Assume a general solution of the form
[tex]y(x)=\displaystyle\sum_{n\ge0}a_nx^n[/tex]
with derivatives
[tex]y'(x)=\displaystyle\sum_{n\ge0}na_nx^{n-1}=\sum_{n\ge1}na_nx^{n-1}[/tex]
[tex]y''(x)=\displaystyle\sum_{n\ge0}n(n-1)a_nx^{n-2}=\sum_{n\ge2}n(n-1)a_nx^{n-2}[/tex]
Substituting into the ODE gives
[tex]\displaystyle\sum_{n\ge2}n(n-1)a_nx^{n-2}-2\sum_{n\ge1}na_nx^n+4\sum_{n\ge0}a_nx^n=0[/tex]
The first sum starts with degree 0; the second starts with degree 1; and the third starts with degree 0. So remove the first term from the first and third sums, then consolidate everything into one sum by shifting indices as needed.
First sum:
[tex]\displaystyle\sum_{n\ge2}n(n-1)a_nx^{n-2}=2a_2+\sum_{n\ge3}n(n-1)a_nx^{n-2}[/tex]
[tex]\displaystyle\sum_{n\ge2}n(n-1)a_nx^{n-2}=2a_2+\sum_{n\ge1}(n+2)(n+1)a_{n+2}x^n[/tex]
Third sum:
[tex]\displaystyle\sum_{n\ge0}a_nx^n=a_0+\sum_{n\ge1}a_nx^n[/tex]
So the ODE can be expressed as
[tex]\displaystyle\left(2a_2+\sum_{n\ge1}(n+2)(n+1)a_{n+2}x^n\right)-2\sum_{n\ge1}na_nx^n+4\left(a_0+\sum_{n\ge1}a_nx^n\right)=0[/tex]
[tex]\displaystyle2a_2+4a_0+\sum_{n\ge1}\bigg((n+2)(n+1)a_{n+2}+(4-2n)a_n\bigg)x^n=0[/tex]
Then the coefficients are given by the recurrence,
[tex]\begin{cases}a_0=a_0\\a_1=a_1\\2(2-n)a_n+(n+2)(n+1)a_{n+2}=0&\text{for }n\ge0\end{cases}[/tex]
Notice that we should have [tex]a_0\neq0[/tex] and [tex]a_1\neq0[/tex] in order to get non-zero solutions.
For [tex]n=2[/tex], we would find that [tex]a_4=0[/tex], which would imply that [tex]a_n=0[/tex] for all even [tex]n\ge2[/tex]. Then the even-indexed terms in the series contribute one solution,
[tex]y_1(x)=a_0+a_2x^2=a_0(1-2x^2)[/tex].
On the other hand, for odd [tex]n\ge1[/tex] we have
[tex]a_{n+2}=\dfrac{2(n-2)}{(n+2)(n+1)}a_n[/tex]
With [tex]n=1[/tex]:
[tex]a_3=\dfrac{2\cdot(-1)}{3\cdot2}a_1=-\dfrac2{3!}a_1[/tex]
With [tex]n=3[/tex]:
[tex]a_5=\dfrac{2\cdot1}{5\cdot4}a_3=-\dfrac{2^2\cdot1}{5!}a_1[/tex]
With [tex]n=5[/tex]:
[tex]a_7=\dfrac{2\cdot3}{7\cdot6}a_5=-\dfrac{2^3\cdot3\cdot1}{7!}a_1[/tex]
So the general pattern for [tex]n=2k+1[/tex], [tex]k\ge1[/tex], is
[tex]a_{2k+1}=-\dfrac{2^k\prod\limits_{i=1}^{k-1}(2i-1)}{(2k+1)!}a_1[/tex]
and we get a second (linearly independent) solution
[tex]y_2(x)=a_1x+\displaystyle\sum_{k\ge1}a_{2k+1}x^{2k+1}[/tex]
What should the balance be in Diane's register?
Will give BRAINLIEST
Answer:
The balance of Dianne's register should be $359.41
Step-by-step explanation:
We can begin at the ending balance on the bank statement at $578.30
Then a check is written for $219.25, so a debit
Next a deposit is made for $140.36, so a credit
Then a withdrawal is made for $140.00, so a debit
This means we can make the equation
[tex]578.30-219.25+140.36-140.00=359.41[/tex]
A manufacturer makes closed cubic containers from sheet metal. How many square centimeters of sheet metal will a 27,000 cm 3 container need?
let's recall that a cube is just a rectangular prism with all equal sides, check picture below.
