Find the volume of the solid formed by revolving the region bounded by the graphs of y = x^2, x = 3, and y = 4 about the y-axis.
25 times pi divided by 2
97 times pi divided by 3
173 times pi divided by 3
None of these
Answers
The volume of the solid formed by revolving the region bounded by the graphs of the given functions is (25π) / 2.
What is integration?Integration is a method of adding or summing up the parts to find the whole. It is a reverse process of differentiation, where we reduce the functions into parts.
The region bound by all three functions is in the first quadrant. If this region were rotated about the y-axis, the solid formed would have washer-shaped cross-sections.
Therefore, we can use the Washer method: V = π∫ (ROuter)² - (RInner)² dy
R(outer) = 3 and R(inner) = √y (Note: the inner radius is formed by the function y = x2 which we solve in terms of y since we are rotating about the y-axis).
V = π∫ (3)2 - (√y)2 dy
(the limits of integration are from y = 4 to y = 9, so a = 4 and b = 9)
= π∫ 9 - y dy where a = 4 and b = 9
Integrate this definite integral to find the volume,
= (25π) / 2.
Hence, the volume of the solid formed by revolving the region bounded by the graphs of the given functions is (25π) / 2.
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Answer:
[tex]\textsf{A)} \quad \dfrac{25\pi}{2}[/tex]
Step-by-step explanation:
To find the volume of the solid of revolution created by rotating the region bounded by the curve y = x², x = 3, and y = 4 about the y-axis, we can use the method of cylindrical shells.
As the region of revolution is bounded by the graphs of two functions, y = x² and y = 4, the volume of a solid of revolution using cylindrical shells is given by the integral:
[tex]\displaystyle V=2\pi\int^b_ax\left(f(x)-g(x)\right)\;\text{d}x[/tex]
where:
a and b are the lower and upper bounds along the x-axis.f(x) is the upper function.g(x) is the lower function.The height of the shell, f(x) - g(x), is determined by the vertical distance between the curve y = x² and the line y = 4. Therefore, the height is:
[tex]f(x)-g(x)=x^2-4[/tex]
The upper bound of the integral is b = 3, since the region is bounded by x = 3.
The lower bound is the x-value of the point of intersection of the two functions y = x² and y = 4. These two functions intersect when x = 2. Therefore, the lower bound of the integral is a = 2.
Substitute these values into the integral:
[tex]\displaystyle V=2\pi\int^3_2x\left(x^2-4\right)\;\text{d}x[/tex]
Expand the brackets:
[tex]\displaystyle V=2\pi\int^3_2\left(x^3-4x\right)\;\text{d}x[/tex]
Evaluate the integral by using the power rule:
[tex]\boxed{\begin{minipage}{4 cm}\underline{Integrating $x^n$}\\\\$\displaystyle \int x^n\:\text{d}x=\dfrac{x^{n+1}}{n+1}+\text{C}$\end{minipage}}[/tex]
Therefore:
[tex]V=2\pi\left[\dfrac{x^4}{4}-\dfrac{4x^2}{2}\right]^3_2[/tex]
[tex]V=2\pi\left[\dfrac{x^4}{4}-2x^2\right]^3_2[/tex]
[tex]V=2\pi\left[\left(\dfrac{(3)^4}{4}-2(3)^2\right)-\left(\dfrac{(2)^4}{4}-2(2)^2\right)\right][/tex]
[tex]V=2\pi\left[\left(\dfrac{81}{4}-18\right)-\left(4-8\right)\right][/tex]
[tex]V=2\pi\left[\left(\dfrac{9}{4}\right)-\left(-4\right)\right][/tex]
[tex]V=2\pi\left(\dfrac{25}{4}\right)[/tex]
[tex]V=\dfrac{25\pi}{2}[/tex]
Therefore, the volume of the solid formed by revolving the region bounded by the graphs of y = x², x = 3, and y = 4 about the y-axis is:
[tex]\Large\boxed{\boxed{\dfrac{25\pi}{2}}}[/tex]
Related Questions
If y=x+2 were changed to y=x-1 how would the graph of the new function compare with the first one
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Answer: C shifted down :)
Answer:
it would be shifted down
Which equation has the steepest graph
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sorry if its wrong, then its c
Answer:
b
Step-by-step explanation:
If you run a 10 kilometer race in 43 minutes 30 seconds, what is your average time per mile? what is your average speed in miles per hour?
