Answer:
x = 1/2, -4/3
Step-by-step explanation:
(2x - 1)(3x + 4) = 0
We know any number multiplied by 0 will be equal 0.
2x - 1 = 0
2x = 1
x = 1/2
3x + 4 = 0
3x = -4
x = -4/3
So, x = 1/2, -4/3
[tex]2x(3x + 4) - 1(3x + 4) = 0[/tex]
[tex]6 {x}^{2} + 8x - 3x - 4 = 0[/tex]
[tex]6 {x}^{2} + 5x - 4 = 0[/tex]
[tex]6 {x}^{2} + 5x = 4[/tex]
Complete each conversion by dragging a number to each box.
Numbers may be used once, more than once, or not at all.
1,20012012,00012
12,000 g =
kg
120 cm =
mm
1. 2 L =
mL
1,200 cm =
m
0. 12 m =
mm
The conversion each unit gives:
12,000 g = 12 kg
120 cm = 1200 mm
1.2 L = 1200 mL
1,200 cm = 12 m
0. 12 m = 120 mm
How to convert from one unit to another?
Conversion of units is the process of converting between different units of measurement for the same quantity through conversion factors.
1000 g = 1 kg. Thus,
12,000 g = 12 kg
1 cm = 10 mm. Thus,
120 cm = 1200 mm
1L = 1000 mL. Thus,
1.2 L = 1200 mL
100 cm = 1 m. Thus,
1,200 cm = 12 m
1 m = 1000 mm. Thus,
0. 12 m = 120 mm
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A regular hexagon is shown. What is the measure of the radius, c, rounded to the nearest inch? use the appropriate trigonometric ratio to solve. 6 in. 10 in. 14 in. 24 in.
The measure of the radius of the hexagon rounded to the nearest inch is 14 inches.
The problem presents a hexagon with a central angle of 60º, and the task is to calculate its radius. To do so, we can use the trigonometric relationship between the radius, apothem, and an angle. The apothem is a line segment from the center of a polygon perpendicular to one of its sides. For a regular hexagon, the apothem length is equal to the radius, which we want to find.
The trigonometric relationship for this case is cos(30) = a/c, where a is the apothem and c is the radius. By rearranging the equation to solve for c, we get c = a/cos(30).
Substituting the value of 12 inches for the apothem, we get c = 12/cos(30). Using a calculator, we can find that cos(30) = 0.866, so c = 12/0.866 = 13.855 inches.
To round to the nearest whole number, we get c = 14 inches.
Correct Question :
A regular hexagon is shown. What is the measure of the radius, c, rounded to the nearest inch? use the appropriate trigonometric ratio to solve. 6 in. 10 in. 14 in. 24 in.
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Imagine that the price per gallon of gas with a $7 car wash is $3. 19 and the price without the car wash is $3. 39. When is it worth it to buy the car wash? When is it worth it if the car wash costs $2?
If we plan to buy more than 10 gallons of gas, it is worth it to buy the car wash, assuming that the cost of the car wash is $2.
The car wash costs $7, and the price per gallon of gas with the car wash is $3.19, while the price per gallon of gas without the car wash is $3.39. Let's assume that we buy x gallons of gas, then the total cost of buying gas with the car wash would be:
Total cost with car wash = 7 + 3.19x
The total cost of buying gas without the car wash would be:
Total cost without car wash = 3.39x
To determine when it is worth it to buy the car wash, we need to find when the total cost with the car wash is less than the total cost without the car wash:
7 + 3.19x < 3.39x
Solving for x:
7 < 0.2x
x > 35
This means that if we plan to buy more than 35 gallons of gas, it is worth it to buy the car wash.
Now, let's consider the scenario where the car wash costs $2. The total cost of buying gas with the car wash would be:
Total cost with car wash = 2 + 3.19x
To determine when it is worth it to buy the car wash in this scenario, we need to find when the total cost with the car wash is less than the total cost without the car wash:
2 + 3.19x < 3.39x
Solving for x:
2 < 0.2x
x > 10
This means that if we plan to buy more than 10 gallons of gas, it is worth it to buy the car wash, assuming that the cost of the car wash is $2.
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18. A singer sells a single as a music download and CD, making a total profit of £246. 64.
