Answers
Answer:
a) 5x/16
b) x = 32
Step-by-step explanation:
a) x/8 + 3x/16 can be simplified by getting a common denominator. x/8 can be changed like this:
x/8 × 2/2 is 2x/16
now we have:
2x/16 + 3x/16
= 5x/16
b) solve x/8+3x/16=10
change the left side of the equation as in part a.
5x/16 = 10
multiply both sides of the equation by 16
5x = 160
divide by 5
x = 32
Related Questions
Is each number rounded correctly to the nearest hundred thousand?
Answers
yes is answer for all option .we can check it by rules of rounding off numbers .
what is rounding ?
Rounding is the process of approximating a number to a nearby value that is easier to work with or more appropriate for a given context. When rounding, we take a number with many decimal places or significant figures and adjust it to a simpler or more convenient value with fewer decimal places or significant figures.
In the given question,
Yes, each number is rounded correctly to the nearest hundred thousand based on the rules of rounding.
To round to the nearest hundred thousand, we look at the digit in the hundred thousand place and the digit to its right (i.e., in the ten thousand place).
If the digit in the ten thousand place is 5 or greater, we round up the digit in the hundred thousand place by adding 1.
If the digit in the ten thousand place is less than 5, we leave the digit in the hundred thousand place as it is.
Using these rules, we can see that:
350000 rounded to the nearest hundred thousand is 400000 because the digit in the ten thousand place is 5, so we round up the digit in the hundred thousand place.
555555 rounded to the nearest hundred thousand is 560000 because the digit in the ten thousand place is 5, so we round up the digit in the hundred thousand place.
137998 rounded to the nearest hundred thousand is 200000 because the digit in the ten thousand place is 7, so we round up the digit in the hundred thousand place.
792314 rounded to the nearest hundred thousand is 800000 because the digit in the ten thousand place is 3, so we leave the digit in the hundred thousand place as it is.
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positive.
an remove
3√5a-8√/35a2
Answers
3√(5a) - 8√(a) / 35a^2
To simplify this expression, we can first separate the two terms in the numerator:
3√(5a) / 35a^2 - 8√(a) / 35a^2
We can then simplify each term separately. For the first term:
3√(5a) / 35a^2 = √(5a) / (35a^2/3)
We can simplify the denominator by using the rule that (a^m)^n = a^(mn):
35a^2/3 = (5a^2/3) * 7 = (a^(2/3))^5 * 7
So the first term simplifies to:
√(5a) / (a^(2/3))^5 * 7
For the second term:
8√(a) / 35a^2 = 8a^(1/2) / (35a^2)
We can simplify the denominator by using the rule that a^-m = 1/a^m:
35a^2 = 5 * 7 * a^2 = a^-2 * 5 * 7
So the second term simplifies to:
8a^(1/2) / (a^-2 * 5 * 7)
Now we can combine the two terms by finding a common denominator:
√(5a) / (a^(2/3))^5 * 7 - 8a^(1/2) / (a^-2 * 5 * 7)
The common denominator is (a^(2/3))^5 * 5 * 7:
√(5a) * 5 * 7 / (a^(2/3))^5 * 5 * 7 - 8a^(1/2) * (a^(2/3))^5 * 5 * 7 / (a^-2 * 5 * 7) * (a^(2/3))^5 * 5 * 7
Simplifying each term, we get:
35√(5a) / 5a^2 - 40a^(5/2) / 5a^3
Now we can simplify further by factoring out a common factor of 5a^(5/2):
5a^(5/2) * (7√(5a) - 8) / 5a^3
The 5's cancel out, and we can remove the common factor of 5a^(5/2) to get:
(7√(5a) - 8) / a^(3/2)
So the simplified expression is:
(7√(5a) - 8) / a^(3/2)
taylor has enough money to buy either 90 granola bars or 78 pop tarts. After returning from the atore, taylor has no money, 75 granola bars and p pop tarts
Answers
Using equations, we can find that the number of pop tarts that Taylor has with him, p = 13.
