Answer:
She needs 300 mililiters of Solution B so that the resulting mixture is a 12% alcohol
Step-by-step explanation:
In this problem you have to take into account that when you are talking about solutions you can't just add the porcentaje because each percentaje represent how many mililiters of the total of the solution are, in this case of alcohol.
So for solving this problem we are first going to establish the variables, because it si solved using a system of equations. In that way we are going to say that:
VT: represents the total volume of the resulting mixture of solution A and solution B at 12% of alcohol
VA: represent the mililiters of solution A, in the problem they say that this value it equals to 400 ml
VB: Represent the mililiters of solution B, that is what we need to find.
From now on, we are just going to use this variables but always keep in mind what does they represent.
VaT: Represent the total volume of alcohol in the resulting mixture solution at 12%
VaA: Represent the volume of alcohol in solution A
VaB: Represent the volume of alcohol in solution B
What comes next? we need to describe the equations from the information we have so that we create a system that can be solve after.
What can we first say about the total volume (VT)? That it is the result of the adition of solution A and B so we can state the following equation:
VT = VA + VB
As we know that VA equals to 400ml we can replace to get:
1) VT = 400ml + VB
But what happens with the other information we have? We now need to take into account the concentration of each solution, so as we can´t add the percentages of alcohol but we can add the volumes of alcohol in each solution we can say that:
2) VaT = VaA + VaB
Now we are going to start to reduce the number of variables changing does that we don't know for those that we do to solve the problem, starting first with the volumes of alcohol.
A porcentaje represents a part of the total volume so to know how much alcohol does each of the solutions has we must do rules of three so that we can leave all the variables in terms of VT, VA and VB:
- VT → 100%
VaT → 12%
VaT = [tex]\frac{12.VT}{100}[/tex] = 0,12.VT
- VA → 100% In this case we know that VA = 400
VaA → 6%
VaA = [tex]\frac{6x400}{100}[/tex]
VaA = [tex]\frac{6x4}{1}[/tex]
VaA=24ml
VaB → 100%
VaB → 20%
VaB = [tex]\frac{20.VB}{100}[/tex] = 0,20.VB
Now we are going to replace this information in the equation number two to get the following expresion:
3) 0,12.VT = 24ml + 0.20VB
At this point we have a system of two equations (remember equation 1) with two variables VT and VB so we are going to do some algebra to clear the variables.
- Replace VT of equation 1 in equation 3
Remeber that VT = 400ml + VB so now we are going to put this information in equation 3) 0,12.VT = 24ml + 0.20VB to get:
4) 0,12 (400ml + VB) = 24ml + 0.20VB
- Use the distributive operation to solve the parentesis
0,12x400ml + 0.12.VB = 24ml + 0.20VB
5) 48ml + 0.12VB = 24ml + 0.20VB
- Organize the information in one side the ones with variables and in the other side just numbers:
0.12VB - 0.20VB = 24ml - 48ml
-0.08 VB = -24ml (do the operations)
As it is a minus in both sides we can divide it and cancel the sign to have:
0,08VB = 24 ml (to clear VB, we must divide in both sides by 0,08)
[tex]\frac{0,08.VB}{0,08} = \frac{24ml}{0.08}[/tex] after doing the division we get:
VB = 300mlwith this you already get the answer of how many mililiters of solution B does she use to get a resulting mixture of 12%.
To verficate we must do the following process:
VT = 300ml + 400ml = 700ml
The total volume of the solution is 700 ml of which 12% equals to:
VaT = 0,12. VT = 0,12(700ml) = 84 ml
VaA = 24ml (Volume of alcohol in solution A, we already calculated)
VaB = 0,20 VB = 0,20(300ml) = 60ml (Volume of alcohol in solution B)
VaT = VaA + VaB (Prove the equation with the values we obtain)
84ml = 24ml + 60ml
84 ml = 84ml
As the equation is the same we have verificated our result.
Robyn needs to use 300 mL of Solution B to achieve a 12% alcohol solution when mixed with 400 mL of Solution A. This was calculated by setting up an equation based on the concentrations and solving for the quantity of Solution B.
To solve this problem, we need to find out how much Solution B (20% alcohol) Robyn needs to add to 400 mL of Solution A (6% alcohol) to get a 12% alcohol solution.
Step-by-Step Solution
First, let's set up the equation assuming she uses x milliliters of Solution B:
Since Solution A is 6% alcohol, in 400 mL of Solution A, there is:
0.06 * 400 = 24 mL of alcoholNext, for Solution B, which is 20% alcohol, the amount of alcohol in x mL of Solution B is:
0.20 * x = 0.2x mL of alcoholWe need the resulting mixture to have a 12% concentration. The total volume of the mixture will be:
400 + x mLThe total amount of alcohol in this mixture will be 12% of the total volume:
0.12 * (400 + x) = 24 + 0.2xSimplify and solve for x:
0.12 * 400 + 0.12 * x = 24 + 0.2x48 + 0.12x = 24 + 0.2x24 = 0.08xx = 300So, Robyn needs to use 300 mL of Solution B.