[tex]\bf \textit{volume of a cube}\\\\ V=s^3~~ \begin{cases} s=&length~of\\ &a~side\\ \cline{1-2} V=&27000 \end{cases}\implies 27000=s^3\implies \sqrt[3]{27000}=s\implies 30=s \\\\[-0.35em] ~\dotfill\\\\ \textit{surface area of a cube}\\\\ SA=6s~~\begin{cases} s=&length~of\\ &a~side\\ \cline{1-2} s=&30 \end{cases}\implies SA=6(30)\implies SA=180[/tex]
Please please help me
Answer:
(a)
Step-by-step explanation:
The line y = x + 1 has a solid circle at x = 2 indicating that x is valid for this value, thus
y = x + 1 for x ≤ 2
The line y = x + 2 has an open circle at x = 2 indicating that x = 2 is not part of the solution but that values greater than 2 are valid, that is
y = x + 2 for x > 2
The definition for the function is (a)
Use this formula to find the value of a house with appreciation: A = V (1+r)Y
When Henry bought his house for $135,700, he was told that it would appreciate at a rate of five percent per year. If this remains true, how much will his house be worth in four years?
Answer:
[tex]A=\$164,944.20[/tex]
Step-by-step explanation:
we know that
[tex]A=V(1+r)^{Y}[/tex]
In this problem we have
[tex]r=5\%=0.05[/tex]
[tex]V=\$135,700[/tex]
[tex]Y=4\ years[/tex]
substitute in the formula and solve for A
[tex]A=\$135,700(1+0.05)^{4}=\$164,944.20[/tex]
Find the volume of the square pyramid below
Answer:
A
Step-by-step explanation:
The volume (V) of a pyramid is
V = [tex]\frac{1}{3}[/tex] area of base × perpendicular height (h)
area of square base = 4² = 16 and h = 4, hence
V = [tex]\frac{1}{3}[/tex] × 16 × 4 = [tex]\frac{64}{3}[/tex] = 21 [tex]\frac{1}{3}[/tex] ft³
Please please help me out
Answer:
The measure of angle y is [tex]m\angle y=54\°[/tex]
Step-by-step explanation:
we know that
The inscribed angle is half that of the arc it comprises.
[tex]m\angle y=\frac{1}{2}(108\°)=54\°[/tex]
Suppose that a coin is tossed 5 times. how many different outcomes include at least two heads
26 different outcomes include at least two heads.
There are [tex]26[/tex] different outcomes that include at least two heads when a coin is tossed [tex]5[/tex] times.
The total number of outcomes when a coin is tossed [tex]5[/tex] times is [tex]\(2^5 = 32\)[/tex], since each toss has [tex]2[/tex] possible outcomes (heads or tails).
To find the number of outcomes with at least two heads, we can find the total number of outcomes with exactly one head and no heads, and subtract that from the total number of outcomes.
1. Number of outcomes with no heads: There is only [tex]1[/tex] outcome with no heads ([tex]5[/tex] tails).
2. Number of outcomes with exactly one head: This can be calculated using combinations. There are [tex]5[/tex] ways to choose which toss will be heads, and for each of these, the remaining [tex]4[/tex] tosses must be tails. So, there are [tex]\(5 \times 1 = 5\)[/tex]outcomes with exactly one head.
Therefore, the number of outcomes with at least two heads is:
[tex]\[ 32 - 1 - 5 = 26 \][/tex]
Jimmy is trying to dive down and touch the bottom of the pool. On his first try he makes it 1/3 of the way to the bottom. On his second try he makes it 3/5 of the way to the bottom. Jimmys second dive was deeper than his first dive by what fraction of the pool?
