The volume of the solid is -5.54 cubic units.
The region is bounded by the parabola x = 1 + (y-2)^2 and the line x = 2. To use the method of cylindrical shells, we imagine rotating this region around the x-axis, which gives us a solid shape.
The shell has thickness Δx and height y, and is located at a distance x from the y-axis. To find the volume of the solid, we'll add up the volumes of all the cylindrical shells.
The volume of a cylindrical shell is given by the formula V = 2πrhΔx, where r is the radius of the shell and h is its height. In this case, the radius of the shell is x - 2 (the distance from the y-axis to the edge of the region), and the height is y. So we have:
V = 2π(x - 2)yΔx
To find the total volume of the solid, we integrate this expression over the range of x values that corresponds to the region we're interested in. The region is bounded by x = 1 + (y-2)^2 and x = 2, so we integrate with respect to x from x = 1 + (y-2)^2 to x = 2:
V = ∫[1+(y-2)²]^2 2π(x - 2)y dx
To evaluate this integral, we can use the substitution u = y - 2, which gives us:
V = ∫1^0 2π(u^2 + 3) (1 + u)^2 du
This integral can be evaluated using the power rule and the substitution v = 1 + u:
V = 2π/5 [(1 + u)^5 - (1 + u)^3] from 1 to 0
V = 2π/5 [1 - 32/5]
V = 2π/5 ( -22/5)
V = - 8.8π/5
So the volume of the solid is approximately -5.54 cubic units. Note that the negative sign means that the solid is oriented in the opposite direction from what we might expect - in other words, it has a hole in it.
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A flower garden is shaped like a circle. Its diameter is 30yd. A ring-shaped path goes around the garden. The width of the path is 5yd. The gardener is going to cover the path with sand. If one bag of sand can cover 6yd, how many bags of sand does the gardener need? Note that sand comes only by the bag, so the number of bags must be a whole number. (Use the value 3.14 for pi.)
The number of bags of sand that has to be used would be 92
How to solve for the bags of sanddiameter = 30 yards
width of path = 5 yards
diameter with path
30 + 2 * 5
= 40 Yd
area of path = area of garden with path - ara of garden
= 3.14 * 40² / 4 - 3.14 * 30² / 4
= 1256 - 706.5
= 549.5
area covered by bag = 549.5 / 6
= 91.58
= 92 bags
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6 There are 2 blue chips for every 3 red chips. If there are 20 total chips, how many blue chips and red chips are in the bucket?
Answer:
12 blue chips 8 red chips.
Step-by-step explanation:
Let's call the number of blue chips "b" and the number of red chips "r". We know that 2 blue chips for every 3 red chips, so we can set up the following equation:
2b = 3r
And we know that there are 20 total chips, so:
b + r = 20
Now we can use substitution to find the value of b. We'll substitute the expression for r from the first equation:
2b = 3(20 - b)
Expanding the right side:
2b = 60 - 3b
Combining like terms:
5b = 60
And finally, solving for b:
b = 12
So there are 12 blue chips and 20 - 12 = 8 red chips.
Solve 4 x + 16 = y for x if y = 35
Answer:
Below
Step-by-step explanation:
So you have this:
4x + 16 = 35 subtract 16 from both sides of the equation
4x = 19 Divide by 4
x = 19/4
Answer:
The value of x is
[tex]x = 4.75[/tex]
Step-by-step explanation:
Greetings!!!
Given expression [tex]4x + 16 = y[/tex]Thus, the value of y is given as 35, substitute known value of y in expression and solve for x.
[tex]4x + 16 = 35[/tex]
Move the constant to the right
[tex]4x = 35 - 16 \\ 4x = 19[/tex]
Divide both sides of the expression by 4
[tex] \frac{4x}{4} = \frac{19}{4} [/tex]
solve
[tex]x = \frac{19}{4} [/tex]
or
[tex]x = 4 \frac{3}{4} [/tex]
or
[tex]x = 4.75[/tex]
Hope it helps!!!
