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Answers
The final answer is: ∫(2x-1)÷[(x-3)(x+2)]dx = ln|x-3| + ln|x+2| + C
What is Integration ?
In calculus, integration is the inverse operation of differentiation. It is a mathematical technique used to find the integral of a function. The integral of a function f(x) is another function F(x), which gives the area under the curve of f(x) from a certain point to another.
To perform the integration of the given function:
∫(2x-1)÷([tex]x^{2}[/tex]-x-6)dx
First, we need to factor the denominator:
[tex]x^{2}[/tex]- x - 6 = (x-3)(x+2)
So we can rewrite the integral as:
∫(2x-1)÷[(x-3)(x+2)]dx
Next, we need to decompose the fraction into partial fractions:
(2x-1)÷[(x-3)(x+2)] = A÷(x-3) + B÷(x+2)
Multiplying both sides by (x-3)(x+2), we get:
2x-1 = A(x+2) + B(x-3)
Substituting x=3, we get:
5A = 5
A = 1
Substituting x=-2, we get:
-5B = -5
B = 1
So we have:
(2x-1)÷[(x-3)(x+2)] = 1÷(x-3) + 1÷(x+2)
Substituting this back into the integral, we get:
∫(2x-1)÷[(x-3)(x+2)]dx = ∫[1÷(x-3) + 1÷(x+2)]dx
Using the first rule of integration, we get:
∫[1÷(x-3) + 1÷(x+2)]dx = ln|x-3| + ln|x+2| + C
where C is the constant of integration.
Therefore, the final answer is: ∫(2x-1)/[(x-3)(x+2)]dx = ln|x-3| + ln|x+2| + C
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[tex] \displaystyle \int \rm \: \dfrac{2x - 1}{ {x}^{2} - x - 6 } dx[/tex]
Answer:
[tex] \underline{\boxed{\rm = ln |x + 2| + ln |x - 3| + C}}[/tex]
Step-by-step explanation:
[tex] = \displaystyle \int \rm \: \dfrac{2x - 1}{ {x}^{2} - x - 6 } dx[/tex]
[tex] = \displaystyle \int \rm \: \dfrac{2x - 1}{ {x}^{2} - 3x + 2x - 6 } dx[/tex]
[tex] = \displaystyle \int \rm \: \dfrac{2x - 1}{ x(x - 3) + 2(x - 3) } dx[/tex]
[tex] = \displaystyle \int \rm \: \dfrac{2x - 1}{ (x + 2) (x - 3)} dx[/tex]
[tex] \rm \: Let : \displaystyle \rm \: \dfrac{2x - 1}{ (x + 2) (x - 3) } = \dfrac{A }{x + 2} + \dfrac{B}{x - 3} [/tex]
[tex]\rm\implies\displaystyle \rm \: \dfrac{2x - 1}{ (x + 2) (x - 3) } = \dfrac{A(x - 3) + B(x + 2) }{(x + 2)(x - 3)} \\ [/tex]
[tex] \rm \implies\displaystyle \rm \: {2x - 1}{ } = {A(x - 3) + B(x + 2) } \\ [/tex]
Put x = 3 , we get
[tex] \rm \implies\displaystyle \rm \: {6 - 1}{ } = {A(3- 3) + B(3 + 2) } \\ [/tex]
[tex] \rm \implies\displaystyle \rm \: {5}{ } = 5 B \\ [/tex]
[tex] \implies \rm \: B = 1[/tex]
Again
put put x = -2
[tex] \rm \implies\displaystyle \rm \: { - 4- 1}{ } = {A( - 2- 3) + B( - 2 + 2) } \\ [/tex]
[tex] \rm \implies\displaystyle \rm \: { - 5}{ } = {A( - 5) } \\ [/tex]
[tex] \rm \implies\displaystyle \rm A = 1 \\ [/tex]
Thus ,
[tex] \displaystyle \int \rm \: \dfrac{2x - 1}{ (x + 2) (x - 3)} dx = \int\dfrac{1}{x + 2} dx + \int \dfrac{1}{x - 3} dx[/tex]
[tex] \rm = ln |x + 2| + ln |x - 3| + C[/tex]
Important formulae:-[tex] \tt\int \dfrac{dx}{ {x}^{2} + {a}^{2} } = \frac{1}{a} { \tan}^{ - 1} \frac{x}{a} + c \\ [/tex]
[tex] \tt\int \dfrac{dx}{ {x}^{2} - {a}^{2} } = \frac{1}{2a} log \frac{x - a}{x + a} + c \\ [/tex]
[tex] \tt\int \dfrac{dx}{ {a}^{2} - {x}^{2} } = \frac{1}{2a} log \frac{a + x}{a - x} + c \\ [/tex]
[tex] \tt\int \: \dfrac{dx}{ \sqrt{ {x}^{2} + {a}^{2} } } = log|x + \sqrt{ {a}^{2} + {x}^{2} } | + c \\ [/tex]
[tex] \tt\int \: \dfrac{dx}{ \sqrt{ {x}^{2} - {a}^{2} } } = log|x + \sqrt{ {x}^{2} - {a}^{2} } | + c \\ [/tex]
[tex] \tt \int \: \dfrac{dx}{ {a}^{2} - {x}^{2} } = { \sin }^{ - 1} \bigg(\dfrac{x}{a} \bigg) + c \\ [/tex]
[tex] \tt \int \: \sqrt{ {x}^{2} + {a}^{2} } dx \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\= \tt \dfrac{x}{2} \sqrt{ {a}^{2} + {x}^{2} } + \dfrac{ {a}^{2} }{2} log |x + \sqrt{ {x}^{2} + {a}^{2} }| + c[/tex]
Related Questions
Write an exponential function in the form y=ab^xy=ab
x
that goes through points (0, 14)(0,14) and (3, 3024)(3,3024).
