The volume of the solid generated by rotating the region bounded by the curves y = 0, y = x² + x, x = 2 and x = 0 about the y-axis is 40π/3 cubic units.
How to find the volume of the solid generated?To find the volume of the solid generated, we use the shell method since the cross-sections are parallel to the axis of rotation.
So, [tex]V = 2\pi\int\limits^a_b {xy} \, dx[/tex]
Now given that the region bounded by the curves y = 0, y = x² + x, x = 2 and x = 0 about the y-axis.
So, we integrate from x = 0 to x = 2.
[tex]V = 2\pi\int\limits^a_b {xy} \, dx\\= 2\pi\int\limits^2_0 {x(x^{2} + x)} \, dx\\= 2\pi\int\limits^2_0 {(x^{3} + x^{2} )} \, dx\\= 2\pi\int\limits^2_0 {x^{3} + 2\pi\int\limits^2_0 x^{2} } \, dx\\= 2\pi[\frac{x^{4} }{4}]_{0}^{2} + 2\pi[\frac{x^{3} }{3}]_{0}^{2} \\= 2\pi[\frac{2^{4} - 0^{4} }{4}}]+ 2\pi[\frac{2^{3} - 0^{3} }{3}]\\= 2\pi[\frac{16 - 0}{4}}]+ 2\pi[\frac{8 - 0}{3}]\\= 2\pi[\frac{16}{4}}]+ 2\pi[\frac{8}{3}]\\= 2\pi (4)+ [\frac{16\pi}{3}]\\= 8\pi + [\frac{16\pi}{3}]\\[/tex]
[tex]= \frac{24\pi + 16\pi}{3} \\= \frac{40\pi}{3}[/tex]
So, the volume of the solid generated by rotating the region bounded by the curves y = 0, y = x² + x, x = 2 and x = 0 about the y-axis is 40π/3 cubic units.
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Please help with pictures below.
The proportion is solved and the variables are a/b = c/d
Given data ,
Let the proportion be represented as A
Now , the value of A is
a / b = c / d
On simplifying the equation , we get
( a + b ) / b = ( c + d ) / d
On dividing the numerator of the fraction by the denominator , we get
( a / b ) + ( b / b ) = ( c / d ) + ( d / d )
On further simplification , we get
( a / b ) + 1 = ( c / d ) + 1
Subtracting 1 on both sides , we get
( a / b ) = ( c / d )
Hence , the proportion is ( a / b ) = ( c / d )
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Given f(x)=4x^2 - 1 and g (x) = 2x- 1 , find each function
1. ( f+g )( x )
2. ( fg ) ( x )
3. ( f^n ) ( x )
The values of the composite functions are
(1) (f + g)(x) = 4x² + 2x - 2
(2) (fg)(x) = 8x³ - 4x² - 2x + 1
(3) (fⁿ)(x) = (4x² - 1)ⁿ
How to evaluate the composite functionsFrom the question, we have the following functions that can be used in our computation:
f(x) = 4x² - 1
g(x) = 2x - 1
Using the above as a guide, we have the following:
(f + g)(x) = f(x) + g(x)
(f + g)(x) = 4x² - 1 + 2x - 1
Evaluate the like terms
(f + g)(x) = 4x² + 2x - 2
(fg)(x) = f(x) * g(x)
(fg)(x) = (4x² - 1) * (2x - 1)
Evaluate the products
(fg)(x) = 8x³ - 4x² - 2x + 1
(fⁿ)(x) = (f(x))ⁿ
So, we have
(fⁿ)(x) = (4x² - 1)ⁿ
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100 Points! Algebra question. Photo attached. Please show as much work as possible. Thank you!
The probability of the digestive tract disease, given a positive test result, is 45%.
