The difference in inches between 7 3/4 and 8 whole numbers is 1/4 inch.
What is Subtraction?
One of the four arithmetic operations, along with addition, multiplication, and division, is subtraction, which is denoted by the minus sign. The operation of subtraction represents the removal of items from a collection. In the adjacent image, for instance, there are 5 2 peaches, which means 5 peaches with 2 subtracted, making a total of 3 peaches.
As a result, 5 minus 2 equals 3, or the difference between 5 and 2. Although subtraction is most often used in mathematics with natural numbers, it can also be used with other types of objects, such as negative numbers, fractions, irrational numbers, vectors, decimals, functions, and matrices, to remove or decrease physical and abstract quantities.
The difference in inches between 7 3/4 and 8 whole numbers is 1/4 inch.
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In Exercises 1 and 2, compute each matrix sum or product if it is defined. If an expression is undefined, explain why.
Let A= [ 2 0 -1; 4 -3 2] B= [7 -5 1; 1 -4 -3] C=[1 2; -2 1] D= [3 5; -1 4] E=[ -5; 3]
1. -2A, B- 2A, AC, CD
The results of the matrix operations are:
-2A = [-4 0 2; -8 6 -4]B - 2A = [11 -5 -1; 9 -10 1]AC = undefinedCD = [1 13; -7 -6]To compute each matrix sum or product, we first need to determine if the operation is defined. Then, we can use the rules of matrix arithmetic to find the result.
-2A is defined, since we can multiply a matrix by a scalar. To find the result, we simply multiply each element of A by -2:
-2A = [-2(2) -2(0) -2(-1); -2(4) -2(-3) -2(2)] = [-4 0 2; -8 6 -4]
B - 2A is defined, since both matrices have the same dimensions. To find the result, we first compute -2A as shown above, then subtract each corresponding element of B and -2A:
B - 2A = [7 -5 1; 1 -4 -3] - [-4 0 2; -8 6 -4] = [7-(-4) -5-0 1-2; 1-(-8) -4-6 -3-(-4)] = [11 -5 -1; 9 -10 1]
AC is undefined, since the number of columns in A (3) does not match the number of rows in C (2).
CD is defined, since the number of columns in C (2) matches the number of rows in D (2). To find the result, we multiply each row of C by each column of D and sum the products:
CD = [(1)(3)+(2)(-1) (1)(5)+(2)(4); (-2)(3)+(1)(-1) (-2)(5)+(1)(4)] = [1 13; -7 -6]
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please help asap i will give brainliest
Answer:
perfect square therefore rational
non perfect square therefore irrational
cuberoot
5
Step-by-step explanation:
the topic is about perfect squares and rationality of numbers
determine the volume of the cone
[tex] \: [/tex]
_________
Given:-[tex] \tt \: Radius=9cm[/tex][tex] \: [/tex]
[tex] \tt \: Height = 12cm[/tex][tex] \: [/tex]
To find:-[tex] \tt \: volume \: of \: cone \: = \: ?[/tex][tex] \: [/tex]
By using formula:-[tex] \small{\star \boxed{ \tt \color{blue} volume \: of \: cone \: = \frac{1}{3}h\pi {r}^{2} }}[/tex]
[tex] \: [/tex]
Solution:-[tex] \tt \: v = \frac{1}{3}h\pi {r}^{2} [/tex][tex] \: [/tex]
[tex] \tt \: v = \frac{1}{3} \times 12 \times \pi \times ( {9})^{2} [/tex][tex] \: [/tex]
[tex] \tt \: v = \frac{1}{3} \times 12\pi \times 81[/tex][tex] \: [/tex]
[tex] \tt \: v = \frac{1}{ \cancel{3} } \times \cancel{972\pi}[/tex][tex] \: [/tex]
[tex] \underline{ \boxed{ \tt{ \color{hotpink} \: 324\pi \: {cm}^{3} }}}[/tex][tex] \: [/tex]
━━━━━━━━━━━━━━━━━━━━━━━━━
hope it helps! :)
simplify – 2b²(3b²–4)
Answer:
I believe the answer you are looking for is [tex]6b^{4}[/tex] - [tex]8b^{2}[/tex]
Step-by-step explanation:
Hope it helps
Sorry if I'm wrong
HELP PLSS!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Answer:
[tex]\frac{y + x}{y\y - x}[/tex]
Step-by-step explanation:
Helping in Jesus' name.
