}{13}\end{array}\right)[/tex]
Now apply the row operations:
[tex]\begin{array}{c} R_{4}\to R_{4}+R_{3}\\R_{5}\to R_{5}+(-1)R_{3}\end{array}[/tex]
The resulting matrix is:
[tex]\left(\begin{array}{ccccc}15&13&4&\vdots&8\\0&\frac{52}{15}&\frac{91}{15}&\vdots&-\frac{73}{15}\\0&0&2&\vdots&-\frac{42}{13}\\0&0&0&\vdots&0\\0&0&0&\vdots&0\end{array} \right)[/tex]
The equivalent linear system associated to this matrix is
[tex]\begin{cases}15x+13y+4z=8\\\frac{52}{15}y+\frac{91}{15}z=-\frac{73}{15}\\2z=-\frac{42}{13}\end{cases}[/tex]
To Solve this last system is very simple by substitution. The solutions are:
[tex]x=-\frac{7}{26}, y=\frac{37}{26}, z=-\frac{21}{13}[/tex]
graph the region of solution of the given linear inequality
x>7
Answer:
The graph of given inequality is shown below.
Step-by-step explanation:
The given linear inequality is
[tex]x>7[/tex]
We need to graph the region of solution of the given linear inequality.
The related equation of given linear inequality is
[tex]x=7[/tex]
We know that x=a is a vertical line which passes through the point (a,0).
Here, a =7. So x=7 is a vertical line which passes through the point (7,0).
Relate line is a dotted line because the sign of inequality is >. It means the points on the line are not included in the solution set.
Shaded region is right side of the related line because the solution set contains all possible values of x which are greater than 7.
17,210,000,000 written in scientific notation?
Answer:
17.21 × 10^7
Step-by-step explanation:
When writting in scientific notation, you only place the decimal as 17.21. Then, you count how many zeros are after of before the numbers. If it's before 17.21, then it would be 17.21^-7, if it is after 17.21, then it would be 17.21^7.
Answer:
[tex]1.721 \cdot 10^{10}[/tex]
Step-by-step explanation:
17,210,000,000 written in scientific notation
To write the number in scientific notation we move the decimal point after the first number.
To move the decimal point after 1, move 10 places to the left
So the number becomes
1.7210000000
Ignore all the zeros. we moved 10 places to the left so multiply with 10^10
[tex]1.721 \cdot 10^{10}[/tex]
Isla estimates that her dinner has 48 grams of fat in it.
What percentage of her total daily allowance of fat does her meal contain?
Answer:
The percentage of her total daily allowance of fat is 68.57%
Step-by-step explanation:
The complete question is:
Isla wants to watch her intake of fat per day. She estimates that the dinner will have 48 grams of fat. What percentage of her total Daily allowance of fat Will the dinner be? Daily allowance is 70g of fat
Solution:
Estimated fat in dinner = 48 grams
Daily allowance of fat = 70 grams
We have to find the percentage of her total Daily allowance of fat = ?
%fat = Estimated fat in dinner × 100 / Daily allowance of fat
%fat = 48×100/70
%fat = 68.57 %
Thus the percentage of her total daily allowance of fat is 68.57%
Without knowing Isla's total daily fat allowance, we can't work out the percentage that her dinner represents. However, if we knew this, we would divide the grams of fat in her dinner by her daily allowance and then multiply by 100 to find the percentage.
Explanation:To answer this question, we would need to have information on what Isla's total daily allowance of fat is. Without this information, it's impossible to calculate the percentage. If we knew her daily allowance, we would divide the amount of fat in her dinner (48 grams) by her total daily fat allowance and multiply by 100 to get the percentage. For example, if her daily allowance was 100 grams, her dinner would comprise 48% of her daily fat intake.
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g You manage an ice cream factory that makes two flavors: Creamy Vanilla and Continental Mocha. Into each quart of Creamy Vanilla go 2 eggs and 3 cups of cream. Into each quart of Continental Mocha go 1 egg and 3 cups of cream. You have in stock 450 eggs and 750 cups of cream. You make a profit of $3 on each quart of Creamy Vanilla and $2 on each quart of Continental Mocha. How many quarts of each flavor should you make to earn the largest profit
Answer:
You should make 200 quarts of Creamy Vanilla and 50 quarts of Continental Mocha to earn the largest profit.