She sells 456 CD singles, earning 35p for every single sold.
She earns 17p for each music download of the single.
How many music downloads did she sell?
The singer sold approximately 512 music downloads for 17p and 456 CDs for 35p and earned a total profit of £246. 64.
Given data:
Total profit = £246. 64
The price for each music download is = 17p
Number of CDs sold = 456 CD
Price for each CD = 35p
Assume that the singer sold x music downloads. when she earns 17p for each music download then her total earnings from music downloads will be 0.17x. The total earnings from selling the CD are,
= 0.35 × 456
= 159.6.
From the above data, we can write the equation to find the value of x by,
0.17x + 159.6 = 246.64
0.17x = 246.64 - 159.6
0.17x = 87.04
x = 87.04/0.17
x = 512
Therefore, the singer sold approximately 512 music downloads.
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1. What is the volume of the sphere?
4
The volume of the given sphere having radius of 4 units is 267.94 units³.
Given the radius of the sphere (r) = 4 units
To find the volume of the given sphere, we have to substitute the radius in the below volume formula of the sphere,
the volume of the sphere = 4/3 * π * r³
the volume of the given sphere = 4/3 * 3.14 * (4)³
[π is approximately equal to 3.14]
the volume of the given sphere = 267.94 units³
So from the above analysis, we can conclude that the volume of the sphere having 4 units radius is 267.94 units³.
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Given question is not having complete information, the complete question is written below:
What is the volume of the sphere having 4 units radius?
Find the slope of the points (-10, -52)
and (-70, -32)
Answer:
Slope= -1/3
Step-by-step explanation:
The slope is found using (y₂ - y₁) / (x₂ - x₁)
(y₂ - y₁)
So let's do the numerator first with the y. -52-(-32). The two negative signs make 32 positive so -52 + 32= -20
(x₂ - x₁)
Now the denominator, x. -10-(-70). Same thing here, the two negative signs make 70 positive so -10 + 70 = 60
(y₂ - y₁) / (x₂ - x₁)
Now put them together so -20/60 which equals -1/3 which the slope
"When a contractor paints a square surface that has a side length of x feet, he needs to know the area of the surface in order to buy the correct amount of paint. Since the contractor always adds 25 square feet to the area, he buys extra paint. Which function can be used to find the totall area in square feet, Ax , that the contractor will use to determine how much paint he needs to buy?
The function that can be used to find the total area is: (x^2 + 25) sq. ft.
What is a square?A square is a type of quadrilateral which has an equal length of sides. So then its area can be calculated as;
area of a square = length x length
We have from the question that; a square surface that has a side length of x feet. So that;
area of the square surface = length * length
= x * x
= x^2 square feet
But since the contractor always adds 25 square feet to the area, he buys extra paint, then the function required is:
total area = (x^2 + 25) sq. ft.
The function is (x^2 + 25) sq. ft.
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Answer:
A(x) = x² + 25----------------------
In order to find the total area, we need to consider both the area of the square surface and the extra paint he always adds.
Find the area of the square surface:
A = x² (since the side length is x feet)Add the extra 25 square feet of paint:
A(x) = A + 25Combining these steps, the function is:
A(x) = x² + 25i need to factor 1/2y-5 1/2 and i can't seem to get it
The factored form of the equation is 1/2(y - √5)
To factor this expression, we need to look for any common factors that can be pulled out of both terms. We can see that both terms contain a factor of 1/2, so we can factor that out:
1/2(y - 5 1/2)
Now we need to see if there are any further factors that we can find. The term inside the parentheses, y - 5 1/2, cannot be factored any further using real numbers. However, we can write the expression as y - √5, which shows that it involves the square root of 5.
So the final factored form of the expression is:
1/2(y - √5)
This means that if we multiply 1/2 by (y - √5), we get back the original expression 1/2y - 5 1/2.
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f (x) = ¹4 - 6. Find the inverse of f(x) and its domain.
O A. f¹(x) =
6 + 4, where x #-6
O B. f¹(x) =
6 +4, where x #4
O c. f¹(x) =
¹6-4, where x 4
OD. f¹(x) = 2¹6-4, where x#-6
The correct option is the first one, and the domain is the set of real numbers except for x = -6.