5. In A LMN, LM = 12, LN = 10. and angle L = 52° What is the length, to the nearest tenth of a unit, of MN?
7.4 units
15.6 units
9.8 units
14.4 units
The length of MN, to the nearest tenth of a unit is equal to: C. 14.4 units.
What is the law of cosine?In order to determine the missing side length of a geometric figure with the adjacent and hypotenuse side lengths given, you should apply the law of cosine:
C² = A² + B² - 2(A)(B)cosθ
Where:
A, B, and C is the length of side of a given triangle.
By substituting the given side lengths and angle into the law of cosine formula, we have the following;
MN² = LM² + LN² - 2(LM)(LN)cosθ
MN² = 12² + 10² - 2(12)(10)cos(52)
MN² = 144 + 100 - 147.76
MN = √96.24
MN = 9.8 units.
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NEED HELP DUE TODAY!!!!!!!!!!!!
Circle B is a dilation of circle A. What is the scale factor of dilation?
The scale factor of dilation is 3.
What is Dilation of a circle?
Dilation is the process of altering an object's or shape's size by reducing or enlarging its dimensions by a certain amount of scale. A circle with a radius of 10 units, for instance, is reduced to a circle with a radius of 5 units. This technique is applied in art and craft, photography, and logo design, among other fields.
Given : Radius of original circle A = 3
Radius of Dilation circle B = 9
We know that, Scale factor of dilation =
Dimension of Dilation shape / Dimension of original shape
So, Scale factor of given dilation = Radius of Circle B / Radius of Circle A
= 9/3
= 3.
Hence, Scale factor of dilation of circle B on Circle A is 3.
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A printing company charges $5,000 for the first 400 page printed. For each page printed over 400, the customer is charged an additional $0.01 per page which equation could be used to calculate the total cost, t, to copy p pages when p is greater than 400
If more than 400 sheets are duplicated, this equation will provide us with the overall expense, t.
What does a basic equation mean?A method that describes how two lines along both side of a sign connect each other. It typically contains an equal symbol and one variable. Similarly, 2x - 4 = 2. The variable x is present in the particular circumstance.
The equation that could be used to calculate the total cost, t, to copy p pages when p is greater than 400 is:
t = $5,000 + ($0.01 × (p - 400))
The base cost for copying the first 400 pages is $5,000. For each additional page above 400, the customer is charged $0.01 per page. Therefore, to calculate the total cost for copying p pages, we need to subtract 400 from p to determine the number of pages that are being charged at the additional rate, and then multiply that by $0.01 to determine the additional cost. This gives us:
Additional cost = $0.01 × (p - 400)
The total cost is then the sum of the base cost and the additional cost:
t = $5,000 + ($0.01 × (p - 400))
This equation will give us the total cost, t, for any number of pages greater than 400 that are copied.
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Find all the real solutions of the equation using the rational root theorem. x^(3)-2x^(2)-5x+6=0
All the real solutions of the equation are x=3, x=1, and x=-2. The rational root theorem states that any rational root of the equation x³-2x²-5x+6=0 can be written in the form p/q, where p is a factor of the constant term (6) and q is a factor of the leading coefficient (1). Therefore, the possible rational roots of the equation are: ±1, ±2, ±3, ±6
We can use synthetic division to test each of these possible roots until we find one that gives a remainder of 0.
1 | 1 -2 -5 6
| 1 -1 4
|_____________
| 1 -1 -6 10
The remainder is not 0, so 1 is not a root of the equation.
-1 | 1 -2 -5 6
| -1 3 2
|_____________
| 1 -3 -2 8
The remainder is not 0, so -1 is not a root of the equation.
2 | 1 -2 -5 6
| 2 0 10
|_____________
| 1 0 -5 16
The remainder is not 0, so 2 is not a root of the equation.
-2 | 1 -2 -5 6
| -2 8 14
|_____________
| 1 -4 3 20
The remainder is not 0, so -2 is not a root of the equation.
3 | 1 -2 -5 6
| 3 3 0
|_____________
| 1 1 -2 6
The remainder is 0, so 3 is a root of the equation. We can use the quotient (1x²+1x-2) to find the remaining roots of the equation.