I seriously am having trouble with this problem! What is f-1(x). [tex]f(x)=\frac{2x-3}{x+1}[/tex]
The answer is [tex]f^-1(x)=\frac{x+3}{2-x}[/tex] but how do I get that answer? My personal answer was [tex]\frac{-(x+3)}{x-2}[/tex]
Please help me and explain how you came up with the actual answer. I greatly appreciate it and thank you!
Answer:
see explanation
Step-by-step explanation:
let y = f(x) and rearrange making x the subject, that is
y = [tex]\frac{2x-3}{x+1}[/tex] ← multiply both sides by (x + 1)
y(x + 1) = 2x - 3 ← distribute left side
xy + y = 2x - 3 ( subtract y from both sides )
xy = 2x - 3 - y ( subtract 2x from both sides )
xy - 2x = - 3 - y ← factor out x from each term on the left side
x(y - 2) = - 3 - y ← divide both sides by y - 2
x = [tex]\frac{-3-y}{y-2}[/tex] factor out - 1 on numerator and denominator
x = [tex]\frac{-(3+y)}{-(2-y)}[/tex]
Change y back into terms of x, thus
[tex]f^{-1}[/tex](x) = [tex]\frac{3+x}{2-x}[/tex] = [tex]\frac{x+3}{2-x}[/tex]
Answer:
f-1(x) = (x + 3)/ (2 - x) or -(x + 3 / (x - 2).
Step-by-step explanation:
Let y = (2x - 3)/(x + 1)
We find x in terms of y:
Cross multiply:
y(x + 1) = 2x - 3
xy + y = 2x - 3
y + 3 = 2x - xy
x(2 - y) = y + 3
x = (y + 3) / (2 - y)
Now replace x by f-1(x) and y by x, we get:
f-1(x) = (x + 3)/ (2 - x).
Your answer was correct. You found the same result written in a different form.
If we multiply the above by -1 / -1 we get
-(x + 3) / -(2 - x)
= -(x + 3) / (x - 2).
There are 10 marbles in a bag: 1 blue, 4 yellow, 3 red, and 2 white. If you choose a marble at random, which is the probability that you will NOT choose white?
The probability of not choosing a white marble from a bag containing 10 marbles of different colors is 80%.
Explanation:To find the probability that you will NOT choose a white marble from a bag containing 1 blue, 4 yellow, 3 red, and 2 white marbles (10 marbles in total), follow these steps. First, calculate the total number of marbles that are not white. There is 1 blue, 4 yellow, and 3 red marbles, summing up to 8 marbles that are not white. The probability of picking a marble that is not white is therefore the number of marbles that are not white divided by the total number of marbles.
Probability = Number of marbles that are not white / Total number of marbles
Probability = 8 / 10
Probability = 0.8 or 80%
Thus, the probability of choosing a marble that is not white is 80%.
The complement of 4 heads in the toss of 4 coins is
(Please explain)! Thanks!
Exactly one tail
At least one tail
All tails
Three heads
By using sample space we can say that complement of 4 heads in the toss of 4 coins is at least one tail .
What is sample space ?set of all possible outcomes experiment is called sample space .
To find compliment of 4 heads we will first find sample space Sample space= S
= { HHHH, HHHT, HHTH, HHTT, HTHH, HTHT, HTTH, HTTT, THHH, THHT, THTH, THTT TTHH, TTHT, TTTH, TTTT }
Given case is 4 heads Say A= { HHHH}
Now we can calculate A complement as
A' = S-A
= { HHHH, HHHT, HHTH, HHTT, HTHH, HTHT, HTTH, HTTT, THHH, THHT, THTH, THTT TTHH, TTHT, TTTH,TTTT }- { HHHH}A'={ HHHT, HHTH, HHTT, HTHH, HTHT, HTTH, HTTT, THHH, THHT, THTH, THTT TTHH, TTHT, TTTH,TTTT }
By using sample space we can say that complement of 4 heads in the toss of 4 coins is at least one tail .
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The complement of 4 heads in the toss of 4 coins is 'at least one tail', which includes any outcome where at least one coin lands on tails, encompassing all possibilities except for the exact outcome of 4 heads.
Explanation:The student is asking about the concept of the complement in probability when flipping coins. Specifically, the complement of getting four heads in the flip of four coins. In probability, the complement of an event is all the possible outcomes that are not part of the event itself. In the case of flipping four coins, the event of getting four heads (HHHH) has a complement that consists of any outcome that does not have four heads. That includes outcomes with at least one tail.
Therefore, the complement of getting four heads is at least one tail, meaning any outcome where at least one coin lands on tails. All the outcomes with 0 heads and 4 tails, 1 head and 3 tails, 2 heads and 2 tails, or 3 heads and 1 tail would be part of this complement.