Answers
The exponential function that passes through the points (0, 14) and (3, 3024) is [tex]$$y = 14\cdot 6^x$$[/tex]
How to find the function using points?To find the values of a and b in the exponential function [tex]$y=ab^x $[/tex]that goes through the given points (0, 14) and (3, 3024), we can use the following system of equations:
[tex]$\begin{align*}a\cdot b^0 &= 14 \a\cdot b^3 &= 3024\end{align*}[/tex]
Simplifying the first equation, we get a=14. Substituting this value into the second equation, we get:
[tex]14(b)^3=3024\\= b^3= 216\\= b=6[/tex]
Therefore, the exponential function that goes through the given points is:
[tex]$$y = 14\cdot 6^x$$[/tex]
We can check that this function satisfies both of the given points:
[tex]$\begin{align*}y &= 14\cdot 6^0 = 14 &&\text{when } x=0 \y &= 14\cdot 6^3 = 3024 &&\text{when } x=3\end{align*}[/tex]
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Paula's Pizza Parlor uses the following ingredients to make pizza.
Number of Pizzas Sauce (oz) Cheese (oz)
2 18 12
5
At this rate, how much sauce and cheese will Paula use to make 5 pizzas?
Paula will use 30 oz of sauce and 45 oz of cheese to make 5 pizzas.
Paula will use 35 oz of sauce and 24 oz of cheese to make 5 pizzas.
Paula will use 45 oz of sauce and 30 oz of cheese to make 5 pizzas.
Paula will use 90 oz of sauce and 60 oz of cheese to make 5 pizzas.
Answers
Answer:
Paula will use 45 oz of sauce and 30 oz of cheese to make 5 pizzas
How to find quantity of ingredients for 5 pizzas?
To find this, you can use the proportion of ingredients used for 2 pizzas and scale it up to 5 pizzas.
For 2 pizzas, Paula uses 18 oz of sauce and 12 oz of cheese.
The proportion of sauce to cheese used is 18/12
To make 5 pizzas, you can use this proportion to find how much sauce and cheese is needed:
2 pizzas = 18 sauce
5 pizzas = 5 / 2 x 18 = 45 sauces.
For cheese required, Paula will use:
2 pizzas = 12 cheese
5 pizzas = 5 / 2 x 12 = 30 cheese.
So the total is 45 oz of sauce and 30 oz of cheese.
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Step-by-step explanation:
Answer:
Paula will use 45 oz of sauce and 30 oz of cheese to make 5 pizzas
Step-by-step explanation:
Ms adams shares out 48 pencils between niamh and jack in the ratio 4:8
Answers
Niamh gets 16 pencils and Jack gets 32 pencils.
The ratio of 4:8 can be simplified to 1:2 by dividing both sides by 4. This means that for every one pencil Niamh gets, Jack gets two pencils.
To find out how many pencils each child gets, we need to divide the total number of pencils by the total number of parts in the ratio, which is 1 + 2 = 3.
So, each part of the ratio represents 48/3 = 16 pencils.
Therefore, Niamh gets 1 part, which is 16 pencils, and Jack gets 2 parts, which is 2 x 16 = 32 pencils.
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The given question is incomplete, the complete question is:
Ms. Adams shares out 48 pencils between Niamh and jack in the ratio 4:8, How many pencil does each child get ?