Step-by-step explanation:
You are going to earn the largest profit when you manage to use all the eggs and cups of cream that you have in stock.
This problem can be solved by a first order equation
I am going to call x the number of quarts of Creamy Vanilla and y the number of quarts of Continental Mocha.
The problem states that each quart of Creamy Vanilla uses 2 eggs and each quart of Continental Mocha uses 1 egg. There are 450 eggs in stock, so:
2x + y = 450.
The problem also states that each quart of Creamy Vanilla uses 3 cups of cream and that each quart of Continental Mocha uses 3 cups of cream. There are 750 cups of cream in stock, so:
3x + 3y = 750
Now we have to solve the following system of equations
1) 2x + y = 450
2) 3x + 3y = 750
The first thing i am going to do is simplify the equation 2) by 3
2) 3(x+y)/3 = 750/3
2) x+y = 250
So now, we have the following system
1) 2x + y = 450
2) x + y = 250
I am going to write y as a function of x in 2), and replace in 1)
y = 250 - x
Replacing in 1)
2x + 250 - x = 450
2x - x = 450 - 250
x = 200
You should make 200 quarts of Creamy Vanilla
From 2), we have
y = 250 - x = 250 - 200 = 50
You should make 50 quarts of Continental Mocha
Write the elements of (a, b,c}-{a, b, d}) {c, d), where a, b, c, and d are distinct
Answer: If we have a set defined as the difference of two sets, this is
A = B - C, then A = B - B∩C
So this is defined as, if you subtract the set C to the set B, then you are subtracting the common elements between C and B, from the set B.
so if A = {a,b,c} - {a,b,d}, the common elements are a and b, so: A = {c}
Suppose that Mr. Warren Buffet and Mr. Zhao Danyang agree to meet at a specified place between 12 pm and 1 pm. Suppose each person arrives between 12 pm and 1 pm at random with uniform probability. What is the distribution function for the length of the time that the first to arrive has to wait for the other?
The distribution function for the length of the time that the first to arrive has to wait for the other is given by a triangular shape. This comes from the continuous uniform distribution nature of their arrival time. More probable waiting times are represented by the peak of the triangle.
Explanation:This problem relates to the mathematical concept of a continuous uniform distribution. In the meeting scenario of Mr. Warren Buffet and Mr. Zhao Danyang, each person arrives between 12 pm and 1 pm at random with a uniform probability. This means there is equal likelihood for each time interval within the hour for them to arrive.
We are interested in the length of time the first person to arrive has to wait for the second. Let's denote the time of arrival of Mr. Buffet by X and that of Mr. Zhao by Y, both from 0 (12:00 pm) to 1 (1:00 pm). The time the first to arrive has to wait is |X - Y|. The distribution function of this waiting time (W = |X - Y|) is a triangle with its peak at W = 0 (no waiting time). The two ends of the base of the triangle fall at -1 and 1. This triangular shape comes from the fact that shorter waiting times (near the peak) are more probable because there is more overlap between X and Y.
Therefore, the cumulative distribution function for the waiting time, W, is given by:
0, for W < 0W^2/2, for 0 <= W < 11 - (1 - W)^2/2, for 1 <= W <= 21, for W > 2Learn more about Continuous Uniform Distribution here:https://brainly.com/question/32614835
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Suppose that the total cost of making and selling x widgets is C = 32x + 560 dollars. What is the meaning of the slope in this case?
Answer: The slope ($32) is giving the rate of each widgets.
Step-by-step explanation:
The equation of a line in intercept form is given by :-
[tex]y=mx+c,[/tex] where the coefficient of x is the slope of line.
Given : The total cost of making and selling x widgets is [tex]C = 32x + 560[/tex] is in intercept form, where the coefficient of x = 32.
Since the slope gives the rate of change of dependent variable w.r.t. independent variable.