Answer:
Jimmy's second dive was [tex]\frac{4}{15}[/tex] of the pool deeper than the first one
Explanation:
We are given that:
First dive was [tex]\frac{1}{3}[/tex] of the way to the bottom
Second dive was [tex]\frac{3}{5}[/tex] of the way to the bottom
We know that the second dive was deeper than the first one since [tex]\frac{3}{5} > \frac{1}{3}[/tex]
To know how much deeper the second dive was compared to the first one, we will simply subtract the depth of the first dive from that of the second one
Therefore:
The second dive was [tex]\frac{3}{5} - \frac{1}{3} = \frac{9}{15} - \frac{5}{15} = \frac{4}{15}[/tex] of the pool deeper than the first one
Hope this helps :)
Final answer:
Jimmy's second dive was 4/15 of the pool deeper than his first dive, calculated by finding a common denominator and subtracting the fractions representing the depth of each dive.
Explanation:
Jimmy's second dive was 3/5 of the way to the bottom of the pool, which is deeper than his first dive at 1/3 of the way. To find out how much deeper the second dive was compared to the first, we subtract the two fractions:
Second dive - First dive = 3/5 - 1/3
To subtract fractions, they must have a common denominator. Multiplying top and bottom of 3/5 by 3 and 1/3 by 5 gives us:
9/15 - 5/15 = 4/15
Therefore, Jimmy's second dive was 4/15 of the pool deeper than his first dive.
Which members are in the sample
Answer:
20, 26, 35, 18
Step-by-step explanation:
So starting at row 129, we look at the sequence two-digits at a time without overlapping. If that number is between 01 and 43, then they get selected.
The first two digits are 20. That fits between 01 and 43, so that member gets selected.
Next, we have 26. That also fits.
After that we have 64. Nope, too high.
98 and 44 are also too high.
35 fits though. So does 18.
So the members that get selected are 20, 26, 35, 18.
(10.02)
The point (−3, 1) is on the terminal side of angle θ, in standard position. What are the values of sine, cosine, and tangent of θ? Make sure to show all work.
Answer:
Part 1) [tex]sin(\theta)=\frac{\sqrt{10}}{10}[/tex]
Part 2) [tex]cos(\theta)=-3\frac{\sqrt{10}}{10}[/tex]
Part 3) [tex]tan(\theta)=-1/3[/tex]
Step-by-step explanation:
we know that
The angle is in the second quadrant so the sine is positive, the cosine is negative and the tangent is negative
step 1
Find the radius r applying the Pythagoras theorem
[tex]r^{2}=x^{2} +y^{2}[/tex]
substitute the given values
[tex]r^{2}=(-3)^{2} +(1)^{2}[/tex]
[tex]r^{2}=10[/tex]
[tex]r=\sqrt{10}\ units[/tex]
step 2
Find the value of [tex]sin(\theta)[/tex]
[tex]sin(\theta)=y/r[/tex]
substitute values
[tex]sin(\theta)=1/\sqrt{10}[/tex]
Simplify
[tex]sin(\theta)=\frac{\sqrt{10}}{10}[/tex]
step 3
Find the value of [tex]cos(\theta)[/tex]
[tex]cos(\theta)=x/r[/tex]
substitute values
[tex]cos(\theta)=-3/\sqrt{10}[/tex]
Simplify
[tex]cos(\theta)=-3\frac{\sqrt{10}}{10}[/tex]
step 4
Find the value of [tex]tan(\theta)[/tex]
[tex]tan(\theta)=y/x[/tex]
substitute values
[tex]tan(\theta)=-1/3[/tex]
Anyone know the answer to this?
Answer:
[tex]2^{n-1}[/tex]
Step-by-step explanation:
Square 1 has 2^0 pennies.
Square 2 has 2^1 pennies.
Square 3 has 2^2 pennies.
Square 4 has 2^3 pennies. The exponent of 2 is 1 less than the square number, so ...
Square n has 2^(n-1) pennies.
PLEASE HELP ME SOLVE THIS QUESTION!!
***quickest and shortest way***
Answer:
1/4 bag for each batch.
Step-by-step explanation:
Start with 4 bags. If the cake requires 1/4 bag, then 3 3/4 bags of flour are left over for the cookies. That's rather a nice number when you are dealing with 15 batches of cookies.
Start by changing the 3 3/4 into a decimal.
3 3/4 = 3.75
Now divide 3.75 by 15
3.75 / 15 = 0.25 bags which is 1/4 bag. You only have to come up with one value so this one will do.