I NEED HELP ASAP
Solve the polynomial
x^3-7x^2-x+7=0
Answers
Answer:
x = 1, x = 7, and x = -1.
Step-by-step explanation:
o solve the polynomial equation x^3 - 7x^2 - x + 7 = 0, we can use a combination of synthetic division and factoring by grouping:
First, we need to find a root of the polynomial using the Rational Root Theorem. The possible rational roots of the polynomial are the factors of 7 (the constant term) divided by the factors of 1 (the leading coefficient), or ±1, ±7. By trying each of these values in the polynomial, we find that x = 1 is a root.
Using synthetic division, we can divide the polynomial by (x - 1) to obtain a quadratic equation:
1 | 1 -7 -1 7
| 1 -6 -7
|_____________
1 -6 -7 0
Therefore, (x - 1) is a factor of the polynomial, and we have:
x^3 - 7x^2 - x + 7 = (x - 1)(x^2 - 6x - 7)
Now we need to solve the quadratic equation x^2 - 6x - 7 = 0. We can factor it as (x - 7)(x + 1) = 0, so the solutions are x = 7 and x = -1.
Therefore, the solutions to the original polynomial equation x^3 - 7x^2 - x + 7 = 0 are x = 1, x = 7, and x = -1.
The following table shows the order in which certain ingredients labeled a - f need to be mixed
Answers
The final component, either before or after C, is F.
Ingredients A through F must be combined (left goes first, right goes last). Also, we are aware that: Ingredients A and D are introduced in succession; B is added before to A; and F is not added immediately prior to or following C.
Which component should be inserted last?
The aforesaid issue is easily resolved by doing the following:
Ingredient labeling: A, B, C, D, E, and F
It is given,
Ingredient A is added before D in the order of adding ingredients A and D.
To A, B is added first.
F is no longer put immediately before or after C.
The leftover ingredients are C, E, and F.
F cannot come before or after C because of this.
C can therefore come after D.
component following C = E
Ingredient list = F
The complete question is-
The following table shows the order in which certain ingredients, labeled A through F, need to be mixed (left goes first, right goes last). Additionally, we know that: Ingredients A and D are added in consecutive steps; B is added before A; F is not added right before or after C. Which ingredient needs to be added last?
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Numbers such as 1, 2, 3,. . . Are also called the _____ numbers
Answers
The numbers 1, 2, 3, and so on are called natural numbers
The numbers 1, 2, 3 and so on are commonly known as natural numbers. They are the most basic and fundamental type of numbers used in mathematics. Natural numbers are positive integers that are used for counting and measuring quantities.
They are called "natural" because they are the numbers that naturally occur when we count objects in the real world. The set of natural numbers is denoted by the symbol N and it is an infinite set that starts from 1 and continues infinitely. Natural numbers form the basis for other types of numbers, such as whole numbers, integers, rational numbers, and real numbers, and they are used in a wide range of mathematical applications.
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I
At an ice cream
stand, there are 4
different types of ice
cream, 3 different
cones, and 3 choices
of toppings.
How many different ways
can an ice cream cone be
ordered?
Answers
Answer: 36
Step-by-step explanation: To determine the number of ways an ice cream cone can be ordered; we must apply the multiplication principle of counting.
Given there are 3 different cone types, 3 choices of toppings, and 4 different types of ice cream, we can construct an equation to identify the maximum number of ways an ice cream cone can be ordered:
4 (types of ice cream) × 3 (types of cones) × 3 (choices of toppings) = 36 ways
Therefore, there are 36 different ways an ice cream cone can be ordered.
The number of different ways an ice cream cone can be ordered is 36.
What is the Permutations?Permutations are different ways of arranging objects in a definite order. It can also be expressed as the rearrangement of items in a linear order of an already ordered set. The symbol nPr is used to denote the number of permutations of n distinct objects, taken r at a time.
Given that, at an ice cream stand, there are 4 different types of ice cream, 3 different cones, and 3 choices of toppings.
We know that, nPr=n!/(n-r)!
Here, ⁴P₁׳P₁׳P₁
= 4!(4-1)! × 3!/(3-1)! × 3!/(3-1)!
= 4×3!/3! × 3×2!/2! × 3×2!/2!
= 4×3×3
= 36
Therefore, in 36 different ways cone can be ordered.
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A bike trail is 5 1/10 miles long. Jade rides 1/4 of trail before stopping for a water break. How many miles does Jade ride before stopping? Show your work.
Answers
Jade travelled 51/40 miles before stopping
What are fractions?Fractions are simply defined as the part of a whole variable or also the part of a whole number that is given.