How to find the inverse?The inverse will be a function such that when we take the composition we get the identity, then we can write:
[tex]f(g(x)) = \frac{1}{g(x) - 4} - 6 = x[/tex]
We need to solve that for g(x), we will get:
[tex]\frac{1}{g(x) - 4} - 6 = x\\\\\frac{1}{g(x) - 4} = x +6\\\\g(x) - 4 = \frac{1}{x + 6} \\g(x) = \frac{1}{x + 6} + 4[/tex]
That is the inverse function, and notice that if x = -6 the denominator becomes zero, so that value is not in the domain.
Then the correct option is the first one.
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The polynomial
y
=
−
0.74
x
4
+
2.8
x
3
+
26.4
describes the billions of flu virus particles in a person’s body
x
days after being infected. find the number of virus particles, in billions, after 1 day.
The number of flu virus particles in a person's body after 1 day is 28.46 billion, based on the given polynomial function.
The polynomial given to us is, y = −0.74x⁴ + 2.8x³ + 26.4 which describes the billions of flu virus particles in a person’s body. To find the number of virus particles in a person's body after 1 day, we need to substitute x = 1 into the given polynomial and evaluate it.
So, we have,
y = -0.74(1)⁴ + 2.8(1)³ + 26.4
= -0.74 + 2.8 + 26.4
= 28.46
Therefore, the number of virus particles in a person's body after 1 day is 28.46 billion.
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Complete question - The polynomial y=−0.74x⁴ + 2.8x³ + 26.4 describes the billions of flu virus particles in a person’s body x days after being infected. find the number of virus particles, in billions, after 1 day.
Two friends larbi and aminu,plays a game of chess with equal amount of money at the beginning (zero sum games) at the end of the game larbi lost 5 elevens of his amount and aminu gains 6 cedis more than one half of what is left for larbi. what total amount of money was left at the beginning of the game
At the beginning of the game, the total amount of money between Larbi and Aminu was 66 cedis.
Let x be the total amount of money at the beginning of the game.
After the game, Larbi lost 5/11x, so he has (1-5/11)x = 6/11x left.
Aminu gained 6 more than 1/2 of what Larbi has left, which is (1/2)(6/11x) + 6 = 3/11x + 6.
The total amount left after the game is the sum of what Larbi and Aminu have, which is (6/11x) + (3/11x + 6) = (9/11x) + 6.
Since this is equal to x (the total amount they started with), we have:
(9/11x) + 6 = x
Solving for x, we get:
x = 66.
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2. How many checks must a customer write per month before the new plan is cheaper than the old plan? and new plan? 3. What formula/equations can be formed to find the cost for any number of checks for the old cheaper for a customer who writes 10 checks per month? 1. Compute the cost of 10 checks under the old plan and under the new plan. Which plan is check will cost 8 cents. The bank claims the new plan will save the customer money. Plus 15 cents for each check announces that it will change its monthly fee to $3 and that each Problem #3 A bank that has been charging a monthly service fee of $2 for checking accounts
The old plan is cheaper for a customer who writes 10 checks per month.
To determine how many checks a customer must write per month before the new plan is cheaper than the old plan, we need to set up an equation to compare the two plans. Let x be the number of checks written per month. The cost of the old plan is given by:
C_old = 0.08x + 2
The cost of the new plan is given by:
C_new = 3 + 0.15x
To find out when the new plan becomes cheaper, we need to set the two costs equal to each other and solve for x:
0.08x + 2 = 3 + 0.15x
0.07x = 1
x ≈ 14.29
Therefore, a customer would need to write 15 checks per month for the new plan to be cheaper than the old plan.
For a customer who writes 10 checks per month, the cost of the old plan is:
C_old = 0.08(10) + 2 = 2.80
The cost of the new plan is:
C_new = 3 + 0.15(10) = 4.50
Therefore, the old plan is cheaper for a customer who writes 10 checks per month.
The formula for the cost of any number of checks for the old plan is:
C_old = 0.08x + 2
where x is the number of checks written per month
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Which one is it please help thank you.
The students who attend Memorial High School have a wide variety of extra-curricular activities to choose from in the after-school program. Students are 38% likely to join the dance team; 18% likely to participate in the school play; 42% likely to join the yearbook club; and 64% likely to join the marching band. Many students choose to participate in multiple activities. Students have equal probabilities of being freshmen, sophomores, juniors, or seniors. If Event A = sophomore or junior, what is Event A'?