A. Two-Way Frequency Table:
Let's create a two-way frequency table to represent the situation:
Test Positive Test Negative Total
Disease Present A B 5,000
Disease Not Present C D 95,000
Total 5,000 95,000 100,000
In this table:
A represents the number of sheep that have the disease and test positive.B represents the number of sheep that have the disease but test negative.C represents the number of sheep that do not have the disease but test positive.D represents the number of sheep that do not have the disease and test negative.B. Probability of Disease Given Positive Test Result:
P(Disease | Positive)
= (P(Positive | Disease) x P(Disease)) / P(Positive)
= 0.94 x 0.05 / P(Positive)
To calculate P(Positive), we can use the law of total probability:
P(Positive) = P(Positive | Disease) x P(Disease) + P(Positive | No Disease) x P(No Disease)
So, P(Positive | No Disease) = 1 - P(Test Negative | No Disease)
= 1 - 0.94 = 0.06
P(No Disease) = 1 - P(Disease) = 1 - 0.05 = 0.95
Now, P(Positive) = (0.94 x 0.05) + (0.06 x 0.95)
= 0.047 + 0.057
= 0.104
Finally, we can calculate P(Disease | Positive):
P(Disease | Positive) = (0.94 x 0.05) / 0.104
= 0.047 / 0.104
≈ 0.452
Therefore, the probability of the digestive tract disease, given a positive test result, is 45%.
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What are the zeros of the function
Answer: i think c
Step-by-step explanation:
Compare:
58,565 ____ 58,566
Answer:
58,565 < 58,566
Step-by-step explanation:
You have to determine which number is greater. I usually start comparing from the farthest left digit (In this case, the ten thousands place.) and work my way up
hope this helped :)
Can someone give me the answer
Between 40 and 60 minutes the target heart rate strictly increasing then strictly decreasing. Therefore, option B is the correct answer.
From the given graph, in the first curve between the interval 0-10 speed is increasing, between the interval 10-30 speed is constant and between the interval 30-40 speed is decreasing.
In the second curve between the interval 40-50 speed is increasing, between the interval 50-60 speed is decreasing, between the interval 60-65 speed is increasing and between the interval 65-73 speed is constant.
In the third curve between the interval 73-85 speed is increasing, between the interval 85-92 speed is constant and between the interval 92-100 speed is decreasing.
Therefore, option B is the correct answer.
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Sneha went to a shopkeeper to purchase sugar.The dishonest shopkeeper uses a weight of 990 g for measuring 1 kg . If Sneha bought 5/2 kg of sugar, find the amount of sugar she actually got.
Sneha actually got 2475 g of sugar instead of 2500 g.
We have,
If the shopkeeper uses a weight of 990 g for measuring 1 kg, it means that he is using a weight that is 10 g less than the actual weight of 1 kg.
So, the actual weight of 1 kg is 1000 g.
Now,
If Sneha bought 5/2 kg of sugar, she should have received:
= (5/2) x 1000 g
= 2500 g
But since the shopkeeper uses a weight of 990 g for measuring 1 kg, the weight of 5/2 kg would be:
= (5/2) x 990 g
= 2475 g
Thus,
Sneha actually got 2475 g of sugar instead of 2500 g.
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Question Find the distance between the two points. (12, 5), (−12, −2)
Answer:
Distance between points is 25 Units.
Step-by-step explanation:
To find the distance between two points, we can use the distance formula. The formula is:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Given the two points: (12, 5) and (-12, -2), we can assign the values as follows:
(x1, y1) = (12, 5)
(x2, y2) = (-12, -2)
Plugging the values into the distance formula, we get:
d = sqrt((-12 - 12)^2 + (-2 - 5)^2)
= sqrt((-24)^2 + (-7)^2)
= sqrt(576 + 49)
= sqrt(625)
= 25
Therefore, the distance between the points (12, 5) and (-12, -2) is 25 units.
in a bag if marbles 1/2 are blue 1/6 are green 1/12 are yellow you pick a marble with out looking what color marble are you most likely going to choose
Answer: Blue
Step-by-step explanation:
Half of the entire freaking bag is blue,it states it in the question.If there is so many,unless by pure blind luck you will get that.
