To prove that the expression (36^5−6^9)(38^9−38^8) is divisible by 30, we need to show that it is divisible by both 2 and 3.
First, we can factor out a 6^9 from the first term:
(36^5−6^9)(38^9−38^8) = 6^9(6^10-36^5)(38^9-38^8)
Notice that 6^10 can be written as (2*3)^10, which is clearly divisible by both 2 and 3. Also, 36 is divisible by 3, so 36^5 is divisible by 3^5. Thus, we can write:
6^9(6^10-36^5) = 6^9(2^10*3^10 - 3^5*2^10) = 6^9*2^10*(3^10 - 3^5)
Since 2^10 is divisible by 2, and 3^10 - 3^5 is clearly divisible by 3, the whole expression is divisible by both 2 and 3, and therefore divisible by 30.
To prove that the expression is divisible by 37, we can use Fermat's Little Theorem. Fermat's Little Theorem states that if p is a prime number and a is any positive integer not divisible by p, then a^(p-1) is congruent to 1 modulo p, which can be written as a^(p-1) ≡ 1 (mod p).
In this case, p = 37, and 36 is not divisible by 37. Therefore, by Fermat's Little Theorem:
36^(37-1) ≡ 1 (mod 37)
Simplifying the exponent gives:
36^36 ≡ 1 (mod 37)
Similarly, 38 is not divisible by 37, so:
38^(37-1) ≡ 1 (mod 37)
Simplifying the exponent gives:
38^36 ≡ 1 (mod 37)
Now we can use these congruences to simplify our expression:
(36^5−6^9)(38^9−38^8) ≡ (-6^9)(-1) ≡ 6^9 (mod 37)
We know that 6^9 is divisible by 3, so we can write:
6^9 = 2^9*3^9
Since 2 and 37 are relatively prime, we can use Euler's Totient Theorem to simplify 2^9 (mod 37):
2^φ(37) ≡ 2^36 ≡ 1 (mod 37)
Therefore:
2^9 ≡ 2^9*1 ≡ 2^9*2^36 ≡ 2^(9+36) ≡ 2^45 (mod 37)
Now we can simplify our expression further:
6^9 ≡ 2^45*3^9 ≡ (2^5)^9*3^9 ≡ 32^9*3^9 (mod 37)
Notice that 32 is congruent to -5 modulo 37, since 32+5 = 37. Therefore:
32^9 ≡ (-5)^9 ≡ -5^9 ≡ -1953125 ≡ 2 (mod 37)
So:
6^9 ≡ 2*3^9 ≡ 2*19683 ≡ 39366 ≡ 0 (mod 37)
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!!PLEASE HELPP!! (check if I’m right pls)
Answer: It's correct
Step-by-step explanation:
Find the absolute extrema of the function, if they exist, over the indicated interval. Also indicate the x-value at which each extremum occurs. If no interval is specified, use the real numbers, (-00,00). f(x) = -0.002x2 + 4.2x - 50 Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. at x= O A. The absolute maximum is at x= and the absolute minimum is (Use a comma to separate answers as needed.) B. The absolute minimum is at x = and there is no absolute maximum. (Use a comma to separate answers as needed.) C. The absolute maximum is at x= and there is no absolute minimum. (Use a comma to separate answers as needed.) D. There is no absolute maximum and no absolute minimum.
The correct choice is: C. The absolute maximum is at x = 1050, and there is no absolute minimum.
To find the absolute extrema of the function f(x) = -0.002x^2 + 4.2x - 50 over the interval (-∞, ∞), we need to find the critical points and then determine if there's a maximum or minimum at each point.
Step 1: Find the derivative of the function f(x) with respect to x. f'(x) = -0.004x + 4.2
Step 2: Set the derivative equal to zero and solve for x. -0.004x + 4.2 = 0 x = 1050
Step 3: Since we have only one critical point, we need to determine if it's a maximum or a minimum. To do this, we can use the second derivative test.
Step 4: Find the second derivative of the function f(x) with respect to x. f''(x) = -0.004
Step 5: Since the second derivative is negative (f''(x) = -0.004 < 0), the critical point x = 1050 corresponds to an absolute maximum. Step 6: Calculate the value of the function f(x) at x = 1050. f(1050) = -0.002(1050)^2 + 4.2(1050) - 50 = 2150
Thus, the absolute maximum is at x = 1050, and the value is 2150. Since the function is a parabola with the "mouth" facing downwards, there is no absolute minimum.