1x²+1x-2=0
Using the quadratic formula, we can find the remaining roots:
x = (-1 ± √(1²-4(1)(-2)))/(2(1))
x = (-1 ± √(1+8))/2
x = (-1 ± √9)/2
x = (-1 ± 3)/2
x = 1 or x = -2
Therefore, the real solutions using rational root theorem and quadratic formula are of the equation are x=3, x=1, and x=-2.
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The cost of 4 shirts is $36. At this rate, what is the cost of 9 shirts?
Answer:
$81
Step-by-step explanation:
36/4 = 9
We can deduce that each shirt costs $9 so...
9x9 = 81
At Blue Coral Bay's annual beach clean-up, volunteers work together to pick up trash that gets washed ashore. This year, the volunteers averaged 1.5 more pounds of collected trash per person than last year. There were 80 volunteers this year, and together they removed a total of 360 pounds of trash from the beach.
The average amount of trash collected is found to be 3 pounds.
Explain about the equation?A formula that expresses the connection between two expressions on each side of a sign. Typically, it has a single variable and an equal sign. Like this: 2x - 4 = 12.Let the average amount of trash collected be 'x'.
As per the given conditions, equation form:
80*(1.5 + x ) = 360
Applying the Distributive Property:
120 + 80x = 360
Simplifying:
80x = 240
x = 240/80
x = 3
Thus, the average amount of trash collected is found to be 3 pounds.
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The complete question is attached.
The first term of geometric sequence is 7 and the common ratio is -3. Find the 7th term
The 7th term of the geometric sequence is 5,103.
To find the 7th term of a geometric sequence with a first term of 7 and a common ratio of -3, we can use the formula:
an = a1 × [tex]r^(n-1)[/tex]
where an is the nth term of the sequence, a1 is the first term of the sequence, r is the common ratio, and n is the number of the term we want to find.
Substituting the given values, we get:
a7 = 7 × [tex](-3)^(7-1)[/tex]
Simplifying the exponent, we get:
a7 = 7 × [tex](-3)^6[/tex]
Evaluating the exponent, we get:
a7 = 7 × 729
Multiplying, we get:
a7 = 5,103
Therefore, the 7th term of the geometric sequence is 5,103.
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A labour economist aims to estimate the variance of unemployed workers' mathematic test scores. Assume that a random sample of 18 scores had a sample standard deviation of 10.4.
Using the information above, form a 90% confidence interval for the population variance.
To have a full mark, you NEED to show your calculation and solution (with all the STEPS) in the midterm exam file posted under Assessment-Assignment (Similar to Weekly Quizzes).
Do NOT use EXCEL
a) The confidence interval is exactly between 46.64 and 202.07
b) The confidence interval is exactly between 26.64 and 182.07
c) The confidence interval is exactly between 86.64 and 222.07
d) The confidence interval is exactly between 66.64 and 212.07
e) None of the answers are correct
A labour economist aims to estimate the variance of unemployed workers' mathematic test scores. Assume that a random sample of 18 scores had a sample standard deviation of 10.4. To form a 90% confidence interval for the population varianceis The confidence interval is exactly between 46.64 and 202.07. The correct answer is option (a)
Find the degrees of freedom (df) by subtracting 1 from the sample size:
df = n - 1 = 18 - 1 = 17
Find the chi-square values for the 90% confidence interval using a chi-square table or calculator. The lower value corresponds to the 5th percentile (0.05) and the upper value corresponds to the 95th percentile (0.95):
χ² lower = 27.488
χ² upper = 8.672
Calculate the lower and upper limits of the confidence interval using the formula:
CI lower = (df * s²) / χ²_upper
CI upper = (df * s²) / χ²_lower
Where s² is the sample variance (the square of the sample standard deviation):
s² = 10.4² = 108.16
Plugging in the values:
CI lower = (17 * 108.16) / 27.488 = 46.64
CI upper = (17 * 108.16) / 8.672 = 202.07
The 90% confidence interval for the population variance is between the lower and upper limits:
CI = (46.64, 202.07)
Therefore, the correct answer is option (a) The confidence interval is exactly between 46.64 and 202.07.