Write the first five terms of the arithmetic sequence. Find the common difference d and write the nth term of the sequence as a function of n.
a1
= 300, ак + 1 = ак - 20
The first five terms of the arithmetic sequence are 300, 280, 260, 240, 220 and the common difference d is 20
What is arithmetic sequence?An arithmetic sequence in algebra is a sequence of numbers where the difference between every two consecutive terms is the same.
Given is an arithmetic sequence, [tex]a_{k+1} = a_k-20[/tex] and [tex]a_1 = 300[/tex],
we need to find the first five terms,
a₂ = a₁-20
a₂ = 300-20
a₂ = 280
a₃ = a₂-20
a₃ = 260
a₄ = a₃ - 20
a₄ = 240
a₅ = a₄ - 20
a₅ = 220
aₙ = aₙ₊₁ - 20
d = 20
Hence, the first five terms of the arithmetic sequence are 300, 280, 260, 240, 220 and the common difference d is 20
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What is the correct answer answer asap for brainlist
According to the information, it can be inferred that the option that the quote explains is: The outcome... is situational irony because the opposite of what was expected is what happening.
How to identify the correct option that explains the quote?To identify the correct option that explains the quote we must read the text carefully and the options. Once we have read and understood this information, we can infer which option explains the quote.
In accordance with the above, option A would be the one that explains the quote because options B and C by discarding are fragments of the quote. So these fragments could not explain the same quote that is being used in the text. On the contrary, the paragraph of option A does explain the situation clearly.
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What are the features of the function f(x) = -2 log_3 (x + 8) graphed below?
The domain of the function is set of all real numbers except 8. Range of the function is set of all real numbers. The given function does not have asymptotes.
What is a function?A mathematical phrase, rule, or law that establishes the link between an independent variable and a dependent variable (the dependent variable).
Domain: The domain of the function is the set of all real numbers greater than -8, since the argument of the logarithm, x + 8, must be positive.
Range: The range of the function is all real numbers.
Asymptotes: The function does not have any vertical asymptotes.
Transformation: The function is a reflection of the standard logarithmic function over the x-axis, with a vertical scaling factor of -2. This means that the function will start from the origin and increase without bound.
Behavior near x = -8: As x approaches -8 from the right, the value of the function approaches negative infinity. As x approaches -8 from the left, the value of the function approaches positive infinity.
The domain of the function is set of all real numbers except 8. Range of the function is set of all real numbers. The given function does not have asymptotes.
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The inequality of the form |x-a| k that has the solution set (5,13) is
The inequality is |x-9| ≥ 4, and its solution set is (5,13).
What is inequality?An Inequality is defined as mathematical statements that have a minimum of two terms containing variables or numbers that are not equal.
The inequality of the form |x-a| ≥ k means that the distance between x and a is greater than or equal to a certain value, k.
If the solution set is (5,13), it means that x is a number between 5 and 13, but not including 5 or 13. In other words, 5 < x < 13.
To find the value of a and k, we need to look for the midpoint of the solution set, which is (5+13)/2 = 9.
The distance between 9 and either endpoint (5 or 13) is 4, which is the value of k.
Thus, the inequality is |x-9| ≥ 4, and its solution set is (5,13).
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Find the value of x
Answer:8
Step-by-step explanation:
Because Angle GJH is 90 degrees, Angle HJZ is also 90 degrees. (Because all right angles are 90 degrees)
Now, find Angle WJZ
90 - 18 = 72 degrees
By the Vertical Angles Theorem, Angle GJL is congruent to Angle ZJW
Now solve for x
72 = 9x
x = 8
A function is given.
g(x) =
2
x
; x = 1, x = a
(a) Determine the net change between the given values of the variable.
(b) Determine the average rate of change between the given values of the variable.
According to the given function net change and average rate of change between the given values of the variable.
A. 2
B. 2/a
What is the net change?A rate of change's integral is taken into account by the net change theorem. According to this, a quantity's new value equals its initial value plus the integral of its rate of change whenever it changes. There are two methods to formulate the equation. The second is more widely known and is just the definite integral.
What does a function's net change look like?The (definite) integral of a function's derivative, in other words, represents the overall change in a function. In specifically, the integral of velocity is the net distance travelled (final position minus initial position). The integral of acceleration is the difference between the final velocity and the starting velocity.