The correct choice is: C. The absolute maximum is at x = 1050, and there is no absolute minimum.
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A spinner with repeated colors numbered from 1 to 8 is shown. Sections 1 and 8 are purple. Sections 2 and 3 are yellow. Sections 4, 5, and 6 are blue. Section 7 is red.
Spinner divided evenly into eight sections with three colored blue, one red, two purple, and two yellow.
Determine the theoretical probability of the spinner not landing on yellow, P(not yellow).
The theoretical probability of the spinner not landing on yellow, would be 75 %.
How to find the probability ?In order to calculate the likelihood of the spinner not landing on yellow, it is necessary to initially identify the quantity of non-yellow partitions and subsequently divide this by the full tally of sections. The spinner comprises a total of 8 individual segments.
Of these, two (i.e., sections 2 and 3) are colored in shades of yellow, hence totaling two yellow sectors. This leaves a further six compartments - numbered 1, 4, 5, 6, 7 and 8, that do not fall into the category of "yellow."
The probability is therefore :
= ( Number of not yellow sections ) / ( Total number of sections )
= 6 / 8
= 3 / 4
= 75 %
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67. 8 x 9. 7 pls someone answer within the next 20 Minutes with work I'm in school lol
Traffic Jam
There are 8 cans of strawberry jam, 7 raspberry jam,
and 5 cherry jam in the cellar. You're trying to sneak
some out, but don't want to attract attention or take
too many. It's dark, so you can't tell what kind of jam
you're taking.
How many cans can you sneak out of the
cellar in the dark with the certainty that there
will still be at least 4 cans of one kind of jam
and 3 cans of another left over?
Answer:
Hey!
You could obviously count how many you're taking, so that's 7 left behind. My guess is that you could taste the jam... but that's the best I've got.
The requreid we can sneak out 9 cans of jam in the dark and still be sure that there will be at least 4 cans of one kind of jam and 3 cans of another left over.
What is arithmetic?It involves the basic operations of addition, subtraction, multiplication, and division, as well as more advanced operations such as exponents, roots, logarithms, and trigonometric functions.
Let's first find the minimum number of cans that need to be left in the cellar to meet the given criteria. We want at least 4 cans of one kind of jam and 3 cans of another leftover. This means we can take a maximum of:
8 - 4 = 4 cans of strawberry jam
7 - 3 = 4 cans of raspberry jam
5 - 3 = 2 cans of cherry jam
So, we can take a maximum of 4 + 4 + 2 = 10 cans in total.
To have certainty that we meet the criteria, we need to take one less than the maximum number of cans, which is 9 cans. So, we can sneak out 9 cans of jam in the dark and still be sure that there will be at least 4 cans of one kind of jam and 3 cans of another left over.
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Lana offered to buy groceries for her roommates, Pam and Cheryl. The total bill was $74. She forgot to save the individual receipts but remembered that Pam's groceries were $0. 05 cheaper than half of her groceries, and that Cheryl's groceries were $2. 10 more than Pam's groceries. How much was each share of the groceries?
Lana paid $36, Pam paid $17.95, and Cheryl paid $20.05, by using substitution or elimination, for the groceries.
Let's start by assigning variables to the unknown quantities in the problem. Let's call the cost of Lana's groceries "L", the cost of Pam's groceries "P", and the cost of Cheryl's groceries "C". We can set up a system of equations based on the information given:
1) P = 0.5L - 0.05 (Pam's groceries were $0.05 cheaper than half of Lana's groceries)
2) C = P + 2.10 (Cheryl's groceries were $2.10 more than Pam's groceries)
3) L + P + C = 74 (the total bill was $74)
We now have three equations with three unknowns, which we can solve using substitution or elimination. Let's use substitution:
Substitute equation 1 into equation 2 for P:
C = (0.5L - 0.05) + 2.10
Simplify:
C = 0.5L + 2.05
Substitute equations 1 and 3 into the equation above:
L + P + C = 74
L + (0.5L - 0.05) + (0.5L + 2.05) = 74
Simplify:
2L + 2 = 74
2L = 72
L = 36
Now that we know the cost of Lana's groceries, we can use equation 1 to find the cost of Pam's groceries:
P = 0.5L - 0.05
P = 0.5(36) - 0.05
P = 17.95
Finally, we can use equation 2 to find the cost of Cheryl's groceries:
C = P + 2.10
C = 17.95 + 2.10
C = 20.05
Therefore, Lana paid $36, Pam paid $17.95, and Cheryl paid $20.05 for the groceries.