Since the question refers to the flip of four coins, only the following outcomes (macrostates) are possible: 0 heads (all tails), 1 head, 2 heads, 3 heads, or 4 heads (all heads). The complement of '4 heads' would comprise all outcomes except for '4 heads', reaffirming that the correct answer is at least one tail.
x+ y+(k −5)z=k where k is an arbitrary constant. For which value(s) of k does this system have a unique solution? For which value(s) of k does the system have infinitely many solutions? For which value(s) of k is the system inconsistent?
Answer: Lets rewrite our equation.
x + y + kz - 5z =k
x + y -5z = k(1-z)
k = [tex]\frac{x + y - 5z}{1 - z}[/tex]
so the only problem with the equation is that z can't be equal to one.
but x and y are free, so you have two free variables and one equation,
that means that for any k the system has infinitely many solutions.
Final answer:
To determine the nature of solutions in a system of equations involving k, we can check the determinant of the coefficient matrix. If the determinant is not zero, there is a unique solution; if it is zero, there are infinitely many solutions; and if contradictory, the system is inconsistent.
Explanation:
For which value(s) of k does this system have a unique solution?
To have a unique solution, the determinant of the coefficients of the variables x, y, and z must not be zero.Therefore, for a unique solution, the determinant of the coefficient matrix must not equal zero.For which value(s) of k does the system have infinitely many solutions?
If the determinant of the coefficient matrix is zero, the system has infinitely many solutions.When the determinant is zero, it implies dependent equations, leading to infinite solutions.For which value(s) of k is the system inconsistent?
The system is inconsistent when the determinant is zero, and the system contains contradictory equations.When the determinant is zero and the equations are contradictory, the system is inconsistent.It takes 16 minutes for a certain bacteria population to double. If there are 6,140,276 bacteria in this population at 9:15 AM what is the best estimate in millions of the number of bacteria at 9:47 AM on the same day
Answer:
The answer to your question is: 24 561 104 bacteria
Step-by-step explanation:
every 16 minutes bacteria population double
There are 6140276 bacteria at 9:15 AM
estimate bacteria at 9:47
First calculate the minutes : 47 - 15 = 32 minutes
and 32 / 16 = 2
then 2 times the bacteria population double
So,
first time = 6140276 x 2 = 1 2280552 bacteria
second time = 1 2280552 x 2 = 24561104 bacteria
"Events A1, A2 and A3 form a partiton of the sample space S with probabilities P(A1) = 0.3, P(A2) = 0.5, P(A3) = 0.2.If E is an event in S with P(E|A1) = 0.1, P(E|A2) = 0.6, P(E|A3) = 0.8, computeP(E) =P(A1|E) =P(A2|E) =P(A3|E) ="
Answer with Step-by-step explanation:
We are given that Events [tex]A_1,A_2 \;and\;A_3[/tex] form a partition of the sample space S .
[tex]P(A_1)=0.3,P(A_2)=0.5,P(A_3)=0.2[/tex]
If E is an event in S
[tex]P(E/A_1)=0.1,P(E/A_2)=0.6,P(E/A_3)=0.8[/tex]
[tex]P(E)=P(A_1)\cdot P(E/A_1)+P(A_2)\cdot P(E/A_2)+P(A_3)\cdot P(E/A_3)[/tex]
Substitute the values then we get
[tex]P(E)=(0.3)(0.1)+(0.5)(0.6)+(0.2)(0.8)=0.03+0.30+0.16=0.49[/tex]
[tex]P(E)=0.49[/tex]
Bayes theorem
[tex]P(E_i/A)=\frac{P(E_i)P(A/E_i)}{\sum_{i=1}^{n}P(E_i)P(A/E_i)}[/tex]
Now, using Bayes theorem
[tex]P(A_1/E)=\frac{P(A_1)\cdot P(E/A_1}{P(A_1)P(E/A_1)+P(A_2)P(E/A_2)+P(A_3)P(E/A_3)}[/tex]
Substitute the values then we get
[tex]P(A_1/E)=\frac{0.3\times 0.1}{(0.1)(0.3)+(0.5)(0.6)+(0.2)(0.8)}=\frac{0.03}{0.49}=\frac{3}{49}[/tex]
[tex]P(A_1/E)=\frac{3}{49}[/tex]
Similarly
[tex]P(A_2/E)=\frac{(0.5)(0.6)}{0.49}=\frac{0.30}{0.49}=\frac{30}{49}[/tex]
[tex]P(A_2/E)=\frac{30}{49}[/tex]
[tex]P(A_3/E)=\frac{(0.2)(0.8)}{0.49}=\frac{0.16}{0.49}=\frac{16}{49}[/tex]
[tex]P(A_3/E)=\frac{16}{49}[/tex]
You can use the Bayes' theorem and the law of total probability with chain rule to find the needed probabilities.
The needed probabilities are:
[tex]P(E) = 0.49[/tex][tex]P(A_1|E) = 0.0612[/tex][tex]P(A_2|E) = 0.612[/tex][tex]P(A_3|E) = 0.327[/tex]What is Bayes' theorem?Suppose that there are two events A and B. Then suppose the conditional probability are:
P(A|B) = probability of occurrence of A given B has already occurred.