According to the given information:g(x) =2/x
x = 1
x = a
A.
g(x) =2/x
g(1) =2/x
g(1) =2
B.
g(x) =2/x
g(a) =2/x
g(a) =2/a
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A scientist has two solutions, which she has labeled Solution A and Solution B. Each contains salt. She knows that Solution A is 65% salt and Solution B is 90% salt. She wants to obtain 130 ounces of a mixture that is 70% salt. How many ounces of each solution should she use?
On solving the provided question we can say that equations are = .60x + .85y = .75(180)
What is equation?A mathematical equatiοn is a formula that joins two statements and uses the equal symbol (=) tο indicate equality. A mathematical statement that establishes the equality of twο mathematical expressiοns is known as an equation in algebra. Fοr instance, in the equatiοn 3x + 5 = 14, the equal sign places the variables 3x + 5 and 14 apart.
The relationship between the twο sentences on either side of a letter is described by a mathematical formula. Often, there is οnly one variable, which also serves as the symbοl. for instance, 2x – 4 = 2.
t x = the number of ounces of Sοlution A
y = the number of ounces of Sοlution B
x + y = 180
y = 180 - x
Solving equation -
0.60 + 0.85y = 0.75(180)
0.60 + 0.85y = 135
60x + 85y = 13500
60x + 85(180-x) = 13500
60x + 15300 - 85x = 13500
-25x = -1800
x = 72ounces
y = 180 - 72
y = 108 ounces
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Welding A welder finds that 2.1 cu ft of acetylene gas is needed to make one bracket. How much gas will be needed to make 27 brackets?
56.7 cubic feet of acetylene gas will be needed to make 27 brackets.
What is Total price?
In order to calculate the total price of something is by using this formula. Purchase Price = Cost Price + Margin.
To find out how much gas will be needed to make 27 brackets, we need to multiply the amount of gas required for making one bracket by the number of brackets to be made.
So, the total amount of gas needed to make 27 brackets would be:
2.1 cu ft * 27 = 56.7 cu ft
Therefore, 56.7 cubic feet of acetylene gas will be needed to make 27 brackets.
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The points K (1,3), L(-4,-3), M (2,-8), and N (7,-2) form quadrilateral KLMN. Where would these points be on the graph?
The location of the points for quadrilateral KLMN have been plotted in the graph shown in the image attached below.
What are the types of quadrilateral?In Geometry, there are different types of quadrilateral and these include the following:
ParallelogramSquaresRectangleRhombusTrapezoidNext, we would use an online graphing calculator to construct (plot) quadrilateral KLMN on a cartesian coordinate based on the given vertices. Also, you should enter the value for each vertices of quadrilateral KLMN on the x-coordinate and y-coordinate respectively as shown in the table.
By critically observing the graph of quadrilateral QRST, we can reasonably infer and logically deduce that it represents a parallelogram.
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4x
Store A $25 for a 45 minute lesson
Store B. $30 for a 1 hour lesson
Store C $45 for a 1.5 hour lesson
Store D. $80 for three 1 hour lessons
Julle wants to purchase swim lessons for her son. There are four different stores that offer swim lessons. Based on the lowest cost per minute, which store has the BEST dea
4x A
4x B
x
C
4x D
Store B
Store A
Store D
Store C
The store with the best deal is given as follows:
Store D.
How to obtain the store with the best deal?The store with the best deal is obtained applying the proportions in the context of the problem.
The best deal is the store with the lowest cost per hour, and the cost per hour is obtained dividing the total cost by the number of hours.
Hence the hourly cost for each store is obtained as follows:
Store A: 25/0.75 = $33.3 per hour. (45 minutes = 45/60 hour = 0.75 hour).Store B: $30/1 = $30 per hour.Store C: $45/1.5 = $30 per hour.Store D: $80/3 = $26.7 per hour -> best deal.More can be learned about proportions at https://brainly.com/question/24372153
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can you please please please help me w this question!!! 40 points!!
which number line shows the solution to x + 8 > 15
7 is not included and hence it is depicted on the number line using an open dot. Hence, solution to the inequality is option B.