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What is the value of 45 nickels as a decimal number ?
Answer:
2.25
Step-by-step explanation:
45 nickels
45*5=225
225 cents
2.25
The value of 45 nickels in decimal number can be 2.25.
In the decimal system, each digit's value depends on its position or place value within the number.
A nickel is worth 0.05 dollars.
To find the value of 45 nickels, multiply the number of nickels by the value of each nickel:
So, Value = Number of nickels × Value of each nickel
= 45 × 0.05
= 2.25
Therefore, the value of 45 nickels is $2.25 as a decimal number.
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Please help
given vectors u and v, find (a)6u (b)6u+4v (c) v-4u
u=(4,5) v=(4,0)
(a) 6u
(b) 6u+4v
(c) v-4u
For the given vectors u and v, (a) 6u = (24, 30). (b) 6u+4v = (40, 30). (c) v-4u = (-12, -20).
Given vectors u and v.
a) To find 6u, we simply multiply each component of u by 6:
6u = 6(4, 5) = (6(4), 6(5)) = (24, 30)
Therefore, 6u = (24, 30).
b) To find 6u + 4v, we first need to find 4v by multiplying each component of v by 4:
4v = 4(4, 0) = (4(4), 4(0)) = (16, 0)
Next, we add 6u and 4v by adding the corresponding components:
6u + 4v = (24, 30) + (16, 0) = (24+16, 30+0) = (40, 30)
Therefore, 6u + 4v = (40, 30).
c) To find v - 4u, we first need to find 4u by multiplying each component of u by 4:
4u = 4(4, 5) = (4(4), 4(5)) = (16, 20)
Next, we subtract 4u from v by subtracting the corresponding components:
v - 4u = (4, 0) - (16, 20) = (4-16, 0-20) = (-12, -20)
Therefore, v - 4u = (-12, -20).
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If donuts are 12 cents a dozen how much does 100 donuts cost.
The cost of 100 donuts is $ 1 if a dozen of donuts cost 12 cents.
This question is solved using the unitary method. The unitary method is a method in which you find the value of a unit and then the value of the required number of units.
1 dozen refers to a group of 12.
Cost of 1 dozen donuts or 12 donuts = 12 cents
Cost of 1 donut = [tex]\frac{12}{12}[/tex] = 1 cent
Cost of 100 donuts = 1 * 100 = 100 cents
100 cents = 1 dollar.
Thus, the cost of 100 donuts is 100 cents or 1 dollar.
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Matthew is saving money for a pet turtle. The data in the table represent the total amount of money in dollars that he saved by the end of each week.
A graph of the points that represent this data are shown on the coordinate plane attached below.
How to construct and plot the data in a scatter plot?In this scenario, the week number would be plotted on the x-axis (x-coordinate) of the scatter plot while the amount of money (in dollars) would be plotted on the y-axis (y-coordinate) of the scatter plot through the use of Microsoft Excel.
On the Excel worksheet, you should right click on any data point on the scatter plot, select format trend line, and then tick the box to display an equation for the line of best fit (trend line) on the scatter plot.
From the scatter plot (see attachment) which models the relationship between the week number and the amount of money (in dollars), a linear equation for the line of best fit is as follows:
y = 1.19x + 1.05
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Given y = 4x² + 3x, find dy/dt when x= -1 and dx/dt = 3(Simplify your answer.)
Given the function y = 4x² + 3x, we will find dy/dt by differentiating y with respect to t. Therefore, the value of dy/dt is -15.
Using the chain rule, we have:
dy/dt = (dy/dx)(dx/dt)
Differentiating y with respect to x, we get:
dy/dx = 8x + 3
Now, we are given that x = -1 and dx/dt = 3. We can substitute these values into our equation:
dy/dt = (8(-1) + 3)(3)
dy/dt = (-5)(3)
dy/dt = -15
So, when x = -1 and dx/dt = 3, the value of dy/dt is -15.
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Find the perimeter of a square that has a side length of 4.3x + 2 inches.
The calculated perimeter of the square from the side length is 17.2x + 8
Finding the perimeter of a square from the side lengthFrom the question, we have the following parameters that can be used in our computation:
A square that has a side length of 4.3x + 2 inches.