P(B|A) = probability of occurrence of B given A has already occurred.
Then, according to Bayes' theorem, we have:
[tex]\rm P(A|B) = \dfrac{P(B|A)P(A)}{P(B)}[/tex]
(assuming the P(B) is not 0)
What is chain rule in probability?For two events A and B, by chain rule, we have:
[tex]P(A \cap B) = P(B)P(A|B) = P(A)P(A|B)[/tex]
What is law of total probability?Suppose that the sample space is divided in n mutual exclusive and exhaustive events tagged as [tex]B_i \: ; i \in \{1,2,3.., n\}[/tex]
Then, suppose there is event A in sample space.
Then probability of A's occurrence can be given as
[tex]P(A) = \sum_{i=1}^n P(A \cap B_i)[/tex]
Using the chain rule, we get
[tex]P(A) = \sum_{i=1}^n P(A \cap B_i) = \sum_{i=1}^n P(A)P(B_i|A) = \sum_{i=1}^nP(B_i)P(A|B_i)[/tex]
Using the above, rules, as we're already given that
[tex]A_1, A_2, A_3[/tex] are forming partition of the sample space (thus they're mutually exclusive and exhaustive events)
Also, its given that
[tex]P(A_1) = 0.3\\P(A_2) = 0.5\\P(A_3) = 0.2[/tex]
[tex]P(E|A_1) = 0.1\\P(E|A_2) = 0.6\\P(E|A_3) = 0.8[/tex]
Thus, by using the chain rule, we get:
[tex]P(E) = \sum_{i=1}^3P(A_i)P(E|A_i) = 0.3 \times 0.1 + 0.5 \times 0.6 + 0.2 \times 0.8 = 0.49\\\\P(E) = 0.49[/tex]
We have:
[tex]P(A_i \cap E) = P(A_i|E)P(E) = P(E|A_i)P(A_i)\\\\\rm P(A_i|E) = \dfrac{P(E|A_i)P(A_i)}{P(E)}[/tex]
Evaluating for i = 1, ,2, 3, we get:
Case 1: i = 1[tex]\rm P(A_i|E) = \dfrac{P(E|A_i)P(A_i)}{P(E)}\\P(A_1|E) = \dfrac{0.1 \times 0.3}{0.49} \approx 0.0612[/tex]
Case 2: i = 2[tex]\rm P(A_i|E) = \dfrac{P(E|A_i)P(A_i)}{P(E)}\\\\P(A_2|E) = \dfrac{0.6 \times 0.5}{0.49} \approx 0.612[/tex]
Case 3: i = 3[tex]\rm P(A_i|E) = \dfrac{P(E|A_i)P(A_i)}{P(E)}\\P(A_3|E) = \dfrac{0.8 \times 0.2}{0.49} \approx 0.327[/tex]
Thus,
The needed probabilities are:
[tex]P(E) = 0.49[/tex][tex]P(A_1|E) = 0.0612[/tex][tex]P(A_2|E) = 0.612[/tex][tex]P(A_3|E) = 0.327[/tex]Learn more about Bayes' theorem here:
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At union station, you notice that three subway lines just arrived at the same time. How long must you wait until all three lines arrive at Union Station at the same time again?
Answer:
You must wait 60 minutes for all three lines to arrive at Union Station at the same time again.
Step-by-step explanation:
The complete question in the attached figure
Step 1
Find the least common multiple (LCM) of the three numbers
List the prime factors of each number
[tex]10=(2)(5)\\12=(2^2)(3)\\15=(3)(5)[/tex]
Multiply each factor the greatest number of times it occurs in any of the numbers to find out the LCM
The LCM is equal to
[tex](2^2)(3)(5)=60[/tex]
therefore
You must wait 60 minutes for all three lines to arrive at Union Station at the same time again.
Help ASAP please!! Find the length of the missing side. If necessary, round to the nearest tenth.
Options:
58
20.5
841
29
Answer:
=20.5
Step-by-step explanation:
1. use the Pythagorean theorem: a^2+ b^2=c^2 (arms+legs squared=hypotenuse)
2. plug in the numbers to the equation:
14^2+15^2=c
225+196=c^2
421=C^2
the square root of 421 is 20.5 which is the hypotenuse of the triangle
The Vilas County News earns a profit of $20 per year for each of its 3,000 subscribers. Management projects that the profit per subscriber would increase by 1¢ for each additional subscriber over the current 3,000. How many subscribers are needed to bring a total profit of $109,725?
Answer:
850
Step-by-step explanation:
Current profit per year = $ 20
Number of subscribers = 3000
For each additional subscriber over 3000, the profit will increase by 1 cent or by $ 0.01. For example, for 3001 (3000 + 1) subscribers, the profit will be $ 20.01 per year. Similarly, for 3002 (3000 + 2) subscribers, the profit will be $20.02 per year and so on.