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27. The rodent population p in a large city is being controlled by a new poison that kills half the population every 2 months m. If there are currently 1,000,000 rodents in the city, how many will there be in 12 months?
Step-by-step explanation:
Since the new poison kills half the population every 2 months, we can say that the remaining half will survive for another 2 months. Therefore, after 2 months, the rodent population will be half of 1,000,000, which is 500,000.
After another 2 months, the remaining half of the 500,000 will survive, which is 250,000.
After another 2 months, the remaining half of the 250,000 will survive, which is 125,000.
After 6 months, the rodent population will be 125,000.
Since 12 months is six 2-month periods, we can repeat this process again. After another 2 months, the rodent population will be 62,500.
After another 2 months, the rodent population will be 31,250.
After another 2 months, the rodent population will be 15,625.
Therefore, after 12 months, the rodent population will be 15,625.
HELP ME IM GIVING 16 BRAINLIEST IF HELP
Answer:
277 milliliters
Step-by-step explanation:
do 945-668
this gives you 277
Answer:
The answer to your problem is, 277 milliliters.
Step-by-step explanation:
The reason I say that is because our problem that you are is a subtraction problem, so.. :
945 - 668 = 277.
Then we just add milliliters.
Thus the answer to your problem is, 277 milliliters.
ACTIVITY (EXPERIMENTAL PROBABILITY) 50 trials
The dice i roll: 2, 3, 4, 4, 4, 4, 6, 5, 2, 4, 2
What is the probability of rolling a sum of even number?
What is the probability of rolling a sum of odd number?
What is the probability of rolling a sum of greater than 5?
This is for statistics ok? Ty!U,,w,,U
The probability of rolling a sum greater than 5 is 13/18.
What is the probability?Probability can be defined as the ratio of the number of favourable outcomes to the total number of outcomes of an event.
We know that, probability of an event = Number of favourable outcomes/Total number of outcomes
There are six different possible outcomes for a dice, the set (S) of all the outcomes can be listed as follows:
(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
a) The probability of rolling a sum of even number
Number of favorable outcomes = 9
Total number of outcomes = 36
Now, probability = 9/36
= 1/4
b) The probability of rolling a sum of odd numbers
Number of favorable outcomes = 9
Total number of outcomes = 36
Now, probability = 9/36
= 1/4
c) The probability of rolling a sum of greater than 5
Number of favorable outcomes = 26
Total number of outcomes = 36
Now, probability = 26/36
= 13/18
Therefore, the probability of rolling a sum greater than 5 is 13/18.
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I’ll mark brainliest
Answer:
θ = 78.69°
Step-by-step explanation:
As this is a right angle triangle we use trigonometric ratios/
The perpendicular i.e the height of the mast = 50 ft
The base i.e the shadow = 10 ft
tanθ = perpendicular / base
tanθ = 50/10
θ = [tex]tan^{-1}[/tex] (50/10)
θ = 78.69°
A tree three-foot tall grows 8% every five years. How tall will the tree be at the
end of 90 years?
Answer:
21.6 feet?
Step-by-step explanation:
8 percent of 3 is 0.24
0.24 x 90 is 21.6
A varies directly as B and inversely as C . A is 12 when B is 6 and C is 2 . What is the value of A when B is 12 and C is 3 .
O 4 O 8 O 12
O 16
If A varies directly as B and inversely as C then The value of A when B is 12 and C is 3 is 16.
To find the value of A, we can use the formula for direct and inverse variation of the given :
A = k*B/C
We are given that A is 12 when B is 6 and C is 2. We can plug these values into the formula to find the value of k:
12 = k*6/2
12 = 3k
k = 4
Now that we know the value of k, we can plug in the new values for B and C to find the value of A:
A = 4*12/3
A = 48/3
A = 16
Therefore, the value of A when B is 12 and C is 3 is 16. The correct answer is O 16.
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Question 6(Multiple Choice Worth 2 points)
(Volume of Cylinders MC)
A helium tank shaped like a cylinder has 2,401π in3 of space inside the tank. If the diameter of the helium tank is 14 inches, what is its height?
49 inches
28 inches
12.25 inches
7 inches
The height of the helium tank is 49 inches.
What is Volume?Volume refers to the amount of space occupied by a three-dimensional object, measured in cubic units.