In mathematics, fractions take different forms, such as;
Mixed fractionsComplex fractionsProper fractionsImproper fractionsSimple fractionsFrom the information given, we have that;
The bike trail is about 5 1/10 miles long
Represent as an improper fraction, we get;
5 1/10 = 51/10 miles long
If Jade travelled 1/4 of the bike trail, then, we have;
1/4 × 51/10
Divide the values
51/40
1 11/40 of the trail.
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Jade rode about 1.275 miles before stopping.
How many miles does Jade ride before stopping?A mile means unit for measuring distance. We have 1,760 yards in a mile which is the same as 5,280 feet or 63,360 inches.
To find out how many miles Jade rode before stopping, we need to multiply the length of the bike trail by the fraction of the trail that Jade rode before stopping.
The length of the bike trail is 5 1/10 miles which is converted to 51/10 miles.
Jade rode 1/4 of the trail before stopping, which we can also express as 5/20.
The number of miles Jade rode before stopping is calculated as:
= (51/10) x (5/20)
= 1.275 miles.
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at point E(-7, -24) lies on the circle whose equation is x^2+y^2=625. if an angle is drawn in standard position and its terminal ray passes through E, what is the value of sine of this angle
Answers
The sine of the angle passing through point E on the circle is given as follows:
sin(x) = 24/25.
What are the trigonometric ratios?The three trigonometric ratios are the sine, the cosine and the tangent, and they are defined as follows:
Sine of angle = length of opposite side to the angle divided by the length of the hypotenuse.Cosine of angle = length of adjacent side to the angle divided by the length of the hypotenuse.Tangent of angle = length of opposite side to the angle divided by the length of the adjacent side to the angle.The equation of the circle is of x² + y² = 625, meaning that the radius, representing the hypotenuse, is of 25 units.
The center of the circle is of (0,0), hence the opposite side to the angle has length of 24 - 0 = 24 units, hence the sine of the angle is given as follows:
sin(x) = 24/25.
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A person who is $5\frac{1}{2}$
feet tall casts a $3\frac{1}{2}$
-foot-long shadow. A nearby flagpole casts a 28-foot-long shadow. What is the height (in feet) of the flagpole?
Answers
The height of the flagpole is [tex]$\frac{28}{3\frac{1}{2}}\times 5\frac{1}{2}=50$[/tex]feet.
[tex]$\frac{28}{3\frac{1}{2}}\times 5\frac{1}{2}$[/tex]
[tex]$\frac{28\times 5\frac{1}{2}}{3\frac{1}{2}}$[/tex]
[tex]$\frac{140+7}{7}$[/tex]
[tex]$\frac{147}{7}$[/tex]
[tex]$50.71\approx50$[/tex] feet
The person and the flagpole are in the same environment, so if the person's height is 5 and a half feet and their shadow is 3 and a half feet, then the flagpole's shadow should be in the same ratio. By calculating the ratio between the flagpole's shadow and the person's height, we can determine the height of the flagpole. The ratio between the flagpole's shadow (28 feet) and the person's height (5 and a half feet) is 28 divided by 3 and a half. Multiplying this ratio by the person's height will give us the flagpole's height. The flagpole's height is approximately fifty feet.
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Mia wants to make a canvas tent as shown. How much canvas is needed to make the tent in inches squared?
Answers
Answer:
429
Step-by-step explanation:
any answers for these two questions?