So, for x additional subscribers over 3000, the profit will increase by 0.01(x). i.e. for (3000 + x) subscribers, the profit will be $(20 + 0.01x)
Since, profit per each subscriber is $(20 + 0.01x), the profit for (3000 + x) subscribers will be:
Total profit = Number of subscribers x Profit per each subscriber
Total profit = (3000 + x)(20 + 0.01x)
We want to calculate how many subscribers will be needed to bring a profit of $109,725. So, we replace Total profit by $109,725. The equation now becomes:
[tex]109725=(3000+x)(20+0.01x)\\\\ 109725=60000+30x+20x+0.01x^{2}\\\\ 0.01x^{2}+50x+60000-109725=0\\\\ 0.01x^{2}+50x-49725=0\\[/tex]
Using quadratic formula, we can solve this equation as:
[tex]\\ x=\frac{-50 \pm \sqrt{50^{2}-4(0.01)(-49725)}}{2(0.01)}\\\\ x=\frac{-50 \pm67}{0.02}\\\\ x=\frac{-50-67}{0.02} , x=\frac{-50+67}{0.02}\\\\ x=-5850, x=850[/tex]
x = -5850 is not a possible solution as this would make the total number of subscribers to be negative. So we reject this value.
Therefore, the answer to this question is 850. 850 more subscribers are needed to being a total profit of $109,725
Final answer:
To bring a total profit of $109,725, the Vilas County News needs an additional 4,969,500 subscribers, totaling 4,972,500 subscribers in all.
Explanation:
The total profit of $109,725 can be calculated as follows:
Current profit = $20 x 3,000 = $60,000
Extra profit per additional subscriber = 1¢ or $0.01
Let x be the number of additional subscribers needed
Total profit equation: $60,000 + $0.01x = $109,725
Solve for x: $0.01x = $49,725
x = 49,725 / $0.01 = 4,972,500 subscribers
To find the additional subscribers needed: 4,972,500 - 3,000 = 4,969,500
Find the coordinates of point Q.
Answer:
Q(2, 6 )
Step-by-step explanation:
Using the midpoint formula
let the coordinates of Q = (x, y ), then
0.5(x + 10) = 6 ( multiply both sides by 2 )
x + 10 = 12 ( subtract 10 from both sides )
x = 2
and
0.5(y + 6) = 6 ( multiply both sides by 2 )
y + 6 = 12 ( subtract 6 from both sides )
y = 6
Coordinates of Q = (2, 6 )
Linda and Juan went shopping. Linda spent $13 less than Juan. If we let z z represent how much Juan spent, write an algebraic expression for how much Linda spent. Preview
Answer:
L= z z -$13
Step-by-step explanation:
We will write "L" for Linda and" z z" for Juan.
As linda spent $13 less than Juan, It is that Linda spent $13 less than zz.
We must substract $13 to what zz spent to obtain what Linda spent. That is,
L = zz - $13
The algebraic expression for how much Linda spent is z - 13, where z represents how much Juan spent.
To represent how much Linda spent, we can use the algebraic expression z - 13.
We will write "L" for Linda and" z z" for Juan.
Since Linda spent $13 less than Juan,
we subtract 13 from the amount Juan spent to get the amount Linda spent.
As linda spent $13 less than Juan, It is that Linda spent $13 less than zz.
We must substract $13 to what zz spent to obtain what Linda spent. That is,
L = zz - $13
So the expression for how much Linda spent is z - 13.
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At the farmer's market, a bag of apples and 3 cartons of strawberries cost $18 total. If a bag of apples costs 50% more than a carton of strawberries, how much does a bag of apples cost?
Answer:
The answer to your question is: 1 bag of apples cost $6.
Step-by-step explanation:
We need to write an equation:
1 bag of apples + 3 cartons of strawberries = $18
but 1 bag of apples = 1.5 cartons of strawberries
So
1.5 cartons of strawberries + 3 cartons of strawberries = $18
simplify 4.5 cartons of strawberries = 18
1 carton of strawberries = 18/ 4.5 = 4
finally 1 bag of apples = 1.5 (4) = 6
Answer:
$6.00
Step-by-step explanation:
WXY and YXZ are supplementary angles. m WXY=8x+12 m YXZ=2x+28. what is YXZ?
Answer:
YXZ= 56 degrees
Step-by-step explanation:
Add the angles and set them equal to 180. Then solve for x. then Pug it in 2x+28.
Answer: [tex]\angle{YXZ}=56^{\circ}[/tex]
Step-by-step explanation:
Given : ∠WXY and ∠YXZ are supplementary angles.
∠WXY= 8x+12 and ∠YXZ = 2x+28
We know that the pair of supplementary angles are added up to 180°.
According to the question, we have
[tex]\angle{WXY}+\angle{YXZ}=180^{\circ}\\\\\Rightarrow\ 8x+12+2x+28=180\\\\\Rightarrow\ 10x+40=180\\\\\Rightarrow\ 10x=180-40\\\\\Rightarrow\ 10x=140\\\\\Rightarrow\ x=\dfrac{140}{10}=14[/tex]
Now, [tex]\angle{YXZ}=2(14)+28=56^{\circ}[/tex]
Hence, the measure of [tex]\angle{YXZ}=56^{\circ}[/tex].