The volume V of a cylinder is given by the formula:
V = πr²h
where r is the radius of the base and h is the height of the cylinder.
Since the diameter of the helium tank is given as 14 inches, the radius r is half of the diameter, which is 7 inches.
We also know that the volume inside the tank is 2,401π cubic inches. Therefore, we can write:
2,401π = π(7)²h
Simplifying the right-hand side, we get:
2,401π = 49πh
Dividing both sides by 49π, we get:
h = 2,401π / 49π
Simplifying, we get:
h = 49
Therefore, the height of the helium tank is 49 inches.
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Answer:
49
Step-by-step explanation:
I took the test and got it right :)
Can someone please help me understand this problem? Solve using
the quadratic formula: y + 1/y = 13/6
The solutions to this problem are y = 5 and y = -1/3.
Sure! The quadratic formula is a useful tool for solving equations of the form ax^2 + bx + c = 0. In this problem, we need to rearrange the equation so that it fits this form before we can use the quadratic formula.
First, let's multiply both sides of the equation by y to get rid of the fraction:
y^2 + 1 = 13y/6
Next, let's rearrange the equation so that it equals zero:
y^2 - 13y/6 - 1 = 0
Now we can use the quadratic formula to solve for y. The quadratic formula is:
y = (-b ± √(b^2 - 4ac))/(2a)
In this problem, a = 1, b = -13/6, and c = -1. Plugging these values into the quadratic formula gives us:
y = (-(-13/6) ± √((-13/6)^2 - 4(1)(-1)))/(2(1))
Simplifying the equation gives us:
y = (13/6 ± √(169/36 + 4))/2
Finally, we can solve for y by simplifying the square root and dividing by 2:
y = (13/6 ± √(289/36))/2
y = (13/6 ± 17/6)/2
y = (30/6)/2 or y = (-4/6)/2
y = 5 or y = -1/3
So the solutions to this problem are y = 5 and y = -1/3. I hope this helps! Let me know if you have any further questions.
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Julie and Barry Spinos purchased a house for $96,400. They made a 25 percent down payment and financed the remaining amount at 5. 50 percent for 30 years. Their monthly payment is $410. 66. How much of the first monthly payment is used to reduce the principal?
The first monthly charge has about $78.78 allotted in the direction of reducing the principal.
The total buy charge of the house is $96,400 and the Spinoses made a 25% down payment, because of this they paid $96,400 x 0.25 = $24,100 in advance.
Therefore, the final quantity that they financed is $96,400 - $24,100 = $72,300.
They financed this amount at 5.50% for 30 years, which gives us the following method for calculating the monthly payment (P):
P = (r * PV) / (1 - (1 + r)^(-n))
in which:
r = month-to-month interest rate PV = present value n = overall range of paymentsSubstituting the values, we get:
P = (0.00458 * 72,300) / (1 - (1 + 0.00458)^(-360))
P ≈ $410.66
We recognise that the monthly payment is $410.66 and we will calculate the interest portion of the primary monthly price as follows:
interest = balance * monthly interest price
interest = $72,300 * (5.50% / 12) ≈ $331.88
To calculate the amount of the primary monthly charge this is used to lessen the fundamental, we subtract the interest component from the total monthly payment:
$410.66 - $331.88 ≈ $78.78
Thus, the first monthly charge has about $78.78 allotted in the direction of reducing the principal.
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The table represents a linear equation. answers?
Option (c) has the same equation as above after simplification, so the answer is (c):y-8=-0.2(x+10).
What is simplification?Simplification involves reducing an expression, equation, or problem to a simpler form. This can involve performing operations such as combining like terms, factoring, or canceling out common factors.
What is point-slope form?Point-slope form is a way of writing the equation of a straight line in two-dimensional space, using a given point on the line and the line's slope.
The point-slope form equation for a line passing through a point (x1, y1) with slope m is:y - y1 = m(x - x1).
In the given question,
To write the equation of a line in point-slope form, we need to use the formula:
y - y1 = m(x - x1)
where m is the slope of the line and (x1, y1) is any point on the line.