I hope it helps!
a librarian has weekends off and has 10 paid vacation days per year, including holidays that fall on weekends. If his salary is 32,500 per year, what is his pay per workday?
his pay per workday will be $127.45.
How to calculate interest?Thus, the formula for calculating simple interest is J = C i t, where J is interest, C is principal, i is interest rate, and t is time. That is, J represents the amount added to the initial value, the capital is the calculation basis, the interest rate is the percentage applied on the capital and time is the capitalization period.
Knowing that:
10 paid vacation days per yearsalary is 32,500 per yearThe total weekends in a year:
52 * 2 = 104 weekend days.
Total paid vacation days, including holidays that fall on weekends:
10 - 4 = 6 paid vacation days.
Total non-working days:
104 + 6 = 110 non-working days.
Total working days in a year:
365 - 110 = 255 working days.
Pay per workday = Annual salary / Total working days
= $32,500 / 255= $127.45
Therefore, the librarian's pay per workday is approximately $127.45.
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Which statement about the location of √7 on the number line is true?
A= It is located at the number 7 on the number line.
B= It is located at the number 3.5 on the number line.
C= It is located between the numbers 2 and 3 on the number line.
D=It is located between the numbers 4 and 9 on the number line
Can someone answer this question
Answer:3rd one
Step-by-step explanation:
What is the median of the following set of numbers? 35, 33, 36, 34, 33, 33, 32, 38, 35
A.35
B. 34
C.33
D.36
Median of the set is 34, as it is the middle value of the ordered set (32, 33, 33, 33, 34, 35, 35, 36, 38).
To find the middle of a bunch of numbers, the initial step is to organize the numbers all together from least to most prominent. For this situation, the arranged set is: 32, 33, 33, 33, 34, 35, 35, 36, 38.
The middle is the center number in the arranged set. On the off chance that there is an odd number of values, the middle is the center worth. On the off chance that there is a considerably number of values, the middle is the normal of the two center qualities.
For this situation, there are nine qualities in the set, which is an odd number. The center worth is the fifth worth, which is 34. Thusly, the middle of the set is B. 34.
To confirm, we can count four qualities before 34 and four qualities after 34 in the arranged set. Since there are an equivalent number of values when the middle, we can infer that 34 is for sure the middle.
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SOMEBODY PLEASE PLEASE HELP ME
ILL GIVE YOU 5 STARS, BRAINLIEST AND GIVE A HEART PLEASE HELP
(Please answer the questions and )
Part A ;
For Bus:
mean = 17.5
median = 15
mode= 15 and 16
Range= 17
For walking:
mean = 20.5
median = 20.5
mode= 20 and 21
Range= 3
How to calculate the various variables given above?For Bus
To calculate the mean:
= 16+14+15+14+31+15/6
= 105/6
= 17.5
To calculate the median;
= 14,14,15,15,16,31
= 15+15/2 = 15
To calculate the range:
= 31-14 = 17
For walking;
To calculate the mean:
= 19+20+20+21+21+22/6
= 123/6
= 20.5 mins
To calculate the median
= 20+21/2
= 20.5
To calculate the mode:
=20 and 21
To calculate the range:
= 22-19
= 3
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What are the side lengths of triangle Graph shows a triangle plotted on a coordinate plane. The triangle is at A(minus 7, 3), B(minus 3, 6), and C(5, 0). Type the correct answer in each box. If necessary, round any decimal to the nearest tenth. units units units
With the coordinate provided, the lengths of the triangle are Side AB = 5 units, Side BC = 10 units and Side CA = 12.4 units.
How do we find the sides of triangles using coordinates?To find the side lengths of the triangle using the coordinates provided, we use the distance formul.
The distance between two points (x1, y1) and (x2, y2) is given by √((x2-x1)² + (y2-y1)²).