Therefore, the meaning of the slope in this case is the the rate of change of cost per unit of widgets x.
i.e Slope gives the cost of each widgets =$32.
Find the derivative of the given function with respect to the independent variable x or t. The symbols a, b and c are constants greater than 1. You are not required to combine like terms, reduce fractions, or otherwise simplify your final answer.
(1) y = (2/[ a+bx])^3
(2) y = (at^3 - 3bt)^3
(3) y = (t^b)e^(b/t)
(4) z = ax^2.sin (4x)
Answer:
(1) [tex]y=(\frac{2}{a+bx})^3[/tex]
By differentiating w.r.t. x,
[tex]\frac{dy}{dx}=3(\frac{2}{a+bx})^2\times \frac{d}{dt}(\frac{2}{a+bx})[/tex]
[tex]=3(\frac{2}{a+bx})^2\times (-\frac{2}{(a+bx)^2})[/tex]
[tex]=-\frac{24}{(a+bx)^4}[/tex]
(2) [tex]y=(at^3-3bt)^3[/tex]
By differentiating w.r.t. t,
[tex]\frac{dy}{dt}=3(at^3-3bt)^2\times \frac{d}{dt}(at^3-3bt)[/tex]
[tex]=3(at^3-3bt)^2 (3at^2-3b)[/tex]
[tex]=9t^2(at^2-3b)^2(at^2-b)[/tex]
(3) [tex]y=(t^b)(e^\frac{b}{t})[/tex]
Differentiating w.r.t. t,
[tex]\frac{dy}{dt}=t^b\times \frac{d}{dt}(e^\frac{b}{t})+\frac{d}{dt}(t^b)\times e^\frac{b}{t}[/tex]
[tex]=t^b(e^\frac{b}{t})\times \frac{d}{dt}(\frac{b}{t}) + bt^{b-1}(e^\frac{b}{t})[/tex]
[tex]=t^be^\frac{b}{t}(-\frac{b}{t^2})+bt^{b-1}e^{\frac{b}{t}}[/tex]
(4) [tex]z = ax^2.sin (4x)[/tex]
Differentiating w.r.t. x,
[tex]\frac{dz}{dt}=ax^2\times \frac{d}{dx}(sin (4x))+sin (4x)\times \frac{d}{dx}(ax^2)[/tex]
[tex]=ax^2\times cos(4x).4+sin (4x)(2ax)[/tex]
[tex]=4ax^2cos (4x)+2ax sin (4x)[/tex]
Find all possible circles that pass through these points: (0,2), (3,1) [Linear Algebra]
I have tried using the equation of a circle to set up an augmented matrix but I'm not sure where to go after I plug in x & y for the equation. I was told that if there is two points, then there will be infinitely many equations.
Answer:
Infinite circles but with values (h,k,r)=(h,3h-3,√(10h²-30h+25))
Step-by-step explanation:
You are correct, if we only have two points, then we will have infinite possibilities, but all those circles will have to be attached to those two points. We have to find a condition to have infinite circles which pass through those points.
Let h be the x coordinate of the center of the circle
Let k be the y coordinate of the center of the circle
Let r be the radius of the circle. Having the circle equation:
[tex](x-h)^{2}+(y-k)^{2}=r^{2}[/tex]
And now expand every binomial and substitute the values of the points:
[tex](x-h)^{2}+(y-k)^{2}=r^{2}\\\\(0-h)^{2}+(2-k)^{2}=r^{2}\\\\(3-h)^{2}+(1-k)^{2}=r^{2}[/tex]
[tex]x^{2}-2xh+h^{2}+y^{2}-2yk+k^{2}=r^{2}\\\\h^{2}+4-4k+k^{2}=r^{2} (I)\\\\9-6h+h^{2}+1-2k+k^{2}=r^{2}[/tex]
Now substract the first and secod equation, and then the first and the third one:
[tex]x^{2}-2xh+y^{2}-4-2yk+4k=0\\\\x^{2}-9-2xh+6h+y^{2}-1-2yk+2k=0[/tex]
Now we substract the second equation from the first one:
[tex]-4+4k+9-6h+1-2k=0[/tex]
Reduce terms and we will have a linear equation, clear k:
[tex]2k-6h+6=0\\\\k=3h-3[/tex]
The last equation let us know the value of k if we asign any value to h.