Using the above as a guide, we have the following:
Perimeter = 4 * side length
Substitute the known values in the above equation, so, we have the following representation
Perimeter = 4 * (4.3x + 2)
When teh brackets are opened, we have
Perimeter = 17.2x + 8
Hence, the perimeter is 17.2x + 8
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If Sarah uses 3/4 yard of ribbon to make a hair bow. How many yards of ribbon will Sarah use to make 9 hair bows?
If Sarah uses 3/4 yard of ribbon to make a hair bow, she will need 6 and 3/4 yards of ribbon to make 9 hair bows.
To find out how many yards of ribbon Sarah will use to make 9 hair bows, we need to multiply the amount of ribbon used for one hair bow (3/4 yard) by the number of hair bows she wants to make (9).
So, the equation we need to use is:
3/4 yard of ribbon per hair bow x 9 hair bows = ? yards of ribbon
To solve for the answer, we can simplify the equation:
3/4 x 9 = 27/4
So Sarah will need 27/4 yards of ribbon to make 9 hair bows.
To convert this fraction to a mixed number, we can divide the numerator (27) by the denominator (4) and write the remainder as a fraction:
27 ÷ 4 = 6 with a remainder of 3
In summary, Sarah will need 6 and 3/4 yards of ribbon to make 9 hair bows, if she uses 3/4 yard of ribbon to make one hair bow.
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(3x^3 y^2)^3 (2x^4 y^2)^2
Answer:
108y^10x^17
Step-by-step explanation:
Use the scatter plot to fill in the missing coordinate of the ordered pair.(,12)
Answer: Pairs (0, 14) and (10, 0
Step-by-step explanation:
From a Word Problem
Jack has $10 in his lunch account. He plans to
spend $2 a week on snacks. How long until
Jack's lunch account reaches zero?
Answer:
Sure, here's the solution to the word problem:
Jack has $10 in his lunch account and plans to spend $2 a week on snacks. To find out how long it will take his lunch account to reach zero, we can divide the total amount of money in his account by the amount he spends each week.
```
$10 / $2 = 5 weeks
```
Therefore, it will take Jack 5 weeks to spend all of the money in his lunch account.
Here's another way to solve the problem:
We can also set up an equation to represent the situation. Let x be the number of weeks it takes Jack's lunch account to reach zero. We know that Jack starts with $10 and spends $2 each week, so we can write the equation:
```
$10 - $2x = 0
```
Solving for x, we get:
```
x = 5
```
Therefore, it will take Jack 5 weeks to spend all of the money in his lunch account.
Answer:
in 5 weeks he will have 0$ in his account
Step-by-step explanation:
If the Math Olympiad Club consists of 14 students, how many different teams of 6 students can be formed for competitions?
The different teams of 6 students that can be formed for competitions is 3003
How many different teams of 6 students can be formed for competitions?From the question, we have the following parameters that can be used in our computation:
Students = 14
Students in the team = 6
Using the above as a guide, we have the following:
n = 14
r = 6
The different teams of 6 students that can be formed for competitions is calculated as
Teams = nCr
substitute the known values in the above equation, so, we have the following representation
Teams = 14C6
So, we have
Teams = 14!/(6! * 8!)
Evaluate
Teams = 3003
Hence, the different teams of 6 students that can be formed for competitions is 3003
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For each of the following equations, • find general solutions; solve the initial value problem with initial condition y(0)=-1, y'0) = 2; sketch the phase portrait, identify the type of each equilibrium, and determine the stability of each equilibrium. (a) 2y" +9y + 4y = 0 (b) y" +2y - 8y=0 (c) 44" - 12y + 5y = 0 (d) 2y" – 3y = 0 (e) y" – 2y + 5y = 0 (f) 4y" +9y=0 (g) 9y' +6y + y = 0
(a) y(x) = c1 e^(-4x/3) cos(2x) + c2 e^(-4x/3) sin(2x), stable node at the origin;
(b) y(x) = c1 e^(2x) + c2 e^(-4x), unstable node at the origin;
(c) y(x) = c1 e^(-x/22) cos(sqrt(119)x/22) + c2 e^(-x/22) sin(sqrt(119)x/22), stable node at the origin;
(d) y(x) = c1 e^(sqrt(3)x/2) + c2 e^(-sqrt(3)x/2), unstable saddle at the origin;
(e) y(x) = c1 e^x cos(2x) + c2 e^x sin(2x), stable spiral at the origin;
(f) y(x) = c1 cos(3x/2) + c2 sin(3x/2), stable limit cycle around the origin;
(g) y(x) = c1 e^(-x/3) + c2 e^(-x), stable node at the origin.