We are given two points on the line: (-10, 8) and (10, 4). We can use these points to find the slope of the line:
m = (y2 - y1) / (x2 - x1)
= (4 - 8) / (10 - (-10))
= -0.15
Now we can use the point-slope form with the point (-10, 8) and slope -0.15:
y - 8 = -0.15(x - (-10))
Simplifying this equation, we get:
y - 8 = -0.15(x + 10)
Option (c) has the same equation as above after simplification, so the answer is (c):
y-8=-0.2(x+10).
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Does someone mind helping me with this question? Thank you!
Answer:
33361
Step-by-step explanation:
The equation appears to be
75 ( 1 + 0.84 )^x
Since it increases by 84% every 2 days, you'd devide 20 by 2, making 10
So to do the equation you'd shove that into a calculator since no one wants to hand do that sort of equation.
75 ( 1 + 0.84 )^10 = 33361.0332844 = 33361
Use the given feasible region to determine the maximum and minimum values of the objective function
P = 50x + 75y,
if they exist. (If an answer does not exist, enter DNE.)
The xy-coordinate plane is given. A 4 sided figure labeled S is located in the first quadrant. The five corner points of the figure are as follows:
(0, 2), (1, 9), (8, 4), and (3, 0)
Determine the minimum value.
Determine where the minimum value occurs.
The minimum value of P occurs at the corner point (8, 4).The minimum value of P occurs at all points on the line segment connecting (0, 2), (3, 0).The minimum value of P occurs at all points on the line segment connecting (0, 2), (1, 9).The minimum value of P occurs at the corner point (3, 0).The minimum value does not exist.
Determine the maximum value.
Determine where the maximum value occurs.
The maximum value of P occurs at all points on the line segment connecting (3, 0), (8, 4).The maximum value of P occurs at the corner point (8, 4).The maximum value of P occurs at all points on the line segment connecting (1, 9), (8, 4).The maximum value of P occurs at the corner point (1, 9).The maximum value does not exist.
The corner point (8, 4) and is equal to 550
The minimum value of P = 50x + 75y exists at the corner point (3, 0) and is equal to 150. The maximum value of P = 50x + 75y exists at the corner point (8, 4) and is equal to 550.
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NEED HELP PLEASE. Graph
The graph of the functions are added as an attachment
How to graph the function using their periodsThe graphs are sinusoidal functions such that they are mathematical functions that exhibit a repetitive pattern over time or space
From the question, we have the following parameters that can be used in our computation:
y = 4cot(4x)
y = -1/2cot(2x - π/4)
To plot y = 4cot(4x) using two periods, we make use of the domain
{0 < x < π/2}
To plot y = 4cot(4x) using two periods, we make use of the domain
{0 < x < π}
See attachement for the graphs of the functions
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Draw the image of the indicated translation of the given pre image please
The new points after the translation are (0, -4), (0, 0), (5, -5) and (5, 0) and the image is added as attachment
Calculating the image of the figureTranslation is the movement of a point either up, left, right or down on the coordinate plane.
If a point A(x, y) is translated a units left and b units down, the new point is A'(x - a, y - b)
From the question, the vertices of the rectangle are (2, 3), (2, 7), (7, 2) and (7, 7).
The translation T<-2, -7>; is a translation of 2 units left and 7 units down.
This means that
The new points are (0, -4), (0, 0), (5, -5) and (5, 0).
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Rationalize the denominator: (a)52+3510(b)33x+3y3xy
For (a): 52 + 35/10m, rationalizing the denominator gives the result (25√2 + 6√5)/10.
For (b): 33x + 3y/3xy, rationalizing the denominator gives the result √3(x+y)/xy.
To rationalize the denominator, we need to multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of a binomial expression is the same expression with the sign between the terms changed. For example, the conjugate of (a+b) is (a-b) and the conjugate of (a-b) is (a+b). By multiplying by the conjugate, we eliminate any radicals in the denominator.
(a) 5/√2 + 3/√5
= (5/√2)(√2/√2) + (3/√5)(√5/√5)
= (5√2)/2 + (3√5)/5
= (25√2 + 6√5)/10
(b) (3√3x + 3√y)/(3√xy)
= (3√3x + 3√y)(3√xy)/(3√xy)(3√xy)
= (9x√3xy + 9y√3xy)/(9xy)
= (9√3xy(x+y))/(9xy)
= √3(x+y)/xy
Therefore, the rationalized denominators are (a) (25√2 + 6√5)/10 and (b) √3(x+y)/xy.