Now we substitute h and k=3h-3 in I:
[tex]h^{2}+4-4(3h-3)+(3h-3)^{2}=r^{2}\\\\10h^{2}-30h+25=r^{2}\\\\r=\sqrt{10h^{2}-30h+25}[/tex]
The last equation let us know the value of r if we asign any value to h.
These equations (in fact functions), will let us know every set for h, k and r to form a circle (which will pass through the mentioned points)
[tex](h,k,r)=(h,3h-3,\sqrt{10h^{2}-30h+25})[/tex]
Where the indepent variable is h for any real, k is any real and the minimum r is √2.5:
[tex]h:R\\k:R\\r:(\sqrt{2.5},infinite)[/tex]
Suppose the Tower of Hanoi rules are changed so that stones may only be transferred to an adjacent clearing in one move. Let In be the minimum number of moves required to transfer tower from clearing A to clearing C? For example, it takes two moves to move a one stone tower from A to C: One move from A to B, then a second move from B to C. So I1 = 2
Answer:
[tex]l_n=3^n-1[/tex]
Step-by-step explanation:
We will prove by mathematical induction that, for every natural n,
[tex]l_n=3^n-1[/tex]
We will prove our base case (when n=1) to be true:
Base case:
As stated in the qustion, [tex]l_1=2=3^1-1[/tex]
Inductive hypothesis:
Given a natural n,
[tex]l_n=3^n-1[/tex]
Now, we will assume the inductive hypothesis and then use this assumption, involving n, to prove the statement for n + 1.
Inductive step:
Let´s analyze the problem with n+1 stones. In order to move the n+1 stones from A to C we have to:
Move the first n stones from A to C ([tex]l_n[/tex] moves).Move the biggest stone from A to B (1 move).Move the first n stones from C to A ([tex]l_n[/tex] moves).Move the biggest stone from B to C (1 move).Move the first n stones from A to C ([tex]l_n[/tex] moves).Then,
[tex]l_{n+1}=3l_n+2[/tex].
Therefore, using the inductive hypothesis,
[tex]l_{n+1}=3l_n+2=3(3^n-1)+2=3^{n+1}-3+2=3^{n+1}-1[/tex]
With this we have proved our statement to be true for n+1.
In conlusion, for every natural n,
[tex]l_n=3^n-1[/tex]
Which of the following events are not mutually exclusive according to the following information: A bowl has 10 pieces of fruit: 4 yellow bananas, 3 red apples, 2 brown kiwis, and 1 red strawberry. a. a banana or an apple c. a strawberry or a red fruit b. a banana or a kiwi d. a kiwi or a red fruit
Answer:
c. a strawberry or a red fruit
Step-by-step explanation:
Since, Banana and apple, banana and kiwi are different fruits. Also, it is given that kiwi is brown, thus kiwi or a red fruit are different sets. So they are mutually exclusive.
Since strawberry is red fruit. Thus a strawberry or a red fruit is not mutually exclusive.
Further, If two sets A and B is Mutually Exclusive, then any units of Set A is not a member of Set B and vice-versa.
How do you multiply two numbers that are written in scientific notation format?
Answer:
To multiply two numbers in scientific notation, multiply their coefficients and add their exponents. To divide two numbers in scientific notation, divide their coefficients and subtract their exponents. In either case, the answer must be converted to scientific notation.
Step-by-step explanation:
Find each of the following functions. f(x) = 5 − x , g(x) = x2 − 9 (a) f + g State the domain of the function. (Enter your answer using interval notation.) (b) f − g State the domain of the function. (Enter your answer using interval notation.) (c) fg Incorrect: Your answer is incorrect. State the domain of the function. (Enter your answer using interval notation.) (d) f/g State the domain of the function. (Enter your answer using interval notation.)
Answer:
a, b and c= (-∞,∞) d = (-∞,3) U (-3,3) U (3,∞)
Step-by-step explanation:
Hi there!