Event A' has a probability of 50% (25% for freshmen + 25% for seniors).
To determine Event A', we need to first identify what Event A represents. Event A is the probability that a student is a sophomore or junior. Since students have equal probabilities of being freshmen, sophomores, juniors, or seniors, the probability of Event A is 50% (25% for sophomores + 25% for juniors).
Event A' is the complement of Event A, which means it includes the other two grade levels not included in Event A, in this case, freshmen and seniors. Therefore, Event A' is the probability that a student is a freshman or a senior. Since students have equal probabilities of being in each grade level, Event A' also has a probability of 50% (25% for freshmen + 25% for seniors).
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The volume of a pyramid is 51 cubic centimeters. The area of the base is 17 square centimeters What is its height?
A point in the figure is chosen at random. Find the probability that the point lies in the shaded region of the circle.
To find the probability that the point lies in the shaded region of the circle, we need to compare the area of the shaded region to the total area of the circle. Let's say the radius of the circle is r.
The area of the shaded region can be found by subtracting the area of the unshaded region from the total area of the circle. The unshaded region is a square with side length equal to the radius of the circle. Therefore, its area is r^2. The total area of the circle is πr^2. So the area of the shaded region is:
πr^2 - r^2 = r^2(π - 1)
Now, if we choose a point at random from the circle, any point has an equal chance of being chosen. So the probability of choosing a point in the shaded region is equal to the area of the shaded region divided by the total area of the circle:
P(shaded) = (r^2(π - 1))/πr^2 = π - 1
Therefore, the probability of choosing a point in the shaded region of the circle is π - 1.
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A, B, and C are points of tangency in the given Circle, a m equals 6, BK equals 4 in the perimeter of mkn is 34
The perimeter of triangle MKN is 34 units.
How to find the length of segment KN?Based on the information provided, we have a circle with three points of tangency: A, B, and C. Let's consider the triangle formed by these points: MKN.
We are given that the length of AM is 6 and the length of BK is 4. We need to find the perimeter of triangle MKN.
To find the perimeter, we need to know the lengths of all three sides. However, the length of side AC is not provided.
Without additional information, we cannot determine the lengths of sides MN and KN or calculate the perimeter of triangle MKN.
Therefore, with the given information, we cannot find the perimeter of triangle MKN or provide a numerical answer
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Area of this shape irregular polygon the high is 4 and width 13
:)
The area of this irregular polygon is 344.5 square units.
To find the area of an irregular polygon, you can divide it into smaller, simpler shapes.
In this case, we can divide the polygon into a rectangle and a right triangle.
The rectangle has a height of 4 and a width of 13, so its area is 4 x 13 = 52 square units.
The right triangle has a base of 13 and a height of (110 - 4 x 13) / 2 = 45, since the total height of the polygon is 110.
Therefore, the area of the right triangle is (1/2) x base x height = (1/2) x 13 x 45 = 292.5 square units.
Adding the areas of the rectangle and the triangle, we get a total area of 52 + 292.5 = 344.5 square units.
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Find the equation for the line that:
passes through (-4,-7) and has slope -6/7
The slope intercept form of the function is:
Answer: [tex]y=\frac{-6}{7} x+\frac{-73}{7}[/tex]
Step-by-step explanation:
The slope intercept form for a line is y=mx+b, where m is slope and b is the y intercept. For this form, we need to know the slope and y intercept.
The slope and one x and y are give, so we can plug in all of these values into the slope intercept equation to solve for b.
Doing so, we get:
[tex]y=mx+b\\-7=\frac{-6}{7} (-4)+b\\b=-7-\frac{24}{7} \\b=\frac{-73}{7}[/tex]
So, knowing the slope and y intercept, our equation is
[tex]y=\frac{-6}{7} x+\frac{-73}{7}[/tex]
At Kennedy High School, the probability of a student playing in the band is 0. 15. The probability of a student playing in the band and playingon the football team is 0. 3. Given that a student at Kennedy plays in the band, what is the probability that they play on the football team?
In order to find the probability of a student playing on the football team given that they play in the band, we'll use conditional probability.