What is inequality?In mathematics, inequalities specify the connection between two non-equal numbers. Equal does not imply inequality. Typically, we use the "not equal sign (≠)" to indicate that two values are not equal. But several inequalities are utilized to compare the numbers, whether it is less than or higher than.
The given inequality is:
x + 8 > 15
Subtracting 8 on both sides of the equation:
x + 8 - 8 > 15 - 8
x > 7
Here, 7 is not included and hence it is depicted using an open dot.
Hence, option B is the correct answer.
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hand consists of 3 cards from a well-shuffled deck of 52 cards.
a. Find the total number of possible 3-card poker hands.
b. A black flush is a 3-card hand consisting of all black cards. Find the number of possible black flushes.
c. Find the probability of being dealt a black flush.
a)
The total number of possible 3-card poker hands is 22100
b)
The number of possible black flushes is 2600
c)
The probability of being dealt a black flush is 0.1176
What is probability?It is the chance of an event to occur from a total number of outcomes.
The formula for probability is given as:
Probability = Number of required events / Total number of outcomes.
We have,
Number of cards = 52
Number of cards in hand = 3
Combination formula.
[tex]^nC_r[/tex] = n! / r! (n -r)!
a)
The total number of possible 3-card poker hands.
= [tex]^{52}C_3[/tex]
= (52 x 51 x 50) / (3 x 2)
= 22100
b.
Number of black cards (spades and clubs) = 13 + 13 = 26
The number of possible black flushes.
= [tex]^{26}C_3[/tex]
= (26 x 25 x 24) / (3 x 2)
= 2600
c.
The probability of being dealt a black flush.
From (a) and (b)
= 2600/22100
= 0.1176
Thus,
The total number of possible 3-card poker hands = 22100
The number of possible black flushes = 2600
The probability of being dealt a black flush = 0.1176
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6. Part D
Another method of proving diagonals of a parallelogram bisect each other
uses a coordinate grid.
A.
P (r, t)
S (0, 0)
R (s, 0)
What could be shown about the diagonals of parallelogram PQRS to com
the proof?
PR and SQ have the same length.
PR is a perpendicular bisector of SQ.
PR and SQ have the same midpoint.
D. Angles formed by the intersection of PR and SQ each measu
B.
Q
(r+s, t)
Answer:
PR and SQ have the same midpoint.
Step-by-step explanation:
PR and SQ have the same midpoint.
Diana wants to build a rectangular
pen for her goats. The length of the pen
should be at least 50 feet, and the perimeter
of the pen should be no more than 190 feet.
What is a viable solution for the dimensions
of the pen? (Lesson 6-5)
A. 29 feet by 40 feet
B. 29 feet by 76 feet
C. 29 feet by 65 feet
D. 29 feet by 29 feet
Copyright © McGraw Hill
Answer:
29 feet by 40 feet to get a viable dimension for the goats pen
Simplify:
9 – 3 x 2 + 4
Answer:
2
:
5STEP
2
:
Trying to factor as a Difference of Squares:
2.1 Factoring: 5-3x2
Theory : A difference of two perfect squares, A2 - B2 can be factored into (A+B) • (A-B)
Proof : (A+B) • (A-B) =
A2 - AB + BA - B2 =
A2 - AB + AB - B2 =
A2 - B2
Note : AB = BA is the commutative property of multiplication.
Note : - AB + AB equals zero and is therefore eliminated from the expression.
Check : 5 is not a square !!
Ruling : Binomial can not be factored as the
difference of two perfect squares
Equation at the end of step
2
:
5 - 3x2 = 0
STEP
3
:
Solving a Single Variable Equation:
3.1 Solve : -3x2+5 = 0
Subtract 5 from both sides of the equation :
-3x2 = -5
Multiply both sides of the equation by (-1) : 3x2 = 5
Divide both sides of the equation by 3:
x2 = 5/3 = 1.667
When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:
x = ± √ 5/3
The equation has two real solutions
These solutions are x = ±√ 1.667 = ± 1.29099
Two solutions were found :
x = ±√ 1.667 = ± 1.29099
Which decimal is represented by the red portion of the picture?