Answers
[tex]\large\textsf{Answer:}[/tex]
[tex]\mathtt{x = 9.4}[/tex]
[tex]\mathtt{y = 14.3}[/tex]
[tex]\textsf{Please see below.}[/tex]
[tex]\large\textsf{Step-by-step explanation:}[/tex]
[tex]\textsf{For these problems, we are asked to solve for the missing variables.}[/tex]
[tex]\textsf{To start, we have \underline{2 exterior angles} and \underline{2 secant lines }that \underline{intersect} at a \underline{vertex}.}[/tex]
[tex]\boxed{\Large\textsf{What is a Vertex?}}[/tex]
[tex]\textsf{A \underline{Vertex} is a \underline{point of intersection} of \underline{2} or more \underline{lines.}}[/tex]
[tex]\boxed{\Large\textsf{What are Exterior Angles?}}[/tex]
[tex]\textsf{\underline{Exterior Angles} are \underline{angles} that are \underline{outside} of a circle.}[/tex]
[tex]\boxed{\Large\textsf{What are Secant Lines?}}[/tex]
[tex]\textsf{\underline{Secent Lines} are \underline{lines} that intersect a circle \underline{twice}.}[/tex]
[tex]\textsf{Because we have Intersecting Secants \underline{outside} of the circle, we should use this formula;}[/tex]
[tex]\mathtt{a(a+b)=c(c+d)}\\[/tex]
[tex]\textsf{Smaller segment is first, then the larger segment is inside the parentheses.}[/tex]
[tex]\underline{\textsf{Substitute values from 10;}}[/tex]
[tex]\mathtt{5(5+x)=6(6+6)}[/tex]
[tex]\underline{\textsf{Multiply:}}[/tex]
[tex]\mathtt{25+5x = 72}[/tex]
[tex]\underline{\textsf{Subtract 25 from both sides:}}[/tex]
[tex]\mathtt{5x = 47}[/tex]
[tex]\underline{\textsf{Divide by 5:}}[/tex]
[tex]\boxed{\mathtt{x = 9.4}}[/tex]
[tex]\textsf{Let's do the same for 11.}[/tex]
[tex]\underline{\textsf{Substitute values from 11;}}[/tex]
[tex]\mathtt{10(10+y)=9(9+18)}[/tex]
[tex]\underline{\textsf{Multiply:}}[/tex]
[tex]\mathtt{100+10y=243}[/tex]
[tex]\underline{\textsf{Subtract 100 from both sides:}}[/tex]
[tex]\mathtt{10y=143}[/tex]
[tex]\underline{\textsf{Divide by 10:}}[/tex]
[tex]\boxed{\mathtt{y=14.3}}[/tex]
The quantities
�
xx and
�
yy are proportional.
�
xx
�
yy
9
99
4.5
4.54, point, 5
14
1414
7
77
30
3030
15
1515
Find the constant of proportionality
(
�
)
(r)left parenthesis, r, right parenthesis in the equation
�
=
�
�
y=rxy, equals, r, x.
�
=
Answers
The constant of proportionality (r) in the table of values of x and y has a value of 0.5.
Calculating the constant of proportionalityTo find the constant of proportionality (r) in the equation y = rx, we need to determine the ratio of y to x for each pair of quantities given in the table.
If x and y are proportional, then this ratio should be the same for all pairs of quantities.
Using the table, we have:
y/x = 4.5/9 = 0.5
y/x = 7/14 = 0.5
y/x = 15/30 = 0.5
Since the ratio of y to x is the same for all pairs of quantities, we can conclude that x and y are indeed proportional, and the constant of proportionality (r) is 0.5.
Therefore, the equation that represents the relationship between x and y is:
y = 0.5x
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Complete question
Rewrite the following properly and solve:
The quantitiesx and y are proportional.
x y
9 4.5
14 7
30 15
Find the constant of proportionality (r) in the equation
y = rx
please help with this
Answers
The original coordinates of triangle NLM include the following:
N (-2 , -1)
L (2 , -3)
M (4,0)
What is a reflection?In Mathematics and Geometry, a reflection over the x-axis is modeled by this transformation rule (x, y) → (x, -y). This ultimately implies that, a reflection over the x-axis would maintain the same x-coordinate while the sign of the x-coordinate changes from positive to negative or negative to positive.
Since triangle NL'M' was reflected over the x-axis, the original coordinates of triangle NLM can be calculated as follows;
(x, y) → (-x, y)
(-x, y) → (x, y)
N'(-2, 1) → N (-2 , -1)
L'(2, 3) → L (2 , -3)
M' (4, 0) → M (4,0)
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The half-life of carbon-14 is 5,730 years. Suppose a fossil is found with 30 percent as much of its carbon-14 as compared to a living sample. How old is the fossil?
Answers
Step-by-step explanation:
Find the number of half lives to get to .3 (which is 30%) then multiply by 5730 years per half life:
.3 = (1/2)^n <====solve for 'n'
log .3 / log (1/2) = n = 1.737 half lives
1.737 X 5730 = ~ 9953 years old
Higher Order Thinking Tanika has 7 toothpicks. She uses them all to create two polygons. Draw two polygons that Tanika could have created. Write the names of your shapes.
Answers
Therefore , the solution of the given problem of polygon comes out to be
hexagon and pentagon.
What is polygon?In Euclidean mathematics, a simple quadrilateral of two sets of equal distances is referred to as a parallelogram. In a specific kind of quadrilateral known as a parallelogram, both set of opposite sides are straight and equal. There are four types of parallelograms, 3 of which are each mutually exclusive. Rhombuses, parallelograms, squares, but also rectangles are the four distinct shapes.
Here,
Here are two examples of shapes Tanika could make with seven toothpicks:
=> First polygon:
This is a hexagon without a triangular. It has seven edges and six sides.