If you have 14 1/2 dozen boxes of envelopes, and you order 3 1/4 dozen more, how many dozen boxes will you have in all?
The answere is 213 boxes
If you add 3 dozens to 14 dozens of boxes of envilopes , then you get 17 dozens of boxes of envilopes.Then of you sum up the 1/2 (2/4 ) with the 1/4.You get 3/4. Finally, you add 17 dozens to 3/4 of a dozen you get 17 3/4 dozens( wich is 213 boxes )
John owns an engine which receives heat input from a reservoir at 600 K and loses heat to a sink at 300 K. What is the maximum possible efficiency of this engine?
A.
25 percent
B.
50 percent
C.
75 percent
D.
100 percent
Answer:
D. 100 percent
Step-by-step explanation:
it is 100 percent because the reservoir is putting out 600k, but loses 300k, which would be easy to think it would be 50 percent because 300k is half of 600k, but however its peek outage is putting out 600k, therefore 600k = 600k, 100%.
Drag each tile to the correct box. Arrange the equations in order from least to greatest bed on their solution. Equation A: 5( x-6)+3x=3/4(2x-8) Equation B: 2.7(5.1x+4.9)=3.2+28.9 Equation C: 5(11x-18)=3(2x+7)
Answer:
B
C
A
Step-by-step explanation:
Find the value of x in each equation then compare the solutions from the least to the greatest
In A
5( x-6)+3x=3/4(2x-8) ------------Open brackets
5x-30+3x=3/2x-6
5x+3x-30=3/2x-6
8x-3/2x=-6+30---------------collect like terms
16x-3x=48
13x=48-----------------dived both sides by 13
x=48/13 = 3.7
In B
2.7(5.1x+4.9)=3.2+28.9-------------open bracket
13.77x+13.23=32.1
13.77x=32.1-13.23---------------collect like terms
13.77x=18.87
x=18.87/13.77----------------------divide both sides by 13.77
x=1.37
In C
5(11x-18)=3(2x+7)--------------------open brackets
55x-90=6x+21
55x-6x=21+90-----------------------collect like terms
49x=111----------------------------------divide both sides by 49 to get x
x=111/49 = 2.27
From the solutions, the least value of x is in B, then C ,and finally A
Equation B:[tex]\(x \approx 1.371\)[/tex]
Equation C:[tex]\(x \approx 2.2653\)[/tex]
Equation A: [tex]\(x = \frac{48}{13}\)[/tex]
Order from least to greatest: B, C, A.
To solve each equation, let's start by simplifying each side of the equation step by step.
Equation A:[tex]\(5(x-6) + 3x = \frac{3}{4}(2x - 8)\)[/tex]
Step 1: Distribute the numbers:
[tex]\(5x - 30 + 3x = \frac{3}{4}(2x) - \frac{3}{4}(8)\)[/tex]
Step 2: Combine like terms:
[tex]\(8x - 30 = \frac{3}{2}x - 6\)[/tex]
Step 3: To get rid of the fraction, multiply both sides by 2:
(16x - 60 = 3x - 12)
Step 4: Move all (x) terms to one side by subtracting (3x) from both sides:
(16x - 3x - 60 = -12)
Step 5: Combine like terms:
(13x - 60 = -12)
Step 6: Add 60 to both sides to isolate (x):
13x = 48
Step 7: Divide both sides by 13 to solve for (x):
[tex]\(x = \frac{48}{13}\)[/tex]
Equation B: 2.7(5.1x + 4.9) = 3.2 + 28.9
Step 1: Distribute the number:
13.77x + 13.23 = 32.1
Step 2: Move the constant to the other side by subtracting 13.23 from both sides:
13.77x = 18.87
Step 3: Divide both sides by 13.77 to solve for (x):
x ≈ 1.371
Equation C: (5(11x - 18) = 3(2x + 7)
Step 1: Distribute the numbers:
55x - 90 = 6x + 21
Step 2: Move all (x) terms to one side by subtracting (6x) from both sides:
55x - 6x - 90 = 21
Step 3: Combine like terms:
49x - 90 = 21
Step 4: Add 90 to both sides to isolate (x):
49x = 111
Step 5: Divide both sides by 49 to solve for (x):
x ≈ 2.2653
Now, let's order these solutions from least to greatest:
[tex]\(x= 1.371\)[/tex] (from Equation B)
[tex]\(x = 2.2653\)[/tex] (from Equation C)
[tex]\(x = \frac{48}{13}\)[/tex] (from Equation A)
So, the order from least to greatest based on their solutions is Equation B, Equation C, and then Equation A.