Define equations?An equation is a mathematical statement that contains the symbol "equal to" between two expressions with identical values. as in 3x + 5 = 15, for example. There are many different types of equations, including linear, quadratic, cubic, and others.
Let the cost of one granola bar = x.
Let the cost of one pop tart = y.
Now, assume Taylor had money, m.
So, as per the question:
90x = m
⇒ x = m/90
78y = m
⇒ y = m/78
75x + py = m
Substituting the values of x and y:
⇒ 75 × m/90 + p × m/78 = m
⇒ 75m/90 + pm/78 = m
⇒ (5850m + 90pm) / 7020 = m
⇒ 5850m + 90pm = 7020m
⇒ 90pm = 7020m - 5850m
⇒ 90pm = 1170m
⇒ p = 1170m/90m
⇒ p = 13.
Therefore, the number of pop tarts, p = 13.
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A rectangle has an area of 108 square
centimeters. Its width is 9 centimeters.
What is the perimeter of the
rectangle?
Answers
Therefore, the perimeter of the rectangle is 42 centimeters.
What is area?The concept of area is used in many areas of mathematics, science, and everyday life. It is used in geometry to calculate the area of various shapes, such as triangles, circles, and polygons. It is also used in physics to calculate the amount of surface area of an object that is exposed to air or water, and in architecture and engineering to determine the amount of material needed to construct a building or structure.
Here,
To find the perimeter of a rectangle, we need to know its length and width. We are given that the width of the rectangle is 9 centimeters, and the area of the rectangle is 108 square centimeters.
We can use the formula for the area of a rectangle:
Area of rectangle = length x width
Plugging in the values we have:
108cm² = length x 9cm
Solving for the length, we can divide both sides by 9cm:
length = 108cm² / 9cm
length = 12cm
So, the length of the rectangle is 12 centimeters.
To find the perimeter of the rectangle, we can use the formula:
Perimeter of rectangle = 2 x (length + width)
Plugging in the values we have:
Perimeter of rectangle = 2 x (12cm + 9cm)
Perimeter of rectangle = 2 x 21cm
Perimeter of rectangle = 42cm
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Clare lives in Iowa and pays 6% in sales tax. She just bought $135 in groceries, but $40 worth of those groceries were not taxable. What is the total amount that Clare paid for the groceries, including sales tax?
Answers
Answer:
$140.70
Step-by-step explanation:
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Determine if the function below is continuous.
A. not continuous at x = 5
B. not continuous at x = 0
C. continuous
D. not continuous x = -2
Answers
Is the function above continuous: B. not continuous at x = 0.
What is a continuous function?In Mathematics and Geometry, a continuous function can be defined as a type of function in which there is no discontinuities or breaks between the intervals for the points plotted on a graph.
What is a discrete function?In Mathematics and Geometry, a discrete function can be defined as a type of function in which the ordered pair of values on the x-axis and y-axis are separate from each other and unconnected.
By critically observing the graph shown above, we can reasonably infer and logically conclude that it represents a discrete function because it is not continuous at x is equal to 0 i.e x = 0.
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PLEASE HELP DUE TODAY ASAP
im pretty sure the answer is B Please tell me if im correct ASAP!
Question 3(Multiple Choice Worth 1 points)
(08.07 LC)
The function g(x) is shown on the graph.
The graph shows an upward opening parabola with a vertex at negative 4 comma 3, a point at negative 6 comma 7, and a point at negative 2 comma 7.
What is the equation of g(x) in vertex form?
g(x) = (x − 4)2 − 3
g(x) = (x − 4)2 + 3
g(x) = (x + 4)2 − 3
g(x) = (x + 4)2 + 3
Answers
Step-by-step explanation:
Vertex at -4,3 would result in vertex form of parabola
g(x) = (x+4)^2 + 3
Miss Wu's class made 36
paper flowers. One half of the
flowers were red. One third of
the flowers were yellow. The
rest were blue. What fraction
of the flowers were blue?