Answer:
D
Step-by-step explanation: I did it
Question 4 of 10
Which of the following is a root of the polynomial shown below?
f(x) = x³ + 2x²-x-2
OA. 0
OB. 2
O C. 1
OD. 3
SUBMIT
Answer:
1
Step-by-step explanation:
f(x)=x3+2x2-x-2
...................
The Roots of the given polynomials shown below f(x) = x^3+2x^2-x-2 are -1, 1, -2.
What is a polynomial?They are mathematical expressions involving variables raised with non negative integers and coefficients(constants who are in multiplication with those variables) and constants with only operations of addition, subtraction, multiplication and nonnegative exponentiation of variables involved.
Example: x² + 3x + 3 is a polynomial.
Since, The given polynomials is,
f(x) = x³ + 2x²-x-2
Now, Solve as;
f(x) = x³ + 2x²-x-2
f (x) = x² (x + 2) - 1 (x + 2)
f (x) = (x² - 1) (x + 2)
Hence, The Roots of the polynomials : -1, 1, -2
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Find the 66th
term in the following
arithmetic sequence
-92, -85, -78, -71, ...
Answer:
The [tex]66[/tex]th term is [tex]363[/tex].
Step-by-step explanation:
To find the [tex]n[/tex]th term, the formula is [tex]7n - 99[/tex]. Since we want to find the [tex]66[/tex]th term, we can plug [tex]n[/tex] for [tex]66[/tex]. So our expression is now [tex]7 \cdot 66 - 99 =[/tex] [tex]66[/tex]th term. Solving this we have
[tex]462 - 99 = 66\text{th term}\\363 = 66\text{th term}\\[/tex]. Therefore, the [tex]66[/tex]th term is [tex]\boxed{363}[/tex].
Answer:
a₆₆ = 363
Step-by-step explanation:
the nth term of an arithmetic sequence is
[tex]a_{n}[/tex] = a₁ + (n - 1)d
where a₁ is the first term and d the common difference
here a₁ = - 92 and d = a₂ - a₁ = - 85 - (- 92) = - 85 + 92 = 7 , then
a₆₆ = - 92 + (65 × 7)
= - 92 + 455
= 363
simplify the expression (m^6)^-2/3 and write your answer using a positive exponent
We have simplified the expression [tex]\mathrm{(m)^6^{(-2/3)}}[/tex] and written answer using a positive exponent that is [tex]\mathrm{\dfrac{1}{m^4}}[/tex] .
What is expression?An expression in mathematics is a combination of numbers, variables, and functions (such as addition, subtraction, multiplication or division etc.)
It is possible to compare expressions to phrases. In language, a phrase by itself may contain an action, but it does not constitute a complete sentence.
The following are some examples of expressions and word phrases that they relate to:
Expression: 3 + 7
Given
⇒ [tex]\mathrm{(m)^6^{(-2/3)}}[/tex]
⇒ [tex]\mathrm{(m)^{(-12/3)}}[/tex]
⇒ [tex]\mathrm{(m)^{(-4)}}[/tex]
⇒ [tex]\mathrm{\dfrac{1}{m^4}}[/tex]
We have simplified the expression [tex]\mathrm{(m)^6^{(-2/3)}}[/tex] and written answer using a positive exponent that is [tex]\mathrm{\dfrac{1}{m^4}}[/tex]
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13. Solve the inequality.
12 ≤ 2x 4 <10
-
-10 ≤x < 4
-2
-4 ≤x≤ 3
-4 ≤x < 7
The solution of the inequality is -4≤x<7. Therefore, option D is the correct answer.
What are inequalities?Inequalities are the mathematical expressions in which both sides are not equal. In inequality, unlike in equations, we compare two values. The equal sign in between is replaced by less than (or less than or equal to), greater than (or greater than or equal to), or not equal to sign.
The given inequality is -12≤2x-4<10.