The formula for conditional probability is P(A|B) = P(A and B) / P(B).
In this case, A represents playing on the football team, and B represents playing in the band.
Given:
P(B) = 0.15 (probability of playing in the band)
P(A and B) = 0.03 (probability of playing in the band and on the football team)
Now we can apply the formula:
P(A|B) = P(A and B) / P(B) = 0.03 / 0.15 = 0.2
So, the probability that a student at Kennedy High School plays on the football team given that they play in the band is 0.2 or 20%.
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Please help it would be amazing if you knew this
The solution of the composite function, (f + g)(x) is 8x + 7
How to solve function?A function relates input and output. In other words, a function is a special relationship among the inputs (independent variable) and their outputs (dependent variable).
A composite function is a function that depends on another function.
Therefore, let's solve the composite function
f(x) = x + 3
g(x) = 7x + 4
Hence,
(f + g)(x) can be solved as follows:
(f + g)(x) = f(x) + g(x)
(f + g)(x) = x + 3 + 7x + 4
combine like terms
(f + g)(x) = x + 7x + 3 + 4
(f + g)(x) = 8x + 7
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A circular mirror has a radius of 3. 4 feet rosalinda is decorating the edge of the mirror with Washington tape if she has exactly enough washi tape which measurement is closest to the length of the piece of washi tape in feet
The measurement closest to the length of the piece of washi tape needed is approximately 21.36 feet.
The circumference of the circular mirror can be calculated using the formula C = 2πr, where r is the radius. Plugging in the given radius of 3.4 feet, we get C = 2π(3.4) = 21.36 feet (rounded to two decimal places). Since Rosalinda is decorating the edge of the mirror with washi tape, she needs a piece of tape that is equal in length to the circumference of the mirror. Therefore, the length of the piece of washi tape needed is closest to 21.36 feet.
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A pathway made of slate tiles measures 5 1/3 yards long. A tile measuring 2 feet is added to the end of the pathway.
What is the total length of the pathway now?
1. 5 1/3 ft
2. 10 2/3 ft
3. 18 ft
4. 22 ft
Answer:
18 feet
Step-by-step explanation:
multiply 5 1/3 yards by 3 to find the measurement in feet.
16/3 x 3 = 16
16 + 2 = 18
Helping in the name of Jesus.
+
ent will
A circle with center (7,3) and radius of 5 is graphed below with a square inscribed in
the circle.
+
Dillon says to write the equation of the tangent line you need the opposite-reciprocal
slope of the slope of the radius and Chelsey says you need to use the same slope as
the radius. Who is correct and why? Write the equation of the tangent line.
Part B: Find the perimeter of BCDE.
Chelsey is correct.
The equation of the tangent line is y = 8
Perimeter of BCDE is 28.28
How to determine tangent line and perimeter?Chelsey is correct. The tangent line at a point on a circle is perpendicular to the radius at that point.
Therefore, it has the same slope as the radius at the point of tangency.
To find the equation of the tangent line at point B(7,8), find the slope of the radius at B.
The radius at B passes through the center of the circle (7,3) and B(7,8), so its slope is:
m = (8 - 3) / (7 - 7) = undefined
This is because the radius is a vertical line. The slope of the tangent line at B is the negative reciprocal of the slope of the radius at B, which is 0.
The equation of the tangent line is:
y - 8 = 0(x - 7)
y = 8
Part B: To find the perimeter of BCDE, we need to find the length of one of its sides and then multiply by 4. Since the square is inscribed in the circle, its diagonal is equal to the diameter of the circle, which is 10 (twice the radius).
Therefore, the length of one side of the square is:
s = 10/√(2) ≈ 7.07
The perimeter of BCDE is:
4s = 4(7.07) ≈ 28.28
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Data-Safe Company leases online data storage
space. A customer can store up to 750
gigabytes of computer files for an annual fee of
$19.95. If the number of gigabytes exceeds 750,
then the equation c=0.05(g - 750) + 19.95
determines the annual cost, c. in dollars in terms
of g, the total number of gigabytes of data
stored. Which statement best describes this
information?
The statement that best describes the information is:
-If the total number of gigabytes stored exceeds 750, then each gigabyte beyond 750 costs 5 cents.