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In how many different ways can a panel of 12 jurors and 2 alternates be chosen from a group of 17 prospective jurors?
Answers
Explanation:
The 12 jurors from 17 prospective can be chosen in C17,12 ways.
That is: 17! / [12! (17 - 12)! ] = (17*16*15*14*13) / (5*4*3*2*1) = 6188 ways
The 2 alternates are chosen from a group of 17 - 12 = 5. Thas is C5,2
C5,2 = 5! / (2! 3!) = 5*4 / 2 = 10
So, the total number of ways is 6188 * 10 = 61880 ways.
Answer: 61880.
Final answer:
To find the different ways of choosing a panel of 12 jurors and 2 alternates from 17 prospective jurors, calculate the combinations separately (C(17, 12) for jurors and C(5, 2) for alternates) and multiply them, yielding a total of 61,880 different ways.
Explanation:
The question asks in how many different ways a panel of 12 jurors and 2 alternates can be chosen from a group of 17 prospective jurors. To solve this, we use the concept of combinations in mathematics. Since the order in which we select the jurors and alternates does not matter, we are dealing with combinations, not permutations.
First, we choose the 12 jurors from the 17, which is the combination of 17 taken 12 at a time. This can be calculated using the formula for combinations:
C(n, k) = n! / (k! * (n - k)!),
where 'n' is the total number of items, 'k' is the number of items to choose from, and '!' denotes factorial.
For the 12 jurors, we compute C(17, 12) which equals 17! / (12! * (17 - 12)!) = 17! / (12! * 5!) = 6188.
Next, we need to choose 2 alternates from the remaining 5 jurors. The calculation is C(5, 2) which equals 5! / (2! * (5 - 2)!) = 5! / (2! * 3!) = 10.
Since choosing jurors and alternates are independent events, we multiply the two combinations to get the total number of different ways to choose the panel:
6188 (ways to choose jurors) * 10 (ways to choose alternates) = 61,880 different ways.
Tyrell does sit-ups every day. On Monday, he did 4 sit-ups. On Tuesday, he did 8 sit-ups, and on Wednesday, he did 12 sit-ups. Based on the pattern, make a conjecture about the number of sit-ups he will do on Friday.
Answers
Answer: 20 sit ups.
Step-by-step explanation: if tyrell does situps everyday on monday he does 4 sit ups and tuesday 8 situps wendsday 12 situps thursday 16 situps friday 20 situps all u do to solve is add 4 every day.
your welcome.
What is the simplified form of the expression? -(4)^-2
Answers
Answer:
The given expression [tex]-\left(4\right)^{-2}[/tex] becomes [tex]-\frac{1}{16}[/tex]
Step-by-step explanation:
Given : Expression [tex]-\left(4\right)^{-2}[/tex]
We have to write the simplified form of the given expression [tex]-\left(4\right)^{-2}[/tex]
Consider the given expression [tex]-\left(4\right)^{-2}[/tex]
Apply exponent rule , [tex]\:a^{-b}=\frac{1}{a^b}[/tex]
[tex]4^{-2}=\quad \frac{1}{4^2}[/tex]
[tex]\frac{1}{4^2} =\frac{1}{16}[/tex]
Thus, The given expression [tex]-\left(4\right)^{-2}[/tex] becomes [tex]-\frac{1}{16}[/tex]
if a job pays $45,000 per year, you live in a state where your total tax burden is 22%, and you receive a weekly paycheck, what is your take-home pay from each paycheck?
Answers
Answer:
$190.38 per week
Step-by-step explanation:
45,000 x .22 (the tax) is 9,900
9,900 / 52 (weeks in a year) is 190.384
Rounded, your answer is $190.38 per week
Which situation CANNOT be represented by this equation? -13x+15=9
A-The water level in a tank measures 1515 inches and is decreasing every 1313 of a minute. At what rate, xx, does the water level need to decrease per minute to reach a depth of 99 feet?
B-The water level in a tank measures 99 inches and is rising 1313 of an inch every minute. How many minutes, xx, will it take for the water level to measure 1515 inches deep?
C-The water level in a tank measures 1515 inches and is decreasing 1313 of an inch every minute. How many minutes, xx, will it take for the water level to measure 99 inches deep?
Answers
THE ANSWER IS: PLEASE MARK AS BRAINLIST
The water level in a tank measures 9 inches and is rising 13 of an inch every minute.