Side AB: Distance between A(-7, 3) and B(-3, 6)
= √((-3 - -7)² + (6 - 3)²)
= √((4)² + (3)²)
= √(16 + 9)
= √25
= 5 units
Side BC: Distance between B(-3, 6) and C(5, 0)
= √((5 - -3)² + (0 - 6)²)
= √((8)² + (-6)²)
= √(64 + 36)
= √100
= 10 units
Side CA: Distance between C(5, 0) and A(-7, 3)
= √((-7 - 5)² + (3 - 0)²)
= √((-12)² + 3²)
= √(144 + 9)
= √153
= 12.4 units
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MA.7.DP.1.4
A group of friends has been given $800 to host a party. They must decide how much money
will be spent on food, drinks, paper products, music and decorations.
Part A. As a group, develop two options for the friends to choose from regarding how to
spend their money. Decide how much to spend in each area and create a circle
graph for each option to represent your choices.
Part B. Mikel presented the circle graph below with his recommendations on how to
spend the money. How much did he choose to spend on food and drinks? How
much did he choose to spend on music?
Party Spending Proposal
Mail
17%
Paper Products
Answer: $130 money did Brenda and Hazel have all together before buying decorations and snacks.
Here, we have,
You want to know Brenda and Hazel's combined money when the ratio of their remaining balances is 1 : 4 after Brenda spent $58 and Hazel spent $37. They had the same amount to start with.
Setup
Let x represent the total amount the two women started with. Then x/2 is the amount each began with, and their fnal balance ratio is ...
(x/2 -58) : (x/2 -37) = 1 : 4
Solution
Cross-multiplying gives ...
4(x/2 -58) = (x/2 -37)
2x -232 = x/2 -37 . . . . . . eliminate parentheses
3/2x = 195 . . . . . . . . . . . . add 232-x/2
x = (2/3)(195) = 130 . . . . . multiply by 2/3
Brenda and Hazel had $130 altogether before their purchases.
Alternate solution
The difference in their spending is $58 -37 = $21.
This is the same as the difference in their final balances.
That difference is 4-1 = 3 "ratio units", so each of those ratio units is $21/3 = $7.
Their ending total is 1+4 = 5 ratio units, or $35.
The total they started with is $58 +37 +35 = $130.
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complete question:
Brenda and Hazel decide to throw a surprise party for their friend, Aerica. Brenda and Hazel each go to the store with the same amount of money. Brenda spends $58 on decorations, and Hazel spends $37 on snacks. When they leave the store, the ratio of Brenda’s money to Hazel’s money is 1 : 4. How much money did Brenda and Hazel have all together before buying decorations and snacks?
Can someone please help me find the domain and range thank you all so much for the help
The domain and the range of the graph are Domain = [-10, -8] and Range = [-10, -1]
Calculating the domain and range of the graphFrom the question, we have the following parameters that can be used in our computation:
The graph
The above graph can be represented as a list of ordered pairs
The rule of a graph is that
The domain is the x valuesThe range is the f(x) valuesUsing the above as a guide, we have the following:
Domain = [-10, -8]
Range = [-10, -1]
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Find the average rate of change of f(x)=x²+2x+1 from x=4 to x=6.
The average rate of change of the function over the interval is 12
Finding the average rate of changeFrom the question, we have the following parameters that can be used in our computation:
f(x) = x² + 2x + 1
The interval is given as
From x = 4 to x = 6
The function is a quadratic function
This means that it does not have a constant average rate of change
So, we have
f(4) = 4² + 2(4) + 1 = 25
f(6) = 6² + 2(6) + 1 = 49
Next, we have
Rate = (49 - 25)/(6 - 4)
Evaluate
Rate = 12
Hence, the rate is 12
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A spinner has three sections. The table shows the results of spinning the arrow on the spinner 80 times.
What is the experimental probability of the arrow stopping over Section 1?