=> Second polygon:
This pentagon has two line segments that stretch from its vertices on either side. It also has 7 edges and 7 sides.
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I need help, thank you.
Answers
Answer:
x ≈ 12.33
Step-by-step explanation:
given 2 secants from an external point to the circle, then the product of the measures of one secant's external part and the entire secant is equal to the product of the measures of the other secant's external part and that entire secant, that is
DC × (DC + x) = BC × (BC + AB) , that is
6(6 + x) = 5 × (5 + 17) = 5 × 22 = 110
36 + 6x = 110 ( subtract 36 from both sides )
6x = 74 ( divide both sides by 6 )
x = 12.33 ( to the nearest hundredth )
Please ASAP Help
Will mark brainlest due at 12:00
Answers
Answer:
The answer is C.
Step-by-step explanation:
A rectangular Inflatable swimming pool is 3 yards long, 14/5 yards wide, and
7/2 yards tall
. What is the volume of the pool? Round to the nearest tenth
Answers
The volume of the rectangular inflatable swimming pool is approximately 29.4 cubic yards.
The volume of an object is the amount of space it occupies in three dimensions, typically measured in cubic units. In this case, we're trying to find the volume of a rectangular pool that is 3 yards long, 14/5 yards wide, and 7/2 yards tall.
To find the volume of a rectangular object, we use the formula:
Volume = length x width x height
In this case, we can plug in the values we have:
Volume = 3 yards x 14/5 yards x 7/2 yards
To simplify the calculation, we can convert the fractions to decimals:
Volume = 3 yards x 2.8 yards x 3.5 yards
Volume = 29.4 cubic yards
In this case, the second decimal place is a 4, so we leave the digit in the first decimal place (9) as-is.
Therefore, we round our answer to 29.4 cubic yards to the nearest tenth.
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help please!!!!!!!!!!!
Answers
The equation of the perpendicular bisector of the line is y = 2x +16
What is slope?Slope of a line is defined as the ratio of the "vertical change" to the "horizontal change" between (any) two distinct points on a line.01 It can also be expressed as a quotient ("rise")
1) the slope (m) = (change in y)/ change in x
m = (6-2)/ 7+3 = 4/10 = 2/5
b) the slope of the perpendicular is given by m₁*m₂ = -1
m₂ = -1/m₁
m₂ = -1 ÷2/5 = -5/2
c) the mid point of the ordered pair is (y₁+y₂)/2 and (x₁+x₂)/2
[(7+3)/2, (6-2)/2]
[10/2 , 4/2 ]
(5,2)
d) equation of the perpendicular is given by
m = (y - y₁)/ (x - x₁)
m( (x - x₁) = (y - y₁)
By substitution we have
2/5(x +3) = y-2
2x + 6 = y - 10
Making y the subject we have
Therefore, y = 2x +16
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Data about annual maximum temperature \( { }^{\circ} \) Celcius of Muscat city for the last ten years since 2011 was collected by a group of STAT2101 students. A summary of the measure of central tendency isA summary of the measure of central tendency is: Based on the given data, what is the distribution of the maximum temperature in Muscat city? Select one: a. The maximum temperature is Another distribution. b. The maximum temperature is No distribution. c. The maximum temperature is Positively Skewed. d. The maximum temperature is Mutually Exclusive.
Answers
The distribution of the maximum temperature in Muscat city based on the given data is positively skewed. The correct option is c.
The distribution of the maximum temperature in Muscat city based on the given data is positively skewed. When a dataset's mean is higher than the median, this is known as a positively skewed distribution. It indicates that there are more lower values than upper values in the dataset.The mean, median, and mode are examples of measures of central tendency. A summary of the measure of central tendency of the annual maximum temperature in Muscat city for the last ten years since 2011 is required as it gives a clear overview of the dataset's central tendency, this is an important statistical summary.
The mean, median, and mode are the three most popular measures of central tendency. The mean is the average of all the data points in a dataset. It's calculated by adding up all the values in the dataset and then dividing by the number of data points.The median is the middle number in a dataset when it's sorted from lowest to highest. If there are two middle numbers, the median is the average of the two.Therefore, the distribution of the maximum temperature in Muscat city based on the given data is positively skewed, as there are more lower values than upper values in the dataset.
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For [tex]x^8-1=0[/tex], find all complex solutions, magnitudes of the roots, and draw them on the complex plane.
Answers
For the equation [tex]x^8-1=0[/tex], the graph with magnitudes of the roots is plotted on complex plane.
What is an equation?
A mathematical definition of an equation is a claim that two expressions are equal when they are joined by the equals sign ("=").