How many minutes, x, will it take for the water level to measure 15 inches deep?
Chris's uncle has a model car. the model is 6 inches long and 3 inches wide. the actual car is 75 inches long. what is the car's actual width, if we assume the model's proportions are accurate? if necessary, round to the nearest tenth of an inch.
Answers
I believe the right answer would be 37.5
solve the system
x+3y=5
x+4y=6
A) (2,1)
B) (1,5)
C) (8,-1)
D (11,-2)
Answers
x + 4y = 6
x + 3y = 5
Subtract the two equations.
You get y = 1.
The only answer with a value of y = 1 is A
A is your answer.
Have a nice day! :)
Answer:
A(2,1)
Step-by-step explanation:
We are given that system of equation
[tex]x+3y=5[/tex]
[tex]x+4y=6[/tex]
We have to solve the system
Subtract equation first from equation second
Then, we get
[tex]y=1[/tex]
Substitute the value of y in equation I
Then, we get
[tex]x+3(1)=5[/tex]
[tex]x=5-3=2[/tex]
Hence, the option A is true.
Answer:A(2,1)
Two pounds is equivalent to approximately how many kilograms
Answers
Solve for the width in the formula for the area of a rectangle.
formulas
1. w = A - L
2. w = AL
3. w = L/A
4. w = A/L
Use the rewritten formula to find the width of a rectangle with an area of 42 square inches and a length of 16.8 inches.
1. 2.5 inches
2. 4.2 inches
3. 12.6 inches
4. 25.2 inches
Answers
A / L = W <==
when A = 42 and W = 16.8
A / L = W
42/16.8 = W
2.5 = W <=== width = 2.5 inches
W=A/L................
2.5 inches...........
Hour hand of a watch rotates 30 degrees every hour, how many rotations makes in 6 days
Answers
Each face of a pyramid is an isosceles triangle with a 82° vertex angle. What are the measures of the base angles? PLZ help
WILL GIVE BRAINLIEST
Answers
base angles are 49°
The value of the base angle will be 49 degrees.
What is an angle?The angle is defined as the span between two intersecting lines or surfaces at or close to the point where they meet.
A vertex is an angle that is connected to a vertex of an n-dimensional polytope in geometry. It refers to the angle created by two lines colliding in two dimensions.
Given that each face of a pyramid is an isosceles triangle with an 82° vertex angle.
The base angle will be calculated by using the formula below:-
Base angles = (180°-vertex)/2=(180-82)/2
Base angles = 98/2=49°
Base angles = 49°
Therefore, the value of the base angle will be 49 degrees.
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The area of a rectangle is 44 ft2 , and the length of the rectangle is 3 ft less than twice the width. find the dimensions of the rectangle.
Answers
Length: 2x-3
44=(x)(2x-3)
44=2x²-3x
0=2x²-3x-44
0=(2x-11)(x+4)
x=+11/2 or x=-4
Since dimensions cannot be negative, we have to use 11/2 or 5.5
The Width is 5.5ft , length is (2(5.5)-3)=8 ft
Hope I helped :)
Final answer:
To find the dimensions of a rectangle where the length is 3 ft less than twice the width and the area is 44 ft², we use the relationship between the width and length to set up a quadratic equation. After solving for the width, we use that width to find the length and confirm that the area is correct.
Explanation:
To find the dimensions of the rectangle with an area of 44 ft2, we need to set up an equation based on the given relationships. Let's denote the width of the rectangle as w and the length as l. According to the problem, the length is three feet less than twice the width, so we have l = 2w - 3. The area of the rectangle is the product of its width and length, so the equation to calculate the area is w × l = 44. Substituting the expression for l into this equation yields w × (2w - 3) = 44.
Solving this quadratic equation, we distribute the width and get 2w2 - 3w = 44. By moving all terms to one side, we have 2w2 - 3w - 44 = 0. Factoring the quadratic equation or using the quadratic formula will result in two values for w, where one will be the width and the corresponding l will be the length when plugged into l = 2w - 3.
Let's find the dimensions:
Width: wLength: 2w - 3Once the values are calculated, we can confirm that the width and length provide an area of 44 ft2.
Therefore, the width of the rectangle is w= 5.5 feet.
Now, we can find the length (l):
l=2w−3=2(5.5)−3=11−3=8
So, the length of the rectangle is l=8 feet.
Use multiplying by 1 to find an expression equivalent to 18/19 with a denominator of 76b.