Correct answers only please! If you don't know the answer, then please don't guess or say what you think it is.
Given a student number system for a county requires that the student number be 6 characters.
The first 4 characters are any single digit number but no character can repeat and the last two characters must be a letter and letters cannot be the same. How many unique student numbers are possible?
A. 6,500,000
B. 3,276,000
C. 3,407,040
D. 6,760,000
Answer: B. 3,276,000
Step-by-step explanation:
Given : A student number system for a county requires that the student number be 6 characters.
Number of digits (0,1,2,3,4,5,6,7,8,9)=10
Number of letters in English alphabet = 26
When repetition of things is not allowed then we use Permutations.
Number of permutations of m things taking n at a time =[tex]^mP_n=\dfrac{m!}{(m-n)!}[/tex]
Similarly, Number of permutations of 10 numbers taking 4 at a time :
[tex]^{10}P_4=\dfrac{10!}{(6)!}=\dfrac{10\times9\times8\times7\times6!}{6!}=5040[/tex]
Number of permutations of 26 letters taking 2 at a time :
[tex]^{26}P_2=\dfrac{26!}{(2)!}=\dfrac{26\times25\times24!}{24!}=650[/tex]
Now, the possible number of numbers can be make = [tex]5040\times650=3,276,000[/tex]
Hence, the correct answer is options (b).
Pasta comes with tomato sauce and can be ordered with some, all, or none of these ingredients in the sauce: {onions, garlic, carrots, broccoli, shrimp, mushrooms, zucchini, green pepper}. How many different variations are available for ordering pasta with tomato sauce?
Answer: Hello!
So the list of ingredients in the sauce are:
{onions, garlic, carrots, broccoli, shrimp, mushrooms, zucchini, green pepper}
there are 8 possible elements in the sauce, so we need to see the combinations for this elements in the sauce.
let's count!
0 ingredients; there is only one combination with 0 ingredients.
1 ingredient: there are 8 combinations with 1 ingredient, one for each.
2 ingredients: now star playing with combinatorics, so the combinatory between A and B is: [tex]\frac{A!}{B!(A-B)!}[/tex], so for 8 and 2 we get:
[tex]\frac{8!}{2!(6)!} = 8*7/2 = 28[/tex]
3 ingredients: Similar as before; the combinatory for 8 and 3 is: [tex]\frac{8!}{3!*5!} = 8*7 =56[/tex]
4 ingredients: for 8 and 4 we have: [tex]\frac{8!}{4!*4!} = \frac{8*7*6*5}{4*3*2} = 2*7*5 = 70[/tex]
5 ingredients: You can think that now we are counting the combinations of 3 ingredients that we are not using, so the combinations are the same as for 3 ingredients: 56
6 ingredients: Similar as before, here are the combinations of the two ingredients that we are not using: 28
7 ingredients: there are 8 options if we decide to not use only one ingredient.
8 ingredient: there is 1 option in this case, same as the case with no ingredients
And now, we need to add all those combinations:
so C = 1 +8 + 28 + 56+ 70+ 55 + 28 +8 +1 = 512 total combinations.
Through the application of combinations in Probability theory, we can determine that there are 256 potential ways to order pasta with tomato sauce, considering the inclusion or exclusion of eight possible ingredients.
Explanation:The question pertains to combinations in Probability theory, specifically involving pasta with options to include or exclude eight different sauce ingredients: onions, garlic, carrots, broccoli, shrimp, mushrooms, zucchini, and green pepper. For each ingredient, we have two choices - either to include it or not, giving us 28 options, where 8 is the total number of ingredients.
To calculate the number of combinations, we use the formula for combinations which is 2n, where 'n' is the number of items. In this case, n equals eight as there are eight ingredients. Therefore, plugging into the formula, we get 28 = 256. So, there can be a total of 256 combinations when ordering pasta with tomato sauce considering the options of eight ingredients to include or not.
Learn more about Combinations here:https://brainly.com/question/30646507
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Six machines, each working at the same constant rate, together can complete a certain job in 12 days. How many additional machines, each working at the same constant rate, will be needed to complete the job in 8 days?A. 2
B. 3
C. 4
D. 6
E. 8
Answer:
B
Step-by-step explanation:
6 machines complete the job in 12 days;
the more machine you have faster the job will be done, inverse proportion
6 machines --- 12 days
x machines -- 8 days
you reduce the number of days by a factor of 1.5 (12/1.5)
sou you hvae to increase the number of machines by 1.5
6 x 1.5 = 9, so you will need 3 additional machines
Write the indicated conditional statements for the given statement.
Given Statement: Gerald bought his first car after saving up $5,000.
Conditional Statement:
Converse Statement:
Inverse Statement:
Contrapositive Statement:
Biconditional Statement:
Please help!!!