Answers
Answer:
1/6 of flowers were blue
Round your answer to the nearest tenth of a degree
Answers
Answer:
57.8°
Step-by-step explanation:
SohCahToa
use sin^1 to get angle
sin^1(11/13)=57.7957725
to the nearest tenth=57.8°
what is 15 to the power of 12
Answers
Answer:
1.2974634e+14
Step-by-step explanation:
ye
The count in a bateria culture was initially 300, and after 35 minutes the population had increased to 1600. Find the doubling period. Find the population after 70 minutes. When will the population reach 10000?
Answers
doubling time = (ln 2) / r
where r is the exponential growth rate.
Using the given information, we can calculate the exponential growth rate as:
r = (ln N1 - ln N0) / t
where N0 is the initial population, N1 is the final population, and t is the time elapsed. Plugging in the values, we get:
r = (ln 1600 - ln 300) / 35
r = 0.5128
Now we can calculate the doubling period as:
doubling time = (ln 2) / r
doubling time = (ln 2) / 0.5128
doubling time = 1.35 hours (rounded to two decimal places)
Therefore, the doubling period is approximately 1.35 hours.
To find the population after 70 minutes, we can use the formula for exponential growth:
N = N0 * e^(rt)
Plugging in the values, we get:
N = 300 * e^(0.5128 * (70/60))
N = 1467.05
Therefore, the population after 70 minutes is approximately 1467.05.
To find when the population will reach 10000, we can use the same formula again:
N = N0 * e^(rt)
Plugging in the given values, we get:
10000 = 300 * e^(0.5128 * t)
Dividing both sides by 300, we get:
e^(0.5128 * t) = 10000 / 300
e^(0.5128 * t) = 33.3333
Taking the natural logarithm of both sides, we get:
0.5128 * t = ln(33.3333)
t = ln(33.3333) / 0.5128
t = 23.37 hours (rounded to two decimal places)
Therefore, the population will reach 10000 after approximately 23.37 hours.
You take out a loan in the amount of your tuition and fees cost $70,000. The loan has a monthly interest rate of 0.25% and a monthly payment of $250. How long will it take you to pay off the loan? Use the formula N= (-log(1-i*A/P))/(log(1+i)) to determine the number of months it will take you to pay off the loan. Let N represent the number of monthly payments that will need to be made, i represent the interest rate in decimal form, A represent the amount owed (total amount of the loan), and P represent the amount of your monthly payment. Be sure to show your work for all calculations made.
Answers
Therefore, it will take 173 months to pay off the loan, or approximately 14 years and 5 months.
What is percentage?A percentage is a way of expressing a number as a fraction of 100. The symbol for a percentage is "%". For example, 50% is the same as 50/100 or 0.5 as a decimal. Percentages are often used to express a portion or share of a whole. For instance, if you scored 90% on a test, it means you got 90 out of 100 possible points. In finance, percentages are commonly used to express interest rates, returns on investments, or changes in stock prices.
First, we need to convert the monthly interest rate from a percentage to a decimal by dividing by 100.
0.25% / 100 = 0.0025
Now we can plug in the values into the formula:
N= (-log (1-0.0025*70000/250))/ (log (1+0.0025))
Simplifying the equation in the parentheses:
N= (-log (1-175))/ (log (1.0025))
N= (-log (0.9964))/ (0.002499)
N= 172.9
Rounding up to the nearest whole number since we can't make partial payments:
N= 173
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The length of the altitude of an equilateral triangle is 4 square root of 3 . Find the length of a side of the triangle.
Answers
[tex]\textit{height of an equilateral triangle}\\\\ h=\cfrac{s\sqrt{3}}{2}~~ \begin{cases} s=\stackrel{length~of}{a~side}\\[-0.5em] \hrulefill\\ h=4\sqrt{3} \end{cases}\implies 4\sqrt{3}=\cfrac{s\sqrt{3}}{2} \\\\\\ 8\sqrt{3}=s\sqrt{3}\implies \cfrac{8\sqrt{3}}{\sqrt{3}}=s\implies 8=s[/tex]
Find the length of the triangle.