(a) The characteristic equation is 2r^2 + 9r + 4 = 0, with roots r1 = -4/3 and r2 = -1/2. The general solution is y(x) = c1 e^(-4x/3) cos(2x) + c2 e^(-4x/3) sin(2x). The equilibrium at the origin is a stable node since both eigenvalues have negative real parts.
(b) The characteristic equation is r^2 + 2r - 8 = 0, with roots r1 = 2 and r2 = -4. The general solution is y(x) = c1 e^(2x) + c2 e^(-4x). The equilibrium at the origin is an unstable node since both eigenvalues have positive real parts.
(c) The characteristic equation is 44r^2 - 12r + 5 = 0, with roots r1 = (3 + sqrt(119))/22 and r2 = (3 - sqrt(119))/22. The general solution is y(x) = c1 e^(-x/22) cos(sqrt(119)x/22) + c2 e^(-x/22) sin(sqrt(119)x/22). The equilibrium at the origin is a stable node since both eigenvalues have negative real parts.
(d) The characteristic equation is 2r^2 - 3 = 0, with roots r1 = sqrt(3)/2 and r2 = -sqrt(3)/2. The general solution is y(x) = c1 e^(sqrt(3)x/2) + c2 e^(-sqrt(3)x/2). The equilibrium at the origin is an unstable saddle since the eigenvalues have opposite signs.
(e) The characteristic equation is r^2 - 2r + 5 = 0, with roots r1 = 1 + 2i and r2 = 1 - 2i. The general solution is y(x) = c1 e^x cos(2x) + c2 e^x sin(2x). The equilibrium at the origin is a stable spiral since both eigenvalues have negative real parts and non-zero imaginary parts.
(f) The characteristic equation is 4r^2 + 9 = 0, with roots r1 = 3i/2 and r2 = -3i/2. The general solution y(x) = c1 cos(3x/2) + c2 sin(3x/2), stable limit cycle around the origin.
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QuestionThe mean monthly salary of the 12 employees of a firm is Rs. 1450. If one more person joins the firm who gets Rs. 1645 per month, what will be the mean monthly salary of 13 employees?ARs. 1465BRs. 1954CRs. 2175DRs. 2569Medium
1465 will be the mean monthly salary .The answer is (A) Rs. 1465.
Let the sum of the 12 employees' salaries be S.
Then, the mean monthly salary of the 12 employees is given by:
S/12 = 1450
S = 12 * 1450
S = 17400
If one more person joins with a salary of Rs. 1645, the new sum of the 13 employees' salaries is:
S' = S + 1645
S' = 17400 + 1645
S' = 19045
The new mean monthly salary of the 13 employees is:
S'/13 = 19045/13
S'/13 = 1465
Therefore, the answer is (A) Rs. 1465.
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La probabilidad de que un vuelo se retrase es 0. 2 (=20%),¿Cuales son las probabilidades de que no haya demoras en un viaje de ida y vueta
La probabilidad de que no haya demoras en un viaje de ida y vuelta es 0.64 (64%).
How to calculate the probabilities?La probabilidad de que no haya demoras en un viaje de ida y vuelta se puede calcular utilizando la probabilidad complementaria. Si la probabilidad de que un vuelo se retrase es 0.2 (20%), entonces la probabilidad de que no haya retrasos en un vuelo individual es 1 - 0.2 = 0.8 (80%).
Para un viaje de ida y vuelta, la probabilidad de que no haya retrasos en ambos vuelos se calcula multiplicando las probabilidades de no retraso de cada vuelo.
Entonces, la probabilidad de que no haya demoras en un viaje de ida y vuelta sería 0.8 * 0.8 = 0.64 (64%), o 64 de cada 100 viajes de ida y vuelta no experimentarían retrasos.
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For each problem, determine what will happen to the first factor.
10x1/2
15 x 7/2
Answer:
52.5
Step-by-step explanation:
15×7=105÷2
=52.5 ans it means fifteen times seven divided by two
How many of the shapes below are trapeziums?
Answer:
2
Step-by-step explanation:
The K and N are the trapeziums and the two lines opposite to them go in a parallel line
Given the center, a vertex, and one focus, find an equation for the hyperbola:
center: (-5, 2); vertex (-10, 2); one focus (-5-√29,2).