Explanation:
Given Statement: Gerald bought his first car after saving up $5,000.
The conditional conjunction after identifies the dependent clause, the one corresponding to the "if" statement in a conditional.
Conditional Statement:
If Gerald saved $5000, then he bought his first car.
__
Converse Statement:
If Gerald bought his first car, then he saved $5000.
__
Inverse Statement:
If Gerald did not save $5000, then he did not buy his first car.
__
Contrapositive Statement: (converse of the inverse)
If Gerald did not buy his first car, then he did not save $5000.
__
Biconditional Statement:
If and only if Gerald saved $5000, then he bought his first car.
A ∠A and ∠ B ∠B are complementary angles. If m ∠ A = ( 2 x − 23 ) ∘ ∠A=(2x−23) ∘ and m ∠ B = ( x + 20 ) ∘ ∠B=(x+20) ∘ , then find the measure of ∠ A ∠A.
Answer:
∠A = 99°
Step-by-step explanation:
Complementary angles sum to 180°, thus
∠A + ∠B = 180 ← substitute values
2x - 23 + x + 20 = 180
3x - 3 = 180 ( add 3 to both sides )
3x = 183 ( divide both sides by 3 )
x = 61
Hence
∠A = 2x - 23 = (2 × 61) - 23 = 122 - 23 = 99°
Clementine & Jake make cookies for the school bake sale.Clementine baked 72 cookies.Jake baked twice as many as clementine.How many cookies did they bake together?
Answer:
216
Step-by-step explanation:
72 + (72×2)=
72 + 144 =
216
If 2x2 + y2 = 17 then evaluate the second derivative of y with respect to x when x = 2 and y = 3. Round your answer to 2 decimal places. Use the hyphen symbol, -, for negative values.
Answer:
y''=-1.26
Step-by-step explanation:
We are given that [tex]2x^2+y^2=17[/tex]
We have evaluate the second order derivative of y w.r.t. x when x=2 and y=3.
Differentiate w.r.t x
Then , we get
[tex]4x+2yy'=0[/tex]
[tex]2x+yy'=0[/tex]
[tex]yy'=-2x[/tex]
[tex]y'=-\frac{2x}{y}[/tex]
Again differentiate w.r.t.x
Then , we get
[tex]2+(y')^2+yy''=0[/tex] [tex](u\cdot v)'=u'v+v'u)[/tex]
[tex]2+(y')^2+yy''=0[/tex]
Using value of y'
[tex]yy''=-2-(-\frac{2x}{y})^2[/tex]
[tex]y''=-\frac{2+(-\frac{2x}{y})^2}{y}[/tex]
Substitute x=2 and y=3
Then, we get [tex]y''=-\frac{2+(\frac{4}{3})^2}{3}[/tex]
[tex]y''=-\frac{18+16}{9\times 3}=-\frac{34}{27}[/tex]
Hence,y''=-1.26
Brielle and Eduwa have cottages on the ocean, but there is a bay that extends between their two homes so they are unable to determine the direct linear distance between their cottages by simply driving and reading their odometers. Markos' house is on the bay that extends between Brielle and Eduwa's cottages. The angle between Brielle and Eduwa's cottages, when measured from Markos' house, is 43∘. The distance between Markos' house and Brielle's cottage is 3.5 miles. The distance between Markos' house and Eduwa's cottage is 2.8 miles. What is the distance between Brielle and Eduwa's cottages? Drawing a picture will help!
Answer: 2.4 miles
Step-by-step explanation: law of cosines
Let's call the distance between Brielle and Eduwa's cottages x. As the angle btw them is 43°, we can use:
x² = 3.5² + 2.8² - 2*3.5*2.8*cos43
x² = 5.755
x = 2.4 miles
Copy center prints black and white or colored copies. Yesterday, they printed 9 black and white copies for every 2 colored. If they printed 200 color copies then how many black and white copies where printed??
Answer:
The answer to your question is 900 black and white copies
Step-by-step explanation:
Data
9 black and white copies for every 2 colored
printed 200 color copies
black and white copies = ?
Process
This is a proportion exercise
BW : C :: bw : c
9 : 2 :: bw : 200
bw = 200(9) / 2
bw = 1800 / 2
bw = 900 copies
Kia has 2 1/4 pounds of peppermint candies. She wants to put them in small bags, each about 3/8 of a pound . How many small bags of candies can she make?
Answer:
Step-by-step explanation:
When I am dealing with fractions I put them in money wise so 2 1/4 would be like 2.25 and 3/8 would be like 0.75.
So that means you divide 2.25 by 0.75 and that will give you three.
Therefore Kia can make 3 small bags of candies.
have a nice day
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How can you express 1/4 as a percent?
Answer: 25%
Step-by-step explanation: To write a fraction as a percent, first remember that a percent is a ratio of a number to 100. If we want to write 1/4 as a percent, we need to find a fraction equivalent to 1/4 with a 100 in the denominator. We can do this by setting up a proportion.