How to make the correct statement based on the informationThis is because the equation c = 0.05(g - 750) + 19.95 applies only when the number of gigabytes (g) exceeds 750. For every gigabyte over 750, an additional 5 cents is charged, while the base cost of $19.95 is for the first 750 gigabytes of storage.
If the customer needs to store more than 750 gigabytes, the annual cost is determined by the equation c = 0.0 ,5(g - 750) + 19.95, where c is the annual cost in dollars, and g is the total number of gigabytes of data stored
The statement that best describes this information is:
The Data-Safe Company offers online data storage services with an annual fee of ,$19.95 for up to 750 gigabytes of storage. If a customer requires more than 750 gigabytes of storage,. an additional fee of $0.05 per gigabyte is charged for the excess storage beyond the initial 750 gigabytes.
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Complete question
Data-Safe Company leases online data storage
space. A customer can store up to 750
gigabytes of computer files for an annual fee of
$19.95. If the number of gigabytes exceeds 750,
then the equation c=0.05(g - 750) + 19.95
determines the annual cost, c. in dollars in terms
of g, the total number of gigabytes of data
stored. Which statement best describes this
information?
-If the total number of gigabytes stored is
more than 750, then every gigabyte stored
costs 5 cents.
-If the total number of gigabytes stored
exceeds 750, then each gigabyte beyond
750 costs 5 cents.
B
-The first 750 gigabytes stored costs 5 cents
each, after which there is an additional cost
of $19.95
-Every gigabyte of data stored costs 5 cents
no matter how many gigabytes are stored.
What are two different ways you can solve 2(x – 3) = 8?
Answer:
There are three methods used to solve systems of equations: graphing, substitution, and elimination.
Step-by-step explanation:
Answer the following questions using what you've learned from this lesson. Write your responses in the
space provided.
For questions 1-4, use the following data to calculate and interpret the linear regression equation
Worldwide carbon dioxide emissions have increased over the years as Earth's population has grown. The
table shows the world carbon dioxide emissions, in millions of metric tons, from 1950 to 1990
Year
1950
1960
Response variable:
1970
1980
1990
1. Which is the explanatory variable.
and which is the response
variable?
Explanatory variable:
Name
Date
Emissions
6000
9500
15,000
19,300
22,500
2. Calculate the linear regression
equation from the above data.
The expected value of emissions when Year is 0, which makes up the intercept of the line (-666708.3) .
what is linear regression ?By fitting a linear equation to the observed data, the statistical technique of linear regression is utilized to describe the connection between two variables. Finding the line that best fits the data and can accurately depict the interaction of the variables is the aim of linear regression. In regression analysis, one variable is regarded as the independent (or predictor) variable, and the other is regarded as the dependent (or response) factor (or response variable). Y = mx + b, where x is the underlying factor, m is the line's slope, and b is the y-intercept, is a linear equation that describes the connection between both variables.
given
The linear regression equation that results is:
Emissions = 666708.3 - 347.6 Year
This equation models the link between Year (the explanatory variable) and Emissions as a straight line (the response variable).
According to the line's slope (347.6), emissions rose by an average of 348 million metric tonnes annually during this time.
The expected value of emissions when Year is 0, which makes up the intercept of the line (-666708.3) .
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If 3r/c=126 what is the value of r/2c
please i have no idea what the hecc is going on.
Examining the expression the value of r/2c is
21How to find r/2cWe can use algebra to calculate the value of r/2c with 3r/c as the variable.
First, let's reduce the expression 3r / c = 126 by multiplying both sides by c / 3:
3r/c * c/3 = 126 * c/3
r/1 = 42c
Now we can obtain r / 2c by dividing both sides by 2c:
solving for the left hand side of the equation
r /2c = (r/1)/(2c)
solving for the right hand side of the equation
42c = (42c)/(2c) = 21
Equation the left hand side to the right hand side of the equation
r/2c = 21
Therefore, r/2c = 21.
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For evrey 500g of reactants, 3. 1 g of catalyst were required. How much catalyst was required for 900g of reactants
5.58g of catalyst is required for 900g of reactants.
How much catalyst for 900g reactants?If 500g of reactants require 3.1g of catalyst, then for 900g of reactants, we can use the following proportion:
500g reactants / 3.1g catalyst = 900g reactants / x
Where x is the amount of catalyst required for 900g of reactants.