The equation is given as [tex]x^8-1=0[/tex].
We can start by factoring the left-hand side of the equation using the difference of squares formula -
[tex](x^4 - 1)(x^4 + 1) = 0[/tex]
Now we have two factors that can each be set equal to zero and solved separately -
[tex]x^4 - 1 = 0[/tex] or [tex]x^4 + 1 = 0[/tex]
The first equation can be factored as a difference of squares again -
(x² - 1)(x² + 1) = 0
Setting each factor equal to zero gives -
x² - 1 = 0 or x² + 1 = 0
These equations have solutions x = ±1 and x = ±i, respectively.
For the second equation, we can use the fact that i^2 = -1 to rewrite it as -
[tex]x^4 + 1 = (x^2)^2 + 1 = (x^2 + i)(x^2 - i) = 0[/tex]
Setting each factor equal to zero gives -
x² + i = 0 or x² - i = 0
These equations have solutions [tex]x = \pm i\sqrt{2}[/tex] and [tex]x = \pm i\sqrt{-2} = \pm i\sqrt{2i} = \pm \sqrt{2}i[/tex], respectively.
Therefore, the complete set of solutions to the original equation is -
x = ±1, ±i, ±i√(2), ±√2 i
To find the magnitudes of these roots, we can use the formula -
|a + bi| = √(a² + b²)
For example, the magnitude of the root x = i is -
|i| = |0 + 1i| = √(0² + 1²) = 1
Similarly, we can find the magnitudes of the other roots -
|x| = 1 for x = ±1, ±i
|x| = √2 for x = ±i√(2)
|x| = √2 for x = ±√2 i
To draw these roots on the complex plane, we can plot them according to their real and imaginary parts.
The roots ±1 and ±i lie on the unit circle centered at the origin, while the roots ±i√(2) and ±√2 i lie on circles centered at the origin with radii equal to √2.
Therefore, the roots are plotted in complex plane.
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3. 8, 15, 100, 123.
4. line the numbers up to multiply them from the y side or x side
5. adding the word problem to find the solution
Answers
There are four numbers: 8, 15, 100 and 123.
What is number?Number is a mathematical object used to count, measure, and label. It is used in many different contexts, from everyday life to scientific research. Numbers can be represented in various forms, such as symbols, digits, and words. They are used to represent quantities, distances, time, and other concepts. Numbers can also be used to represent relationships, such as equations, which are statements that describe how two or more things are related.
To solve this problem, it is best to line the numbers up on the y axis or x axis. To multiply the numbers, start by multiplying the numbers on the y axis first. So, 8 x 15 = 120. Then, multiply the result by the number on the x axis which is 100. 120 x 100 = 12,000. Finally, multiply 12,000 by the last number on the x axis which is 123. 12,000 x 123 = 1,476,000. Therefore, the answer to the problem is 1,476,000.
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Enter an equation for the balance,b, in the account after n months
Answers
An equation for the balance, b, in the student's savings account after n months is b = d - wn.
What is an equation?An equation is a mathematical or algebraic statement showing that two or more mathematical or algebraic expressions are equal or equivalent.
An equation is denoted using the equal symbol (=), which an algebraic expression lacks.
Mathematical expressions combine variables with numbers, constants, or values using mathematical operands.
The amount in the savings account = d
Monthly withdrawals = w
The period of withdrawals = n
Let the balance in the account after n months = b
Therefore, the balance after n months will be represented by the equation, b = d - wn.
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Complete Question:A student has a savings account with d dollars in it. The student plans to withdraw w dollars each month for n months. Enter an equation for the balance, b, in the account after n months.
Write the equation of the trigonometric graph. Use a positive coefficient on cosine for this activity.
Answers
The cosine function's positive coefficient ensures that the curve moves trigonometry downward from its highest point rather than upward from its lowest point.
what is trigonometry?The area of mathematics called trigonometry examines how triangle side lengths and angles relate to one another. The subject first came to light in the Hellenistic era, roughly in the third century BC, as a result of the use of geometry in astronomical investigations. The area of mathematics known as exact techniques deals with several trigonometric functions and possible computations using them. There are six common trigonometric functions in trigonometry. These go by the designations sine, cosine, tangent, cotangent, secant, and cosecant, respectively (csc). The study of triangle properties, particularly those of right triangles, is known as trigonometry. Consequently, studying geometry entails learning about the characteristics of all geometric shapes.
A trigonometric equation involving a cosine function with a positive coefficient looks like this:
y = 2cos(x) (x)
If this equation were graphed, a cosine curve with a period of 2 would be visible, oscillating between a maximum value of 2 and a minimum value of -2. The cosine function's positive coefficient ensures that the curve moves downward from its highest point rather than upward from its lowest point.