Here, -12≤2x-4
Add 4 on both the sides of an inequality, we get
-8≤2x
Divide 2 on both the sides of an inequality, we get
-4≤x
Now, 2x-4<10
Add 4 on both the sides of an inequality, we get
2x<14
Divide 2 on both the sides of an inequality, we get
x<7
So, the solution of the inequality is -4≤x<7
Therefore, option D is the correct answer.
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Only using positive exponents, what is the simplified form of the expression p−6q7r/p2q−2
Step-by-step explanation:
Here is it. I hope it will help
Line CD passes through points (0, 2) and (4, 6). Which equation represents line CD?
Oy=2x-2
Oy = 2x + 2
Oy=x+2
Oy=x-2
The required equation of the line representing line CD is y = x + 2. Option C is correct.
The equation of the line passing through two points is given by the equation,
y - y₁ = (y₂ - y₁) / (x₂ - x₁) (x - x₁) - - - -- -(1)
Here,
Line CD passes through points (0, 2) and (4, 6)
Let (0, 2) and (4, 6) equal to (x₁, y₁) and (x₂, y₂) respectively.
Substitute this point in equation 1 to get the equation of the line representing CD,
So,
y - 2 = (6 - 2)/(4-0) (x - 0)
y = x + 2
Thus, the required equation of the line representing line CD is y = x + 2. Option C is correct.
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What is the volume 
Answer:
384 cm
Step-by-step explanation:
show that the following function are continuos and discontinuos
A.(f×)={×+1 ,for×less than or equai to 2
{ 2×–1 ,for 1<×<2
{×–1 ×<1
f(x) is discontinuous.
Since LHS ≠ RHS ≠ f(x).
What is a function?A function is a relationship between inputs where each input is related to exactly one output.
Example:
f(x) = 2x + 1
f(1) = 2 + 1 = 3
f(2) = 2 x 2 + 1 = 4 + 1 = 5
The outputs of the functions are 3 and 5
The inputs of the function are 1 and 2.
We have,
f(x) = x + 1, for x ≤ 2
f(x) = 2x - 1, for 1 < x < 2
f(x) = x - 1, for x < 1
Now,
A x = 1
LHS of f(x) = [tex]\lim_{x\to1^-}[/tex] f((h - 1)) = [tex]\lim_{h \to 0}[/tex] (h - 1 - 1) = 0 -2 = -2
RHS of f(x) = [tex]\lim_{x\to1^+}[/tex] f(h + 1) = [tex]\lim_{h \to 0}[/tex] (2(h + 1) - 1) = 2h + 2 - 1 = 0 + 1
f(1) = x + 1 = 1 + 1 = 2
We see that,
LHS ≠ RHS ≠ f(x)
So,
f(x) is not continuous, it is discontinuous.
Thus,
f(x) is discontinuous.
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The complete question.
Show that the following functions are continuous or discontinuous at x = 1.
f(x) = x + 1, for x ≤ 2
f(x) = 2x - 1, for 1 < x < 2
f(x) = x - 1, for x < 1
Let (8,-6) be a point on the terminal side of an angle 0 in standard position. Find the exact value of cos 0.
The exact value of cosine is 4/5
How to determine the exact value of cosineFrom the question, we have the following parameters that can be used in our computation:
(8,-6) be a point on the terminal side of an angle
Using the above as a guide, we have the following:
Cos(angle) = x/(√x² + y²)
In this case
x = 8 and y = -6
substitute the known values in the above equation, so, we have the following representation
Cos(angle) = 8/(√8² + (-6)²)
So, we have
Cos(angle) = 8/10
Simplify
Cos(angle) = 4/5
Hence, the value is 4/5
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One snowy day 3/4 Ms staples class was late to school 1/3 of the late students brought notes what fraction of Mrs. Stanford class bra late notes that day
The fraction of Mrs. Stanford's class that came late and brought notes that day was 1 / 4
How to find the fraction of students ?To find the fraction of Mrs. Stanford's class that came late and brought notes that day, the relevant formula would be:
= Fraction of students who came late x Fraction of late students who brought notes
Fraction of students who came late = 3 / 4
Fraction of students who were late and brought notes = 1 / 3
The fraction of total students who were late and brought notes were ;
= 3 / 4 x 1 / 3
= 3 / 12
= 1 / 4
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