The length of the unknown side of the triangle is __________
Answers
Answer:
The answer is 2√10
Step-by-step explanation:
Hyp²=opp²+adj²
let hyp be x
x²=6²+2²
x²=36+4
x²=40
square root both sides
√x²=√40
x=2√10
What kind of angle is
∠mah?
What is the value of x?
What is the measure of
∠mat ?
What is the measure of
∠tah ?
Answers
So according to the given figure ∠mah is right angle(90°), The value of x is 17, ∠mat is 38° and ∠tah is equal to 52°.
What is the source of this answer and define a graph?(A) ∠mah is right angle(90°) [GIVEN IN THE FIGURE]
(B) (2x+4)+(3x+1) = 90
5x+5 = 90
x= 17
(C) 2x+4 = 2×17 + 4 = 38°
(D) 3X + 1 = 3 × 17 + 1 = 52°
A graph is a visual representation or diagram that displays facts or values in an organised manner in mathematics. The points on a graph are typically used to depict the relationships between two or more things.
A graph in discrete mathematics is made up of vertices, which are groups of points, and edges, which are the connections between those vertices. There are numerous different types of graphs besides linked and disconnected graphs, weighted graphs, bipartite graphs, directed and undirected graphs, and simple graphs.
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What is a solution to the system of equations that includes quadratic function f(x) and linear function g(x)?
(x is -2,-1,0,1,2 and g(x) is 1,3,5,7,9?
f(x) = 2x^2 + x + 4
Answers
We see that f(-1) = g(1) = 5. Therefore, the solution to the system of equations is x = -1.
Describe System of equation?A system of equations is a set of two or more equations that are to be solved simultaneously. Each equation in the system represents a relationship between variables, and the solution to the system is the set of values for the variables that satisfy all of the equations in the system.
One way to solve a system of equations is by substitution. We can solve one of the equations for one of the variables in terms of the other variable, and then substitute that expression into the other equation to get an equation in one variable. We can then solve that equation for the variable, and use that value to find the value of the other variable.
Another way to solve a system of equations is by elimination. We can add or subtract the equations in the system to eliminate one of the variables, and then solve for the other variable. We can then use that value to find the value of the eliminated variable.
To find a solution to the system of equations that includes the quadratic function f(x) and the linear function g(x), we need to find the value of f(x) for each value of x and compare it to the corresponding value of g(x). If f(x) and g(x) are equal for any value of x, then that value is a solution to the system.
Substituting each value of x into the quadratic function f(x), we get:
f(-2) = 2(-2)² + (-2) + 4 = 10
f(-1) = 2(-1)² + (-1) + 4 = 5
f(0) = 2(0)² + 0 + 4 = 4
f(1) = 2(1)² + 1 + 4 = 7
f(2) = 2(2)² + 2 + 4 = 14
Comparing these values to the corresponding values of g(x), we see that f(-1) = g(1) = 5. Therefore, the solution to the system of equations is x = -1.
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the 3rd and 6th term in fibonacci sequence are 7 and 31 respectively find the 1st and 2nd terms of the sequence
Answers
Answer:
Step-by-step explanation:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,144,233,377,610,987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, ... Can you figure out the next few numbers?
Find tan(Q).
A 1
3 9
"5
135°
P
para os a
was a s
Save and Exit
Next
Submit
Answers
Answer:
tanQ = 1
Step-by-step explanation:
the exterior angle of a triangle is equal to the sum of the 2 opposite interior angles.
∠ OPQ is an exterior angle of the triangle , then
∠ R + ∠ Q = 135°
90° + ∠ Q = 135° ( subtract 90° from both sides )
∠ Q = 45°
Then
tanQ = tan45° = 1
please help me with this pentomino question.