Can someone help me¿
Answer:
Step-by-step explanation:
number3
Express in scientific notation 1,789
Answer:
[tex]1.789*10^{3}[/tex]
Answer: 1.789 × 10 to the third power
Step-by-step explanation: the answer would be 1.789 x 10 to the third power because the A term has to be between 1 and 10
this is a geometry I question please explain your answer ty
Answer:
B. <Q = <R
Step-by-step explanation:
Angles are related to their intercepted arcs. An intercepted arc is found by finding the arc segment on a circle whose endpoints connect with the segments that make up an angle.
In this case, TQ and SQ make up <Q, so TS is the intercepted arc of <Q. However, TR and SR make up angle <R as well, making TS the intercepted arc of <R as well.
This means that because the angles share an intercepted arc, they are congruent.
Information about the recycling drive at school is shown in the table. Let A be the event that the item pulled out of the recycling bin is a plastic bottle, and let B be the event that a tenth grader recycled that item. Which statement is true about whether A and B are independent events? A and B are independent events because P(A∣B) = P(A). A and B are independent events because P(A∣B) = P(B). A and B are not independent events because P(A∣B) ≠ P(A). A and B are not independent events because P(A∣B) ≠ P(B).
Answer:
the answer is c
Step-by-step explanation:
The events are illustrations of probability, and the events A and B are not independent events because P(A∣B) ≠ P(A)
How to determine the true statement?From the complete table, we have the following parameter:
P(A∣B) ≠ P(A)
Two events A and B are independent if
P(A∣B) = P(A)
Given that:
P(A∣B) ≠ P(A)
It means that the events are not independent.
Hence, the events A and B are not independent events because P(A∣B) ≠ P(A)
Read more about probability at:
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What is the product?
8(–1)
8(-1) = -8
When a positive and a negative number is being multiplied the product is always negative, but when a negative and a negative number is being multiplied the product is positive
Hope this helped!
~Just a girl in love with Shawn Mendes
A candle manufacturer sells cylindrical candles in sets of three. Each candle in the set is a different size. The smallest candle has a radius of 0.5 inches and a height of 3 inches. The other two candles are scaled versions of the smallest, with scale factors of 2 and 3. How much wax is needed to create one set of candles? A. 27 π cubic inches B. 36 π cubic inches C. 53 π cubic inches D. 86 π cubic inches E. 98 π cubic inches
Answer:
A. 27 π cubic inches
Step-by-step explanation:
The volume of a cylinder is calculated using the formula;
[tex]Volume=\pi r^2h[/tex]
From the given information, the smallest candle has a radius of 0.5 inches and a height of 3 inches.
We substitute [tex]r=0.5[/tex] and [tex]h=3[/tex] into the given formula.
The vlume of the smallest candle is
[tex]Volume=\pi \times0.5^2\times 3[/tex]
[tex]Volume=\frac{3}{4}\pi in^3[/tex]
from the given information, the other two candles are scaled versions of the smallest, with scale factors of 2 and 3.
The volume of the other two candles will be [tex]2^3\times \frac{3}{4}\pi=6\pi in^3[/tex] and [tex]3^3\times \frac{3}{4}\pi=\frac{81}{4}\pi in^3[/tex]
The wax needed to create one set of candle is
[tex]\frac{3}{4}\pi+6\pi+\frac{81}{4}\pi=27\pi\: in^3[/tex]
The correct answer is A
Answer:
27 pi in³
Step-by-step explanation:
I just took a test on Plato/Edmentum with this question and this was the right answer
~Please mark me as brainliest :)
A copy machine depreciates at the rate of 15% each year. If the original cost of the copy machine was $20,000, what is the approximate value of the machine at the end of 3 years?
$12,282.5 (20000x0.85x0.85x0.85)
The value of the copy machine depreciates at a rate of 15% each year. After calculating the depreciated value for 3 consecutive years, the copy machine that initially cost $20,000 is worth approximately $12,282.50 after 3 years.