To solve for x, we can cross-multiply:
500g reactants * x = 3.1g catalyst * 900g reactants
Then, we can divide both sides by 500g reactants to isolate x:
x = (3.1g catalyst * 900g reactants) / 500g reactants
Simplifying this expression gives:
x = 5.58g catalyst
Therefore, 5.58g of catalyst is required for 900g of reactants.
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You deposit $2000 earned at a summer job in an account that pays 4. 2% simple interest. What is the balance in the account in 3 years? Estimate to the nearest whole number
A deposit of $2000 earning 4.2% simple interest for 3 years will have a balance of $2252. The estimated balance rounded to the nearest whole number is $2252.
To calculate the balance in the account after 3 years, we can use the formula
balance = principal x (1 + interest rate x time)
Plugging in the values, we get
balance = 2000 x (1 + 0.042 x 3)
balance = 2000 x (1 + 0.126)
balance = 2000 x 1.126
balance = 2252
Therefore, the balance in the account after 3 years is $2252.
As for the estimate, since the interest is simple, we can approximate it by multiplying the interest rate by the number of years and adding it to the principal. So, the estimate would be
estimate = principal x (1 + interest rate x time)
estimate = 2000 x (1 + 0.042 x 3)
estimate = 2000 x (1 + 0.126)
estimate = 2000 x 1.126
estimate = 2252
Rounding to the nearest whole number, the estimate is also $2252.
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evaluate the integral tan inverse v(x+2 ) dx by making substitution
and then table of integrals
To evaluate the integral of tan inverse v(x+2) dx, we need to make a substitution. Let u = x + 2, then du/dx = 1 and dx= du. Therefore, the final answer is: ∫ tan inverse v(x+2) dx = (x+2) tan inverse v(x+2) - tan inverse v(x+2) / v'(x+2) + C
Substituting this back into the integral, we get:
∫ tan inverse v(x+2) dx = ∫ tan inverse v(u) du
Using the formula from the table of integrals, we have:
∫ tan inverse v(u) du = u tan inverse v(u) - ∫ u / (1 + v(u)^2) du
Substituting back u = x + 2, we get:
∫ tan inverse v(x+2) dx = (x+2) tan inverse v(x+2) - ∫ (x+2) / (1 + v(x+2)^2) dx
Now, we can use another substitution, let t = v(x+2), then dt/dx = v'(x+2) and dx = dt / v'(x+2).
Substituting this back into the integral, we get:
∫ (x+2) / (1 + v(x+2)^2) dx = ∫ (x+2) / (1 + t^2) dt / v'(x+2)
Using the formula from the table of integrals, we have:
∫ (x+2) / (1 + t^2) dt = tan inverse t + C
where C is the constant of integration.
Substituting back t = v(x+2), we get:
∫ (x+2) / (1 + v(x+2)^2) dx = tan inverse v(x+2) / v'(x+2) + C
Therefore, the final answer is:
∫ tan inverse v(x+2) dx = (x+2) tan inverse v(x+2) - tan inverse v(x+2) / v'(x+2) + C
where C is the constant of integration.
To evaluate the integral of tan inverse v(x+2) dx using substitution, we'll first make a substitution:
Let u = x+2. Then, du = dx.
Now, we can rewrite the integral as:
∫tan^(-1)(v(u)) du
Next, we'll look up the integral of tan^(-1)(v(u)) in a table of integrals. Unfortunately, there isn't a direct formula for this specific integral. However, we can use integration by parts to proceed further.
Let I = ∫tan^(-1)(v(u)) du. Let's choose:
f(u) = tan^(-1)(v(u)) and df(u) = du,
g'(u) = 1 and dg(u) = u du.
Using integration by parts formula:
I = f(u)g(u) - ∫g(u)df(u)
I = u*tan^(-1)(v(u)) - ∫u(1/(1+v^2(u))) du
Now, we'll need to substitute back x+2 for u:
I = (x+2)*tan^(-1)(v(x+2)) - ∫(x+2)(1/(1+v^2(x+2))) dx
This integral doesn't have a simple closed-form solution, so the final answer will remain in the form shown above.
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