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In a city there are 500 households.
Houses have house numbers ranging from 1-500. This city has organized
activity coming up by bringing a signboard that has 2 sides One side is red, the other side is blue. to stick to the front
door of each house and will give the number of residents 500 people go back to the signboard that will be in the area in front of each house's door, respectively, by the occupants Residents can turn over the sign only for the sign of the house. that their entry order can divide the number of the house
Yes, after the residents have returned all the nameplates. 500 people, who guessed the number of blue plates
All correct will be rewarded. ask when the end How many blue tiles will there be at the end of the event? At the beginning, all plates were red.
There will be 22 blue signs at the end of the event.
Describe Probability?The probability of an event can be determined by dividing the number of favorable outcomes by the total number of possible outcomes. For example, the probability of rolling a six on a standard six-sided die is 1/6, as there is only one favorable outcome (rolling a six) out of six possible outcomes (rolling a one, two, three, four, five, or six).
To solve this problem, we need to consider the factors that determine whether a house's sign will be flipped or not. A sign will only be flipped if the number of residents can divide the house number. For example, the sign on house number 12 will be flipped by the residents of that house because 12 is divisible by 2, 3, 4, and 6 (the factors of 12).
At the beginning, all signs are red, which means that none of the houses have had their sign flipped yet. Let's start with house number 1 and consider which houses will flip its sign.
For house number 1, its sign will not be flipped because 1 is only divisible by 1.
For house number 2, its sign will be flipped because 2 is divisible by 2.
For house number 3, its sign will not be flipped because 3 is only divisible by 1 and 3.
For house number 4, its sign will be flipped because 4 is divisible by 2 and 4.
For house number 5, its sign will not be flipped because 5 is only divisible by 1 and 5.
And so on, continuing this process for all 500 houses.
We can see that the signs on the houses with numbers that have an odd number of factors will end up blue, and the signs on the houses with numbers that have an even number of factors will end up red. This is because a factor always comes in pairs, except when the number is a perfect square, in which case the factor pair is identical (e.g. 4 has factors 1, 2, and 4, which is an odd number of factors).
So we need to count the number of integers from 1 to 500 that have an odd number of factors. These are the perfect squares, which are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, and 484.
Therefore, there will be 22 blue signs at the end of the event.
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You have $39,000 you would like to invest in two different stocks, Stock Boll and Stock Coff. The buying limit on Stock Coff is $12,900 and you want to spend at least twice as much money on Boll as Coff. If Stock Boll earns 5% annual interest and Stock Coff earns 8% annual interest, how much money should you invest in each stock to maximize your annual interest earned? What is the maximum annual interest?
You should invest $26,100 in Stock Boll and $12,900 in Stock Coff to maximize your annual interest earned, and the maximum annual interest is $2,337.
To maximize your annual interest earned, you should invest as much money as possible in the stock with the higher interest rate, which is Stock Coff. However, there are two constraints that you need to consider: the buying limit on Stock Coff and the requirement to spend at least twice as much money on Stock Boll as Stock Coff.
Let's use algebra to solve this problem. Let x be the amount of money you invest in Stock Boll and y be the amount of money you invest in Stock Coff. Then we have the following equations:
x + y = $39,000 (the total amount of money you have to invest)
y <= $12,900 (the buying limit on Stock Coff)
x >= 2y (the requirement to spend at least twice as much money on Stock Boll as Stock Coff)
To maximize your annual interest earned, you want to maximize the expression 0.05x + 0.08y (the sum of the interest earned from Stock Boll and Stock Coff).
If you invest the maximum amount of money in Stock Coff ($12,900), then you have $39,000 - $12,900 = $26,100 left to invest in Stock Boll. This satisfies the requirement to spend at least twice as much money on Stock Boll as Stock Coff ($26,100 >= 2 * $12,900).
So the optimal solution is to invest $26,100 in Stock Boll and $12,900 in Stock Coff. The maximum annual interest earned is 0.05 * $26,100 + 0.08 * $12,900 = $1,305 + $1,032 = $2,337.
Therefore, you should invest $26,100 in Stock Boll and $12,900 in Stock Coff to maximize your annual interest earned, and the maximum annual interest is $2,337.
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