Explanation:The original cost of the copy machine is $20,000. Given that the copy machine depreciates, or loses value, at a rate of 15% each year, we need to calculate the value of the copy machine each year for 3 years by subtracting 15% of its current value.
In the first year, the value of the copy machine would be $20,000 - (15% of $20,000) = $20,000 - $3,000 = $17,000. In the second year, we will take 15% off $17,000, so the value will be $17,000 - (15% of $17,000) = $17,000 - $2,550 = $14,450. Finally, in the third year, we will take 15% off $14,450, so the final value is $14,450 - (15% of $14,450) = $14,450 - $2,167.50 = $12,282.50. So, the approximate value of the machine at the end of 3 years is $12,282.50.Learn more about Depreciation here:https://brainly.com/question/17827672
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Hotdogs and corndogs were sold at last night’s football game. Use the information below to write equations to help you determine how many corndogs were sold.
The number of hotdogs sold was three fewer than twice the number of corndogs sold. Write an equation relating the number of hotdogs and corndogs. Let h represent the number of hotdogs and c represent the number of corndogs. A hotdog costs $3 and a corndog costs $1.50. If $201 was collected, write an equation to represent this information. How many corndogs were sold? Show how you calculated your answer.
Answer:
28
Step-by-step explanation:
If we set up our equation using the unknown number of hot dogs and corn dogs with their individual prices attached to them, we can set the sum of them equal to $201. We know that a hot dog costs $3, so we can represent hot dogs monetarily by attaching the cost of a single hot dog to the h. For example, if a hot dog costs $3, and we represent the expression as 3h, with h being the number of hot dogs sold, if we sell 4 hot dogs at $3 apiece, we make $12. If we sell 6 hot dogs we will make $18. The same goes for the corn dogs. We don't know how many corn dogs or hot dogs we sold, but we do know that the sales of both made $201. So our expression for that is
3h + 1.50c = 201
That's great, but we have too many unknowns, and that's a problem. So let's look back up to where we are told that the number of hot dogs is 3 less than 2 times the number of corn dogs. "3 less than" is -3 algebraically. "Twice the number" is 2times and the words "is" and "was" represent the = sign. So putting those words into an algebraic equation looks like this:
h = 2c - 3
That says "the number of hot dogs was twice the number of corn dogs less 3". Now that we have an expression for hot dogs we can sub it into our money equation in place of h:
3h + 1.5c = 201 becomes 3(2c - 3) + 1.5c = 201
Now we have an equation with only c's in it.
Distribute through the parenthesis to get
6c - 9 + 1.5c = 201
Simplify to 7.5c = 210
Now divide by 7.5 to get that c = 28.
Now that we know that, we go back with that number and sub it in for c in
h = 2c - 3 --> h = 2(28) - 3 gives us that the number of hot dogs sold was 53
The total number of corn dogs sold is 28 corn dogs and number of hotdogs is 53
let
h = number of hotdogs
c = number of corndogs.
cost of hotdog = $3
corn dog = $1.50
The equation:
3h + 1.50c = 201 (1)
h = 2c - 3 (2)
substitute (2) into (1)
3(2c - 3) + 1.50c = 201
6c - 9 + 1.50c = 201
7.50c - 9 = 201
7.50c = 201 + 9
7.50c = 210
c = 210/7.50
c = 28
Therefore,
h = 2c - 3
= 2(28) - 3
= 56 - 3
= 53
Read more:
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Solve for x.
1/10(x - 3) = -40
A) -403
B) -397
C) -7
D) -1
Answer:
-397Step-by-step explanation:
This may look daunting, but let us approach it step by step.
Step 1: Remove the Parenthesesmultiply 1/10 by x - 3
0.1x - 0.3 = -40
Step 2: Add 0.3In algebra, the goal is always to undo all the operations and get back to the original problem so that the mystery value can be determined. In this case since 0.3 was removed, we must add it back.
0.1x = -39.7
Step 3. Divide by 0.10.1x/0.1 = x
39.7/0.1 = -397
Step 4. Preliminary AnswerAnswer seems to be B. -397, but we should still check it.
Step 5: Check0.1(-397) - 0.3 = -40
-39.7 - 0.3 = -40
-40 = -40 Correct
If the answer was incorrect, this would show that there had been a flaw in our calculations. But everything checks out, so we are done!