Answers
a. There are twelve unique pentominos possible in all.
b. In total, only 6 pentominos have the refection symmetry.
What is a pentomino?
A pentomino is an order 5 polyomino, or a plane polygon formed of 5 identical squares joined edge to edge. This is just linked with a certain arrangement of squares.
a. There are twelve unique pentominos possible in all which on reflection and rotation can create multiple more pentominos.
b. In total, 6 pentominos have refection symmetry which means that now the total pentominos are 18 of which 6 have reflection symmetry.
Hence, the required solution has been obtained.
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A conic has vertices at (0,8) and (0,-8) and foci at F(0, √48) and F(0,-√48). (a) Write the standard form of the equation of this conic.
Answers
The standard form of the equation of the conic is x² + y² - 16x - 16y - 192 = 0.
What is conic?It is a type of curve with a curved line in the middle and two straight lines on the sides. Conic sections include circles, ellipses, parabolas, and hyperbolas.
The standard form of the equation of a conic is
Ax² + Bxy + Cy² + Dx + Ey + F = 0, where A, B, C, D, E, and F are constants.
In this case, the equation of the conic is x² + y² - 16x - 16y - 192 = 0.
To derive this equation, we'll use the distance formula to calculate the distance between the vertices and the foci. The distance between the vertices and the foci is given by
d = √((x_1 - x_2)² + (y_1 - y_2)²)
where (x_1, y_1) is the coordinates of the first vertex and (x_2, y_2) is the coordinates of the first foci.
For the first vertex, (x_1, y_1) = (0, 8). For the first foci, (x_2, y_2) = (0, √48). Thus, the distance between the two points is
d = √((0 - 0)² + (8 - √48)² )
= √(0 + (-8 - √48)² )
= √(-16 - 2√48 + 48)
= √(32 -2√48)
= 2√(16 - √48)
= 2√(16 - 6.9)
= 2√(9)
= 2(3)
= 6
So, the distance between the vertices and the foci is 6.
The equation of the conic can then be written as x² + y² - 16x - 16y - (-8)² = 0, which simplifies to x² + y² - 16x - 16y - 192 = 0.
Thus, the standard form of the equation of the conic is x² + y² - 16x - 16y - 192 = 0.
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The reflector of a flashlight is in the shape of a paraboloid of revolution. Its diameter is 8 centimeters and its depth is 4 centimeters. How far from the vertex should the light bulb be placed so that the rays will be reflected parallel to the axis?
Answers
Answe: The distance of light bulb can be calculated using equation of parabola. The parabola is a plane curve which is U-shaped.
Step-by-step explanation:
Given F(x) = 4x - 8 and g(x) = -3x + 1, what is (f - g)(x)?
A) 7x-9
B) 7x - 7
C) x-9
D) x - 7
Answers
Therefore, the answer is (A) 7x-9 when it is given that function F(x) = 4x - 8 and g(x) = -3x + 1.
What is function?In mathematics, a function is a relationship between two sets of values, where each input (or domain element) is associated with a unique output (or range element). In other words, a function is a rule or a process that takes an input (or inputs) and produces a corresponding output. Functions can be expressed using various mathematical notations, such as algebraic formulas, graphs, tables, or even verbal descriptions. They are widely used in many fields of mathematics, science, engineering, economics, and computer science, to model and solve problems that involve relationships between variables or quantities.
Here,
To find (f - g)(x), we need to subtract g(x) from f(x), so we get:
(f - g)(x) = f(x) - g(x)
Substituting the given functions, we get:
(f - g)(x) = (4x - 8) - (-3x + 1)
Simplifying the expression by distributing the negative sign, we get:
(f - g)(x) = 4x - 8 + 3x - 1
Combining like terms, we get:
(f - g)(x) = 7x - 9
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Aeronautical researchers have developed three different processes to pack a parachute. They want to compare the different processes in terms of time to deploy and reliability. There are 1,200 objects that they can drop with a parachute from a plane. Using a table of random digits, the researchers will randomly place the 1,200 items into three equally sized treatment groups suitable for comparison. Which design is the most appropriate for this experiment
- Randomly number each item with 1, 2, or 3. Assign the items labeled 1 to the process 1 group, assign the items labeled 2 to the process 2 group, and assign the items labeled 3 to the process 3 group.