A} 1/28
B} 7/20
C} 7/13
D} 4/5
Answer:
B} 7/20
Step-by-step explanation:
total: 80
section 1: 28
p(section 1) = 28/80 = 7/20
Find the sum of the finite series:
10
Σ2· (3)n-1
n=1
Answer: 59,048
Step-by-step explanation:
To find the sum of the given finite series:
Σ2· (3)n-1
n=1
where n ranges from 1 to 10, we can use the formula for the sum of a geometric series.
The formula for the sum of a geometric series is:
S = a * (r^n - 1) / (r - 1)
In this case, a = 2 (the first term of the series) and r = 3 (the common ratio).
Using the formula, we can calculate the sum:
S = 2 * (3^10 - 1) / (3 - 1)
= 2 * (59049 - 1) / 2
= (118098 - 2) / 2
= 118096 / 2
= 59048
Therefore, the sum of the given finite series is 59,048.
Using the image below, answer the following question: you are asked to pick 2
marbles out of the bag, what is the probability of picking a blue marble and then a
green marble, without replacing the blue one?
Please
The probability of picking a blue marble and then a green marble without replacing the blue one is 2/15.
From the figure we can write
Number of Blue marbles = 2
Number of Green marbles = 6
Number of Purple marbles = 2
Total number of marbles = 2 + 6 + 2 = 10
Now, Probability of picking a blue marble
= Number of Blue marbles / Total number of marbles
= 2 / 10
= 1/5
After removing the blue marble, the total number of marbles becomes 9
Probability of picking a green marble
= 6 / 9
= 2/3
So, the probability of picking a blue marble and then a green marble
Probability = (1/5) x (2/3) = 2/15.
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find the area of a triangle whose base is 8 inches and whose height is 12 in.
(a) 96in.²
(b) 10in ²
(c) 20in.²
(d) 48in.²
Answer:
48in.²
Step-by-step explanation:
The formula for the area of a triangle is 1/2 x b x h
Since the b = 8 and the h = 12, substitute 8 and 12 for b and h.
1/2 x 8 x 12
4 x 12
48in.²
Answer:
d
Step-by-step explanation:
area= ½ × b × h
= ½ × 8 × 12
= ½ × 96
= ⁹⁶/²
= 48in²
FInd the measure of the arc using the picture provided , please and thanks
The measure of arc angle QTP is 204 degrees.
How to find arc angle?When a tangent and a secant intersect outside a circle then the measure of the angle formed is one-half the positive difference of the measures of the intercepted arcs.
Therefore, let's find the measure of arc angle QTP as follows:
90 = 0.5(x - 68)
90 = 0.5x - 34
0.5x = 124
x = 124 / 0.5
x = 248 degrees
Where
x = arc angle TSQ
Therefore,
arc QP = 44 degrees
Therefore,
arc QTP = 248 - 44
arc QTP = 204 degrees
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50 Points! Multiple choice geometry question. Photo attached. Thank you!
Answer:
The correct answer:
(B) reduction
Teena uses 1/4 cup of oil for a cake. How many cakes can she make if she has 6 cups of oil?
Answer:
24 cakes.
Step-by-step explanation:
6 cups of oil divided by 1/4 cup oil per cake = 24 cakes
6/(1/4) = 24
or 6/(0.25) = 24
She can make 24 cakes with 6 cups of oil.
how do you find out the area of a tent, floor included
To find the area of a tent, including the floor, we need to measure or determine the dimensions of both the tent's floor and any additional areas such as vestibules or extensions.