Step 6: Final AnswerOur final answer is B. -397.
PLEASE MARK BRAINLIEST
Answer:
The answer is -397
Step-by-step explanation:
A bag contains 26 tiles showing a different letter from A to Z. Each player draws a letter tile at random. Player 1 wins if the letter is in his or her name. Player 2 wins if the letter is in his or her name. If the letter is in both their names then no one wins. Nathan and Katie play the game. Is this a fair game? If not, who has the advantage?
Answer:
no
Step-by-step explanation:
Nathan has 6 letters in his name so he has a higher chance of winning
No, Katie has the advantage.
Katie has 5 distinct letters.
Nathan has 4 distinct letters.
They have one overlap (a).
Katie can expect to win 4 out of 26 games.
Nathan can expect to win 3 out of 26 games.
Since Katie has more distinct letters, she has the advantage, so it is not a fair game.
HELP PLEASE!!
Question 1 (2 points)
Generalize the pattern by finding the nth term.
6, 10, 14, 18, 22,
A. 4n
B. 4n + 2
C. 4n + 10
D. 6n + 4
Answer:
B. 4n+2
Step-by-step explanation:
You are given the pattern 6, 10, 14, 18, 22
Rewrite it as
[tex]a_1=6\\ \\a_2=10\\ \\a_3=14\\ \\a_4=18\\ \\a_5=22[/tex]
Note that
[tex]a_2-a_1=a_3-a_2=a_4-a_3=a_5-a_4=4[/tex]
This means that given pattern is a part of arithmetic sequence with
[tex]a_1=6\\ \\d=4[/tex]
So, the nth term of this arithmetic sequence is
[tex]a_n=a_1+(n-1)d\\ \\a_n=6+4(n-1)\\ \\a_n=6+4n-4\\ \\a_n=4n+2[/tex]
At Eagle Rock High School, the probability that a student takes theatre and choir is 0.078. The probability that a student takes choir is 0.26. What is the probability that a student takes theatre given that the student is taking choir?
Final answer:
The probability that a student takes theatre given that the student is taking choir is found using conditional probability and is calculated to be 0.3, or 30%.
Explanation:
To find the probability that a student takes theatre given that the student is taking choir, we use the definition of conditional probability. In this scenario, the probability of a student taking theatre and choir (joint probability) is given as 0.078, and the probability of a student taking choir (marginal probability) is 0.26.
The formula for conditional probability is:
P(A | B) = P(A and B) / P(B)
Let A represent the event of a student taking theatre, and B represent the event of a student taking choir. Substituting the given values into the formula yields:
P(A | B) = 0.078 / 0.26
Performing the division gives us:
P(A | B) = 0.3
Therefore, the probability that a student takes theatre given that the student is taking choir is 0.3, or 30%.
The City Zoo collected $100 in one morning. An adult ticket is $5 each, and a child's ticket is $3. How many different combinations of adult and children's tickets would have totaled $100?
5
6
7
8
Final answer:
To calculate the number of different combinations of adult and children's tickets that would total $100, we can set up an equation: 5x + 3y = 100. We can find possible values of 'x' and 'y' that satisfy the equation. There are 6 different combinations of adult and children's tickets that would have totaled $100.
Explanation:
To calculate the number of different combinations of adult and children's tickets that would total $100, we can set up an equation:
5x + 3y = 100
Where 'x' represents the number of adult tickets and 'y' represents the number of children's tickets. We need to find whole number solutions for 'x' and 'y'.
We can start by finding the possible values of 'x' and 'y' that satisfy the equation and add up to $100. The possible combinations are:
x = 0, y = 33
x = 5, y = 31
x = 10, y = 29
x = 15, y = 27
x = 20, y = 25
x = 25, y = 23
Therefore, there are a total of 6 different combinations of adult and children's tickets that would have totaled $100.
Anyone mind helping me out?:)
The answer is does matter.
through (-1,2) parallel to y=-4x+3
Answer:
y = -4x - 2
Step-by-step explanation:
Parallel has same slope
so
y - 2 = -4(x + 1)
y - 2 = -4x - 4
y = -4x - 2
Equation
y = -4x - 2