Answers
so we multiply by 4b/4b ( = 1) :-
18 * 4b 72b
---------- = ----- <------ Answer
19 * 4b 76b
A gardener charges an initial fee of $37.50 plus a rate of $44.90 per hour. Write a function that shows the total fees for H hours of work
Answers
44.9 = slope/rate
In problems like these, just use y = initial rate + rate(variable)
In this case, the answer would be y = 37.5 + 44.9h
Identify an equation in slope-intercept form for the line parallel to y = 5x + 2 that passes through (–6, –1).
A. y = 5x + 29
B. y = –5x – 11
C. y= 1/5 x+1/6
D. y = 5x – 29
Answers
That's both option A and option D
Substitute (-6, -1) to find the value of the y-intercept
-1 = 5 * -6 + b
-1 = -30 + b
29 = b
So y = 5x + 29 (Option A)
The equation of line which is parallel to y = 5x + 2 and passing through (-6, -1) is y = 5x + 29. Then the correct option is A.
What is a linear equation?A relationship between two or more parameters that, when shown on a graph, produces a linear model. The degree of the variable will be one.
The linear equation is given as,
y = mx + c
Where m is the slope of the line and c is the y-intercept of the line.
Identify an equation in slope-intercept form for the line parallel to y = 5x + 2 that passes through (–6, –1).
The slope of the parallel lines is the same.
Then the equation will be
y = 5x + c
And the line is passing through (-6, -1), then the value of c will be
-1 = 5(-6) + c
c = 30 - 1
c = 29
Then the equation of line which is parallel to y = 5x + 2 and passing through (-6, -1) is y = 5x + 29.
Then the correct option is A.
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In the above figure, ∠AOB = 80°. What does ∠ACB equal?
A. 80°
B. 10°
C. 160°
D. 40°
Answers
ACB would be half of AOB
80/2 = 40
answer is D 40 degrees
The length of a rectangle is 2 in. more than twice its width. if the perimeter of the rectangle is 28 in., find the dimensions of the rectangle.
Answers
Width = w
Perimeter = 2length + 2width
28 = 2(2+w) + 2(w)
28= 4+ 2w + 2w
24= 4w
6 =w
width =6
length = 8
What is the coefficient of x7 in the expansion of (1 + x)11?
Answers
11C(something)(1)(x)⁷ ----> This gives the coefficient of x⁷
11C7(1)⁴(x)⁷ ----> You know that pascal's triangle is equal on both sides
11C7×1⁴ = coefficient of x⁷ (just ignore x⁷)
330x1⁴=330
This means that the coefficient of x⁷ is 330. You can double check by expanding the equation. If you do not understand what I did there, you can ask me how I got to the solution.
Hope I helped :)
Final answer:
Using the binomial theorem, the coefficient of x⁷ in the expansion of (1 + x)¹¹ is found by calculating '11 choose 7', which is 330.
Explanation:
The coefficient of x⁷ in the expansion of (1 + x)¹¹ can be found using the binomial theorem. To find the coefficient of x⁷, we look for the term that comes from choosing x seven times in the expansion. This corresponds to the 8th term, because the terms are indexed from 0, where the binomial coefficient is found by calculating "11 choose 7" or C(11, 7).
To calculate "11 choose 7", use the formula:
C(n, k) = n! / [k!(n-k)!], which gives us:
C(11, 7) = 11! / [7!(11-7)!] = 11! / (7!4!) = (11×10×9×8) / (4×3×2×1) = 330
Therefore, the coefficient of x⁷ in the expansion of (1 + x)¹¹ is 330.
Patricia donated 1 pint of blood to the blood bank.The lab assistant divided patricia's blood equally into 3 bags.How much blood was in each bag?
Answers
Each bag contained [tex]\(\frac{1}{3}\)[/tex] pint of blood.
To solve the problem, we need to divide the total amount of blood donated by Patricia equally into 3 bags. Patricia donated 1 pint of blood.
Since there are 3 bags, we divide 1 pint by 3 to find out how much blood is in each bag. The division is straightforward:
[tex]\[ \frac{1 \text{ pint}}{3 \text{ bags}} = \frac{1}{3} \text{ pint per bag} \][/tex]
Therefore, each bag contains [tex]\(\frac{1}{3}\)[/tex] pint of blood. This is the simplest form of the fraction, and there is no need to convert it to a different unit or simplify it further. The answer is [tex]\(\frac{1}{3}\)[/tex] pint per bag.