The equation of the hyperbola is -(x + 5)²/71 + (y - 2)² = -71
How to calculate the valueWe can also find the distance between the center and the given focus, which is the distance between (-5, 2) and (-5 - √29, 2):
d = |-5 - (-5 - √29)| = √29
Substituting in the known values, we get:
c² = a² + b²
(√29)² = (10)² + b²
29 = 100 + b²
b² = -71
(x - h)²/a² - (y - k)²/b² = 1
where (h, k) is the center of the hyperbola.
Substituting in the known values, we get:
(x + 5)²/100 - (y - 2)²/-71 = 1
Multiplying both sides by -71, we get:
-(x + 5)²/71 + (y - 2)²/1 = -71/1
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For the class party, Josue and Pho each brought 1 3/5 liters of lemonade. How many liters of lemonade did they bring altogether?
Josue and Pho brought 3 1/5 liters of lemonade altogether
Josue and Pho brought 1 3/5 liters of lemonade each, so the total amount of lemonade they brought is:
1 3/5 + 1 3/5 = 3 1/5
To add the two mixed numbers, we first need to find a common denominator. In this case, the common denominator is 5. Then we convert both mixed numbers into fractions with a denominator of 5:
1 3/5 = (5 × 1 + 3) / 5 = 8/5
1 3/5 = (5 × 1 + 3) / 5 = 8/5
Now we can add the fractions:
8/5 + 8/5 = (8 + 8) / 5 = 16/5
Finally, we can convert the fraction back to a mixed number:
16/5 = 3 1/5
Therefore, Josue and Pho brought 3 1/5 liters of lemonade altogether.
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Complete the sentences about the expressions 3x+4 –2x
, and 5x+2x+x
.
CLEAR CHECK
In the expression 3x+4 –2x
, you can combine
like terms, and the simplified expression is
.
In the expression 5x+2x+x
, you can combine
like terms, and the simplified expression is
For the expressions 3x+4 –2x, and 5x+2x+x the simplified expression after combining like terms is x+4 and 8x.
The given expressions are 3x+4 –2x, and 5x+2x+x
We have to simplify these expressions by combining the like terms
For the expression 3x+4 –2x
We have to combine like terms
x+4
Now for expression 5x+2x+x
Combine the like terms to get
8x
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tim can paint a room in 6 hours . bella can paint the same room in 4 hours . how many hours would it take tim and bella to paint the room while working together y=kx+b
please help me now.
Answer: 3
Step-by-step explanation:
Answer:
2hrs 24 mins
Step-by-step explanation:
Ok so let's make this problem a bit simpler by splitting it up.
Tim paints a room in 6 hours.
So, we can also say that she paints 1/6 of that room in 1 hour
Bella paints it in 4 hours
So, we can also say that she paints 1/4 of that room in 1 hour
Now, lets see what we have:
Bella: 1/4 every hour
Tim: 1/6 every hour
The problem states that they are working together, so we need to add the values we have:
1/4 + 1/6
We cannot just add them, we must make them have the same common denominator.
LCD is 12, you can find that by just doing the times tables for 4 and 6 and seeing what number they match on first.
3/12 + 2/12 = 5/12
So, tim and bella working together paint 5/12 of a room in 1 hour.
They paint 5/12 of a room in 60 minutes
They paint 1/12 of the room in 12 minutes(divide both values by 5)
So if they paint 1/12 of the room in 12 minutes, we can multiply both values by 12 to get our answer.
They paint the full room in 144 minutes(12*12).
144 minutes is 2 hours and 24 minutes
How much must be deposited today into the following account in order to have a $110,000 college fund in 17 years? Assume no additional deposits are made.
An account with quarterly compounding and an APR of 4.9%
Therefore, an initial deposit of $37,728.66 is required to have a college fund of $110,000 in 17 years with quarterly compounding and an APR of 4.9%.
What is a deposit used for?An amount held in an account is referred to as a deposit. It might be put up in a bank as collateral for goods that are being rented out or bought. A deposit is used in many different sorts of economic transactions.