- Number each item from 1 to 1,200.
Reading from left to right from a table of random digits, identify 800 unique numbers from 1 to 1,200. Assign the items with labels in the first 400 numbers to the process 1 group. Assign the items with labels in the second 400 numbers to the process 2 group. Assign the remaining items to the process 3 group.
- Number each item from 0000 to 1199.
Reading from left to right on a random number table, identify 800 unique four-digit numbers from 0000 to 1199. Assign the items with labels in the first 400 numbers to the process 1 group. Assign the items with labels in the second 400 numbers to the process 2 group. Assign the remaining items to the process 3 group.
- Select an item, and identify the first digit reading from left to right on a random number table. If the first digit is a 1, 2, or 3, assign the item to the process 1 group.
If the first digit is a 4, 5, or 6, assign the item to the process 2 group. If the first digit is a 7, 8, or 9, assign the item to the process 3 group. If the first digit is a 0, skip that digit and move to the next one to assign the item to a group. Repeat this process for each item.
Answers
Answer: The most appropriate design for this experiment is the third option:
- Number each item from 0000 to 1199.
- Reading from left to right on a random number table, identify 800 unique four-digit numbers from 0000 to 1199. Assign the items with labels in the first 400 numbers to the process 1 group. Assign the items with labels in the second 400 numbers to the process 2 group. Assign the remaining items to the process 3 group.
This design ensures that the groups are equally sized and selected randomly without any biases. The use of a random number table to assign the groups helps to avoid any systematic patterns or preferences that might arise from numbering or labeling the items directly.
Step-by-step explanation:
$2000 are invested in a bank account at an interest rate of 5 percent per year.
Find the amount in the bank after 7 years if interest is compounded annually.
Find the amount in the bank after 7 years if interest is compounded quaterly.
Find the amount in the bank after 7 years if interest is compounded monthly.
Finally, find the amount in the bank after 7 years if interest is compounded continuously.
Answers
The amount in the bank after 7 years increases as the compounding frequency increases, and it is highest when interest is compounded continuously.
Simple interest calculation.
Using the formula A = P(1 + r/n)^(nt), where:
A = the amount in the account after t years
P = the principal (initial amount)
r = the annual interest rate (as a decimal)
n = the number of times the interest is compounded per year
t = the number of years
a) If interest is compounded annually:
A = 2000(1 + 0.05/1)^(1*7) = $2,835.08
b) If interest is compounded quarterly:
A = 2000(1 + 0.05/4)^(4*7) = $2,888.95
c) If interest is compounded monthly:
A = 2000(1 + 0.05/12)^(12*7) = $2,905.03
d) If interest is compounded continuously:
A = Pe^(rt) = 2000e^(0.05*7) = $2,938.36
Therefore, the amount in the bank after 7 years increases as the compounding frequency increases, and it is highest when interest is compounded continuously.
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Angel made a table runner that has an area of 80 square inches. The length and width of the table runner are whole numbers. The length is 5 times greater than the width. What are the dimensions of the table runner?
Answers
the dimensions of the table runner are [tex]20[/tex] inches in length and [tex]4[/tex] inches in width.
What are the dimensions?Let's denote the width of the table runner as "w" inches. Since the length is 5 times greater than the width, the length would be 5w inches.