David gained 16 pounds over the winter. he went on a diet and lost 25 pounds. then he regained 4 pounds and weighed 177 pounds. how much did he weigh originally?
Answers
x+16-25+4 = 177
x-5=177
x= 177+5= 182 pound
David's original weight was 182 pounds, as we determined by retracing his weight changes in reverse order.
Explanation:To find out how much David originally weighed, we need to work backwards from his ending weight, taking his weight changes into account. Currently David weighs 177 pounds. However, before gaining the last 4 pounds, he would have weighed 177 - 4 = 173 pounds.
Going back further, before losing 25 pounds, he would have weighed 173 + 25 = 198 pounds. Finally, before David gained 16 pounds over the winter, his original weight would have been 198 - 16 = 182 pounds.
So, David's original weight was 182 pounds.
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Select all that justify the following statement.10 + x • 12 = 12x + 10
Answers
10 + x*12 = 12x+10
since the left side can be re-written as 12x+10 this does equal the right side
so any real number ( any value of x ) will work
A coach needs both small and large cones to set up an obstacle course. The coach knows that there are fewer than 30 cones in the storage room. The coach needs at least a dozen large cones.
Let x represent the number of small cones and y represent the number of large cones.
Which inequalities model the situation? Select each correct answer.
x+y≤30
x>0
x+y<30
y≥12
y>12
Answers
He needs AT LEAST a dozen (12), so that means that 12 will work or more than 12:
[tex]y \geqslant 12[/tex]
Answer: The correct options are,
x+y < 30
y ≥ 12
Step-by-step explanation:
Here, x represent the number of small cones and y represent the number of large cones.
∵ Large cones are at least 12,
⇒ y ≥ 12,
Now, total cones = x + y
∵ There are fewer than 30 cones,
I.e. x + y < 30,
Now, number of cones can not be negative,
⇒ x, y ≥ 0
Formulate the recursive formula for the following geometric sequence.
{-16, 4, -1, ...}
Answers
Answer: divide by -4
Step-by-step explanation:
is 2xy = -8 a function
Answers
You can find a value of x and a value for y that would make that true, so YES it is a function.
Example:
2(2)(-2) = -8
The equation 2xy = -8 is a rectangular hyperbola which is an example of the function.
What is a function?A function is an assertion, concept, or principle that establishes an association between two variables. Functions may be found throughout mathematics and are essential for the development of significant links.
When the transverse and conjugate axes of a hyperbola are both the same length, the diagnosis and appropriate are said to be rectangular.
The equation is given below.
2xy = -8
Separate the variable 'y', then we have
2xy = -8
xy = -4
y = -4/x
The equation 2xy = -8 is a rectangular hyperbola which is an example of the function.
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A ship leaves port at 1:00 p.m. and travels s35°e at the rate of 26 mi/hr. another ship leaves the same port at 1:30 p.m. and travels s20°w at 20 mi/hr. approximately how far apart are the ships at 3:00 p.m.?
Answers
A fisherman separated 1/10 of his catch into 2 equal bins.What fraction is the entire catch was in each bin?A. 20/1 B. 5/1 C. 1/20 D. 1/5 Plz help me plz i need help with this right now.
Answers
1/10 divided by 2 is 1/20
Given the parent functions f(x) = 3x + 2 and g(x) = 5x − 10, what is g(x) − f(x)?
g(x) − f(x) = 5x − 8 − 3x
g(x) − f(x) = 5x − 12 − 3x<--------------------
g(x) − f(x) = 3x − 8 − 5x
g(x) − f(x) = 3x + 12 − 5x
Points earned on this question: 2
Answers
(PROOF)
f(x) = 3x + 2
g(x) = 5x - 10
g(x) - f(x) = (5x - 10) - (3x + 2)
g(x) - f(x) = (5x - 3x) + (-10 - 2)
g(x) - f(x) = 2x - 12
Answer:
Option B. g(x) - f(x) = 5x - 3x - 12
Step-by-step explanation:
In this question the given functions are
f(x) = 3x + 2
g(x) = 5x - 10
Now we have to calculate the value of g(x) - f(x)
g(x) - f(x) = (5x - 10) - (3x +2)
= 5x - 10 - 3x - 2
= (5x - 3x) - (10 + 2)
= 5x - 3x - 12
Option B. (5x - 3x - 12) is the answer.