Compound interest can be calculated using the following formula to determine the required down payment:
A = P(1 + r/n)(nt)
where:
A = the future value of the account (in this case, $110,000)
P = the principal or initial deposit
r = the annual interest rate (4.9%)
n = the number of times the interest is compounded per year (4 for quarterly compounding)
t = the number of years (17)
When we enter the specified numbers into the formula, we obtain:
$110,000 = P(1 + 0.049/4)(4*17)
$110,000 = P(1.01225)⁶⁸
$110,000 = P * 2.9126
Dividing both sides by 2.9126, we get:
P = $37,728.66
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Use the definition of the laplace transform to show that if f(x) = 0 then
[tex]l[f(x)] = 0[/tex]
show that f(x)= 1 then
[tex]l[f(x)] = \frac{1}{s} [/tex]
show that f(x)= x then
[tex]l[f(x)] = \frac{1}{ {s}^{2} } [/tex]
show that f(x)= e^ax then
[tex]l[f(x)] = \frac{1}{s - a} [/tex]
provide the steps by using the definition and evaluating the integral.
Answer:
Step-by-step explanation:
the Laplace transform of the function f(x) = e^(ax) is 1/(a-s).
The definition of the Laplace transform of a function f(t) is given by:
L{f(t)} = F(s) = ∫_0^∞ e^(-st) f(t) dt
where s is a complex number.
If f(x) = 0, then we have:
L{f(x)} = L{0} = ∫_0^∞ e^(-st) 0 dt = 0
Therefore, the Laplace transform of the zero function is zero.
If f(x) = 1, then we have:
L{f(x)} = L{1} = ∫_0^∞ e^(-st) dt
Using integration by parts, we get:
L{1} = ∫_0^∞ e^(-st) dt = [-e^(-st)/s]_0^∞ = [0 - (-1/s)] = 1/s
Therefore, the Laplace transform of the constant function 1 is 1/s.
If f(x) = x, then we have:
L{f(x)} = L{x} = ∫_0^∞ e^(-st) x dt
Using integration by parts again, we get:
L{x} = ∫_0^∞ e^(-st) x dt = [(-e^(-st) x)/s]_0^∞ + (1/s) ∫_0^∞ e^(-st) dt
Since e^(-st) x approaches zero as t approaches infinity, the first term evaluates to zero. We can then simplify the second term using the result from part 2:
L{x} = (1/s) ∫_0^∞ e^(-st) dt = 1/s * (1/s) = 1/s^2
Therefore, the Laplace transform of the function f(x) = x is 1/s^2.
If f(x) = e^(ax), then we have:
L{f(x)} = L{e^(ax)} = ∫_0^∞ e^(-st) e^(ax) dt
Simplifying the integrand, we get:
L{e^(ax)} = ∫_0^∞ e^((a-s)t) dt
We can evaluate this integral using the formula:
∫_0^∞ e^(-bx) dx = 1/b
Setting b = a - s, we get:
L{e^(ax)} = ∫_0^∞ e^((a-s)t) dt = 1/(a-s)
Therefore, the Laplace transform of the function f(x) = e^(ax) is 1/(a-s).
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Helppp translation and reflection
The images of points B and C are B'(x, y) = (- 2, 6) and C'(x, y) = (- 1, 7), respectively.
How to compute the image of a point by translation
In this problem we find must determine the image of two points by translation, whose formula is introduced below:
T(x, y) = P'(x, y) - P(x, y)
Where:
P(x, y) - Original point.P'(x, y) - Resulting point.T(x, y) - Translation vector.First, determine the translation vector:
T(x, y) = (1, 4) - (0, 0)
T(x, y) = (1, 4)
Second, determine the images of points B and C:
B'(x, y) = (- 3, 2) + (1, 4)
B'(x, y) = (- 2, 6)
C'(x, y) = (- 2, 3) + (1, 4)
C'(x, y) = (- 1, 7)
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G(x)=5−2xg, left parenthesis, x, right parenthesis, equals, 5, minus, 2, x Determine for each x xx-value whether it is in the domain of g gg or not
The domain of function is the set of all possible values of x for which the function is defined. In this case, the function G(x) = 5 - 2x is defined for all real numbers of x.
The collection of all feasible input values (x-values) for which a function may be defined is known as the domain of function. In other words, it is the collection of all x values that may be passed into the function and provide a legitimate result (a y-value).
The function G(x) = 5 - 2x in this instance is a linear function, meaning it is defined for all real values of x. This is so that we may plug in any real value for x, and the function will output a real number for G(x) that corresponds. As an illustration, when we enter x = 0, we obtain G(0) = 5 - 2(0) = 5, and when we enter x = 2, we obtain G(2) = 5 - 2(2) = 1.
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