1) Firstly, let's recap the sum of functions rule:
f(x) + g(x) = (f+g)(x)
Applying it to those functions, we have:
f(x)=5-x +g(x)=[tex]x^{2} -9[/tex] = [(5-x) +([tex]x^{2}[/tex]-9)](x)
(f+g)(x)=[tex]5-x+x^{2} -9[/tex]
2) To State the Domain is to state the set which is valid the quantities of x, of a function. In this case,
a) (f+g)(x)= 5-x+[tex]x^{2}[/tex]+9
Simplifiying
(f+g)(x) =[tex]x^{2} -x+4=0[/tex]
Since there are no restrictions neihter discontinuities, this function has a Domain which can expressed this way:
X may assume infinite quantities, negatives or positives one in the Real set.
(-∞< x <∞+) or simply (-∞,∞)
Or simply put, x ∈ R. Remember, ∞ is not a number, it's a notation meaning infinite values. That's why it's not a closed interval.
Check the graph below.
b) (f-g)(x) =(5-x) -(x²-9)
(f-g)(x)= 5-x-x²+9
Domain of (f-g)(x) =x²-x+14
Similarly to a) this function (f-g)(x) has not discontinuity, nor restrictions on its Domains.
Since there are no restrictions either discontinuities, this function has a Domain which can be expressed this way:
X may assume infinite quantities, negatives or positives one in the Real set.
(-∞< x <∞ +) then finally, the answer: (-∞,∞)
c) (f*g)(x)=(5-x)(x²-9)
(f*g)(x)=5x²-45-x³+9x
Again, this function has no discontinuities, nor restrictions in its Domain as you can check it on its graph.
Then, the Domain of (f*g)(x)=(5-x)(x²-9) is also (-∞,∞)
d) (f/g)(x) =(5-x)/(x²-9)
Highlighting the denominator, we can calculate the Domain.
We can see a restriction here. There is no denominator zero, defined for the Set of R.
Then, let's calculate
[tex]x^{2} -9>0\\ \sqrt{x^2} >\sqrt{9}\\x>3[/tex] and x < -3
In the Numerator, no restrictions.
So the Domain will be the union between the Numerator's Domain and the Denominator's Domain with Restrictions.
Check the graph below.
Finally
D = (-∞,3) U (-3,3) U (3,∞)
Final answer:
The sum, difference, product, and quotient of two functions f(x) = 5 - x and g(x) = x^2 - 9 can be found by adding, subtracting, multiplying, and dividing their corresponding terms. The domain of each resulting function can be determined by considering the domain of the original functions and any restrictions imposed by the operations involved.
Explanation:
(a) f + g
To find the sum of two functions, we simply add the corresponding terms. So, f + g = (5 - x) + (x^2 - 9).
Simplifying, we get f + g = x^2 - x - 4.
The domain of the function f + g is the intersection of the domains of f and g. Since both f and g are defined for all real numbers, the domain of f + g is also all real numbers, which can be represented as (-∞, ∞) in interval notation.
(b) f - g
To find the difference of two functions, we subtract the corresponding terms. So, f - g = (5 - x) - (x^2 - 9).
Simplifying, we get f - g = -x^2 + x + 14.
Similar to part (a), the domain of the function f - g is all real numbers, which can be represented as (-∞, ∞) in interval notation.
(c) fg
To find the product of two functions, we multiply the corresponding terms. So, fg = (5 - x)(x^2 - 9).
Simplifying, we get fg = -x^3 + 9x - 5x^2 + 45.
Similar to parts (a) and (b), the domain of the function fg is all real numbers, which can be represented as (-∞, ∞) in interval notation.
(d) f/g
To find the quotient of two functions, we divide the corresponding terms. So, f/g = (5 - x)/(x^2 - 9).
However, we need to consider the values of x that make the denominator zero, since division by zero is undefined. In this case, the denominator x^2 - 9 is equal to zero when x = ±3.
Therefore, the domain of the function f/g is all real numbers except x = ±3. In interval notation, this can be represented as (-∞, -3) ∪ (-3, 3) ∪ (3, ∞).