The area of a rectangle is calculated by multiplying its length by its width. Given that the area of the table runner is 80 square inches, we can set up the following equation:
Length × Width = Area
[tex](5w) \imes w = 80[/tex]
Simplifying further:
[tex]5w^2 = 80[/tex]
Dividing both sides by 5:
[tex]w^2 = 16[/tex]
Taking the square root of both sides:
w = ±4
Since the width cannot be negative in this context, we discard the negative value. Therefore, the width (w) of the table runner is [tex]4[/tex] inches.
Substituting this value back into the equation for length:
Length [tex]= 5w = 5 \times 4 = 20[/tex] inches
So, the dimensions of the table runner are [tex]20[/tex] inches in length and 4 inches in width.
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Factor completely. 5x ^2-5y ^2
Answers
In this expression, the GCF is 5. Therefore, we can factor out a 5 to get 5(x² - y²).
What is difference of squares formula?The difference of squares formula states that the difference of two squares can be factored into the product of two terms, one of which is the sum of the terms and the other is the difference of the terms.
When factoring polynomials, it is important to factor out the greatest common factor (GCF).
In this expression, the GCF is 5.
Therefore, we can factor out a 5 to get 5(x² - y²).
Now, we can expand the parentheses and factor out the remaining terms. Since both terms are perfect squares, we can factor them using the difference of squares formula:
5(x² - y²) = 5(x + y)(x - y)
In this case, the terms are x and y, resulting in the factorization of 5(x² - y²) into 5(x + y)(x - y).
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I need help in like 5 minutes please
Answers
Answer: 3/2 or 1 1/2 or 1.5
Step-by-step explanation:
9/8 times 2 is 2.25 or 9/4
9/4-6/8= 3/2 or 1 1/2 or 1.5
Part of the proceeds from a garage sale was $290 worth of $5 and $20 bills. If there were 8 more $5 bills than $20 bills, find the number of each denomination.
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Answer:
18 5-dollar bills
10 20-dollar bills
11. Triangles can be classified by their side lengths or by their angle measures. Sometimes triangles are classified using both classifications (side length and angle measure) but these can be redundant and unnecessary. Give an example of a classification based on both side length and angle measure that is unnecessary and explain why.
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By answering the presented question, we may conclude that As a result, adding the isosceles triangle categorization is superfluous and redundant in this circumstance.
What is triangle?A triangle is a polygon since it has three sides and three vertices. It is a basic geometric shape. Triangle ABC refers to a triangle with the vertices A, B, and C. In Euclidean geometry, a single plane and triangle are obtained when the three points are not collinear. If a triangle has three sides and three corners, it is a polygon. The triangle's corners are the spots where the three sides meet. The sum of three triangle angles equals 180 degrees.
An isosceles triangle has two sides of equal length, but a right triangle has one angle that measures 90 degrees. As a result, an isosceles right triangle is one with two equal-length sides and one 90-degree angle.
The Pythagorean theorem states that in any right triangle, the two legs (the sides next to the right angle) are always of equal length. Hence, if we know that a triangle has one 90-degree angle and two equal-length sides, we already know that it is a right triangle and that the two legs are equal. As a result, adding the isosceles triangle categorization is superfluous and redundant in this circumstance.
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An average newspaper contains at least 9 pages and at most 46 pages. How many newspapers must be collected to be certain that at least two newspapers have the same number of pages?
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We need to collect 46 newspapers to be certain that at least two newspapers have the same number of pages.
What is the minimum number of newspapers needed to guarantee that two newspapers have the same number of pages?According to the Pigeonhole Principle, if we have n+1 pigeons and n holes, then there must be at least one hole with two or more pigeons. Similarly, if we have n+1 newspapers with n possible page counts, then there must be at least one page count that appears in two or more newspapers.
In this case, we have a range of 38 possible page counts (46 - 9 + 1), so we need at least 39 newspapers to guarantee that each possible page count appears in at least one newspaper.
However, to guarantee that at least two newspapers have the same number of pages, we need one more newspaper than the number of possible page counts, so we need a total of 46 newspapers. Therefore, if we collect 46 newspapers, we can be certain that at least two of them have the same number of pages.
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