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Reflect (-3, -8) across the y-axis.
Then reflect the result across the x-axis.
What are the coordinates of the final point?
Answers
Answer:
(3, 8)
Step-by-step explanation:
Coordinate (-3, -8)
Reflect across the y-axis. The x will change to the opposite, and the y will remain the same. So, the coordinate is (3, -8)
Then reflect the result across the x-axis. The y will change to the opposite, and the x will remain the same. So, the coordinate is (3, 8)
So, the coordinate of the final point is (3, 8)
It’s argent
How much would you have to deposit now to be able to withdraw $2,400 at the end of each year for 10 years from an account that earns 4% compounded annually?
Answers
Answer:
The amount that would need to be deposited now to withdraw $2,400 at the end of each year for 10 years from an account that earns 4% compounded annually is $60,058.50.
Step-by-step explanation:
To determine how much would need to be deposited now to withdraw $2,400 at the end of each year for 10 years from an account that earns 4% compounded annually, we can use the present value formula for an annuity:
PMT x [(1 - (1 + r/n)^(-nt)) / (r/n)] = PV
Where:
PMT = the periodic payment
r = the annual interest rate (as a decimal)
n = the number of times the interest is compounded per year
t = the total number of years
PV = the present value (the amount to be deposited now)
In this case, we have:
PMT = $2,400
r = 4% = 0.04 (decimal)
n = 1 (compounded annually)
t = 10 years
Plugging these values into the formula, we get:
PV = $2,400 x [(1 - (1 + 0.04/1)^(-1*10)) / (0.04/1)]
PV = $2,400 x [(1 - 0.5537) / 0.04]
PV = $60,058.50
Therefore, the amount that would need to be deposited now to withdraw $2,400 at the end of each year for 10 years from an account that earns 4% compounded annually is $60,058.50.
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please help me again
Answers
Answer:
75
Step-by-step explanation:
[tex]10x^2 - 3x - 6, x=3\\[/tex]
substitute all values of x.
[tex]10(3)^2 -3(3)-6[/tex]
then simplify the equation.
[tex]90-9-6[/tex]
=[tex]75[/tex]
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Answer:
75
Step-by-step explanation:
10(3)^2-3x3(or 9)-6
by applying the compound angles and without using a calculator. Determine the value of sin105
Answers
Therefore, the value of sin105° is: = 0.7123
What is value?Value is a measure of the worth or importance of something. It is a subjective measure, as it is based on a person's individual beliefs, experiences, and values. Value can be seen in the monetary cost of a product or service, the quality of a product or service, the time spent on a task, or the amount of effort put into creating something. Value can also be intangible, such as the feeling of having achieved a goal or a sense of accomplishment. Value can be used to assess the worth of something, as well as to determine whether something is worth pursuing or investing in.
The value of sin105° can be determined by applying the compound angles formula. The formula states that the sine of an angle is equal to the product of the sine and cosine of the other two angles that make up the original angle.
For the angle 105°, the two other angles are 75° and 30°. Therefore, the sine of 105° can be calculated as follows:
sin105° = sin75° x cos30° + cos75° x sin30°
Using the values of sin75° and cos30° from a trigonometry table, we can calculate the sine of 105° as:
sin105° = 0.9659 x 0.5 + 0.2588 x 0.5
Therefore, the value of sin105° is:
sin105° = 0.7123
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Model Real Life A recipe calls for (1)/(2) cup of soy sauce. You only have a quarter cup for measuring. How many quarter cups do you need for the recipe?
Answers
This implies that since (1/2) cup is equal to 2 quarter cups, the amount of soy sauce required for the recipe is 2 quarter cups.
what is unitary method ?By first determining the value of one unit and afterwards multiplying or dividing to determine the value of another unit, the unitary method is a mathematical strategy used to solve proportional problems. The "single unit" or "one unit" technique is another name for it. The unitary technique entails segmenting a problem into manageable pieces before determining the value of one unit of the specified quantity.
given
We must establish how many quarter cups are equal to (1/2) cup in order to quantify soy sauce using a quarter cup.
Starting with the knowledge that 1 cup is equal to 4 quarter cups, we can write:
4 quarter glasses equal 1 cup.
(1/2) cup is therefore equal to:
1/2 cup equals 1/2 * 4 quarter cups, or 1/2 cup equals 2 quarter cups.
This implies that since (1/2) cup is equal to 2 quarter cups, the amount of soy sauce required for the recipe is 2 quarter cups.
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