Floor Area: Measure the length and width of the tent's floor in the same unit of measurement (e.g., feet or meters). Multiply the length by the width to calculate the floor area. For example, if the length is 8 feet and the width is 6 feet, the floor area would be 8 feet * 6 feet = 48 square feet.Additional Areas: If the tent has vestibules, extensions, or any other separate areas, measure each area's dimensions and calculate their individual areas separately.Total Area: Once you have calculated the individual areas of the floor and any additional areas, simply add them together to find the total area of the tent, including the floor. For example, if the floor area is 48 square feet and there is an additional vestibule area of 10 square feet, the total area would be 48 square feet + 10 square feet = 58 square feet.Learn more about the area of the tent here:
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Solve and graph. |3x+6/9|>=2
The solution to the inequality expression is -8/9 ≤ x ≥ 4/9
How to solve and graph the inequality expressionFrom the question, we have the following parameters that can be used in our computation:
|3x + 6/9| ≥ 2
Expand the inequality expression
So, we have
-2 ≤ 3x + 6/9 ≥ 2
Subtract 6/9 from both sides
So, we have
-24/9 ≤ 3x ≥ 12/9
Divide through the equation by 3
-8/9 ≤ x ≥ 4/9
Hence, the solution to the inequality expression is -8/9 ≤ x ≥ 4/9
The graph is attached
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Question 5
TABLE 4 below shows the distribution of revenue among the different spheres of
government in South Africa from the 2017/2018 to 2021/2022 financial year. Some values
have been omitted.
TABLE 4: Distribution of Revenue among the different government spheres for
the financial years 2017/2018 to 2021/2022.
R(billion)
2019/20
546.1
471,4
Government
BHOJDIN
National
Provincial
Local
TOTAL
2017/18
453.4
410.6
**
2018/19
490,00
439,5
87.6
1 017,1
98.3
1:15.8
2020/21
557,5
500,4
103.3
1 161.2
2021/22
1 240.5
(adapted from DBE 2014 MLQP)
Use the table above to answer questions that follows
When the revenue of the local government sector was compared to the provincial sec
2017/18, it was found to be 20, 12% of the provincial sector.
Calculate
the missing value E, the total revenue for the period 2017/18.
ont sector always receives a larger share than t
The missing value E, the total revenue for the period 2017/18, is approximately 82.61 billion.
How to calculate the valueLocal government sector's revenue in 2017/18 (L): Missing value (to be calculated)
Provincial sector's revenue in 2017/18 (P): 410.6 (billion)
According to the information provided, L is 20.12% of P. Mathematically, we can write this as:
L = 20.12/100 * P
Substituting the known value of P, we get:
L = 20.12/100 * 410.6
L ≈ 82.61 (rounded to two decimal places)
Therefore, the missing value E, the total revenue for the period 2017/18, is approximately 82.61 billion.
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how many triangles?(urgent)
There are 75 triangles that can be formed using these 9 vertices.
Here we have,
We have 9 vertices arranged in a pattern where 5 vertices are above and 4 vertices are below in a parallel direction.
And we want to know how many triangles can be formed using these vertices.
In order to this,
Count the number of ways to choose 3 vertices from the 9 vertices.
This can be done using the combination formula, which is:
[tex]^{9}C_{3}[/tex] = 9! / (3! * (9-3)!)
= 84.
Since,
A degenerate triangle is one where all three vertices lie on the same line. To count the number of degenerate triangles,
We need to count how many ways we can choose 3 vertices that lie on the same line.
There are 5 lines with 3 points each (the 5 lines with the upper vertices). So the number of degenerate triangles is:
5 [tex]^{3}C_{3}[/tex] + 4 [tex]^{3}C_{3}[/tex] = 9.
Subtract the number of degenerate triangles from the total number of triangles to get the number of non-degenerate triangles.
Therefore, the number of non-degenerate triangles is:
84 - 9 = 75.
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Given f(x) = x^2 what is the range of g(x) = f(x+2)-5
The range of the function g(x) = f(x+2)-5 is:
R = [0, ∞)
What is the range of g(x) = f(x+2)-5?For a function y = f(x), we define the range as the set of possible outputs of the function.
Here we want to find the range of:
g(x) = f(x+2)-5
Where f(x) = x²
Replacing f(x) in the rule for g, we will get the composition:
g(x) = (x + 2)² - 5
The first part can only be a positive number or 0, then the minimum of the range is -5
And then the function only grows, then the range of g(x) is:
R = [0, ∞)
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