[tex]\frac{1}{4}[/tex] = [tex]\frac{n}{100}[/tex]
Now, we can use cross products to find the missing value.
4n = 100
÷4 ÷4 ← divide by 4 on both sides
n = 25
Therefore, 1/4 is equivalent to 25%.
the line of reflection for the given pre-image and image. Explain how you found the correct line of reflection.
Answer:
x = -2
Step-by-step explanation:
The line of reflection is the perpendicular bisector of the segment between a pre-image point and its image. For example, the segment HH' is bisected by x=-2, the line of reflection.
In measuring reaction time, a psychologist estimates that a standard deviation is .05 seconds. How large a sample of measurements must he take in order to be 95% confident that the error in his estimate of mean reaction time will not exceed 0.01 seconds?
Answer:
97
Step-by-step explanation:
We are asked to find the size of sample to be 95% confident that the error in psychologist estimate of mean reaction time will not exceed 0.01 seconds.
We will use following formula to solve our given problem.
[tex]n\geq (\frac{z_{\alpha/2}\cdot\sigma}{E})^2[/tex], where,
[tex]\sigma=\text{Standard deviation}=0.05[/tex],
[tex]\alpha=\text{Significance level}=1-0.95=0.05[/tex],
[tex]z_{\alpha/2}=\text{Critical value}=z_{0.025}=1.96[/tex].
[tex]E=\text{Margin of error}[/tex]
[tex]n=\text{Sample size}[/tex]
Substitute given values:
[tex]n\geq (\frac{z_{0.025}\cdot\sigma}{E})^2[/tex]
[tex]n\geq (\frac{1.96\cdot0.05}{0.01})^2[/tex]
[tex]n\geq (\frac{0.098}{0.01})^2[/tex]
[tex]n\geq (9.8)^2[/tex]
[tex]n\geq 96.04[/tex]
Therefore, the sample size must be 97 in order to be 95% confident that the error in his estimate of mean reaction time will not exceed 0.01 seconds.
Calculate the acceleration of gravity on the surface of the following worlds. How much would you weigh, in pounds, on each of the following worlds? A. Mars (mass 0.11 M_Earth) (radius 0.53 R_Earth) B. Venus (mass 0.82 M_Earth) (radius 0.95 R_Earth) C. Jupiter (mass 317.8 M_Earth) (radius 11.2 R_Earth)
Answer:
Mars: 0.103 *(Real weigh)
Venus: 0.9 * (Real weigh)
Jupiter 2.53 * (Real weigh)
Step-by-step explanation:
The formula is this:
GM/r**2
M= Mass
r = radius
You do not need G for comparing between different masses or planets.
A class has 35 students, of which 16 are male and 19 are female. If 6 of the students are selected at random to form a committee, what is the probability that exactly 2 male students are selected?
Answer:
The probability of choosing exactly 2 male and 4 female students =[tex]\frac{\binom{16}{2}\times \binom{19}{4}}{\binom{35}{6}}[/tex]
Step-by-step explanation:
We are given that a class has 35 students
Number of male=16
Number of female=19
We have to choose 6 students for committee
We have to find the probability that exactly 2 male students are selected
Probability=P(E)=[tex]\frac{number\;of\;favorable\;cases}{total\;number\;of\;cases}[/tex]
If we have to choose total 6 student in which 2 male and 4 female
Combination formula:
[tex]nC_r=\frac{n!}{r!(n-r)!}[/tex]
Using the formula
The probability of choosing exactly 2 male and 4 female students =[tex]\frac{\binom{16}{2}\times \binom{19}{4}}{\binom{35}{6}}[/tex]
Which of the following functions is graphed below?
Answer:
C
Step-by-step explanation:
For this problem, the point where the line changes its slope is the focus point. The trick for this is to find the x point for the point. You have to look at he absolute value parenthesis, for this just put the opposite negative value of the number being added or subtracted inside the parenthesis. The "b" (y=mx+b) for the equation, isn't changed at all, so -2 would be the y value.
Tyler went to the supermarket to buy food for a food pantry. He has $36, and can carry up to 20 pounds of food in his backpack.
Pasta costs $1 for a 1-pound package. Pasta sauce costs $3 for a 1.5 pound jar.
Let x = the number of packages of pasta and y = the number of jars of pasta sauce.
Identify each point as either a solution to the system or not a solution to the system of inequalities.
(1, 12)
(2, 10)
(4, 5)
(6, 10)
(12, 8)
(18, 6)
Answer:
solutions: (2, 10), (4, 5)non-solutions: all other pointsStep-by-step explanation:
It can be useful to graph the inequalities related to Tyler's limits of money and weight.
1x +3y ≤ 36 . . . . . . the limit on the cost of the items Tyler can afford
1x +1.5y ≤ 20 . . . . . the limit on the weight of the items Tyler can carry
Then plotting the given points shows that only (2, 10) and (4, 5) are in the doubly-shaded area that is the solution space. The other points are not a solution.