The first quadrant that lies between the circle x² + y² =4 is 1.09587
The graph of x² +y² = 4 is a circle of radius 2, centered at (0, 0). In polar coordinates, this equation becomes simply r = 0.
The second equation requires a bit more work. To convert it to a polar equation, we
substitute, r cos θ for x and r sin θ for y:
r²cos ²θ + r²sin ²θ
= 2r cos Ф
⇒r² = 2r cos θ
⇒r = 2 cos θ
This is the polar equation for a circle of radius 1, centered at the point (x, y) = (1, 0). (This can be seen directly from the original equation by completing the square, which results in the equation (x − 1)²+ y² = 1.)
To describe the region between the circles in polar coordinates, we can let θ range from 0 to π/2. Our values for r should range from the smaller circle to the larger one
Recalling again that x = r cos θ and dA = rdrdθ, our integral becomes
∫ ∫ₐ x DA = 8/3 - π/2
We can get a rough check of this answer as follows. The area of the region D is easily computed to be π/2. The average value of the function x in the region is somewhat less than 1, since D is thicker on the left. Doing these sorts of visual estimates can be very tricky, but let’s say the average value of x in D is about 0.70. Thus
∫∫ₐ x dA ∼=π/2· 0.70 ∼= 1.09956
Since 8/3 − π/ 2∼= 1.09587, this estimate agrees quite well with our answer.
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Simplify the expression below as a simplified fraction.
(1+1/3)²−2/9
(1 + 1/3)^2 - 2/9
= (3/3 + 1/3)^2 - 2/9
= (4/3)^2 - 2/9
= 16/9 - 2/9
= 14/9
a rectangle is inscribed in a right isosceles triangle with a hypotenuse of length . what is the largest area the rectangle can have?
The largest area of the rectangle is 2 square units.
We have a right angled isosceles triangle with hypotenuse 4 units.
so its base will be equal to the perpendicular , let it be B
So we know ,
B² + P² = H²
2B² = H²
B² = 16/2
B = 2√2
So base is equal to perpendicular = 2√2 units
Now let consider a rectangle inside the triangle with length 2x and the width y , and the area will become
A = 2xy ....(1)
we see from the figure the length of the perpendicular AD drawn on the hypotenuse is
BD² + AD² = AB²
AD² = AB² - BD²
= 8 - 4
AD² = 4
AD = 2 units
Now we see triangle ABD and AEG are similar to each other , So
AD/BD = AG/GE
2/2 = 2-y/x
therefore
x = 2-y
put this value in eqn (1)
A = 2(2-y)y
A = 4y - 2y²
Now to maximize the area we take the derivative of the equation and then put it equal to zero
∴ 4 - 4y = 0
y = 1
∴ x = 2-y = 2-1 = 1 units
So area will be maximum when length and width are 1 units each.
therefore , largest area (A) of the rectangle is = 2xy = 2 square units.
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Please help this is hard
The equation that represents each of the figure in slope intercept form are
equation of the left side - y = 3x + 2
equation of the right side - y = 3x - 4
equation of the top side - y = 5
equation of the bottom side - y = -1
How to write the linear equation in slope intercept formThe the slope intercept form has the formula
y = mx + c
where
m = slope
c = intercept
x = input variables
y = output variables
The left and right side's slopes require calculation however the top and bottom slope is zero
equation of the left side
The points used for the slope is gotten from the graph as (1. 5) (0, 2)
m = (y₂ - y₁) / (x₂ - x₁)
substituting the values
= (2- 5) / (0 - 1)
= -3 / -1
= 3
writing the equation using points (0, 2)
(y - y₂) = m(x - x₂)
y - 2 = 3(x - 0)
y = 3x + 2
equation of the right side
The points used for the slope is gotten from the graph as (5. 5) (3, -1)
m = (y₂ - y₁) / (x₂ - x₁)
substituting the values
= (-1 - 5) / (3 - 5)
= -6 / -2
= 3
writing the equation using points (3, -1)
(y - y₂) = m(x - x₂)
y - -1 = 3(x - 3)
y + 1 = 3x - 3
y = 3x - 4
equation of the top side
the top side is a horizontal line and the equation is written by picking the point where the line cut the y axis. from the graph the point is 5 hence
y = 5
equation of the bottom side
the bottom side is a horizontal line and the equation is written by picking the point where the line cut the y axis. from the graph the point is -1 hence
y = -1
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how do i graph this?? it doesnt make any sense to me :(
Answer:
Basically you need to show on a graph the cost half of the Drinks and the Popcorn and then when everything is graphed and then the rest of the instructions are stated below:
A) Sketch the graph that represents the situation and label the intercepts. Use one axis to represent the number of bags of popcorn and the other axis to represent the number of drinks. (3 points - 1 for x-intercept, 1 for -intercept, 1
PLEASE GIVE BRAINLIEST THANK YOU!!!
kozol observes the strain on fremont's students at lunchtime, when all of the 3,300 students in attendance are served in one 30-minute meal period. one teacher calculates that the extended walk to the cafeteria and long food lines create a 10-minute window for students to eat. what often results is that many students go all day without a meal (176).
Kozol notices the stress on the 3,300 kids at Fremont during lunch, when the entire meal is provided in a single 30-minute period. One teacher estimates that the lengthy distance to the cafeteria and the lengthy queues for food provide children a 10-minute window during which to eat. As a result, many students frequently skip meals entirely (176).
Given,
Lunch was served to the pupils in this school in a single sitting, unlike the staggered luncheon sessions I saw at Walton High. 3,300 children cannot be fed at once due to physical limitations, the teacher explained. "The session lasts barely 30 minutes, and there is a very large line for children to obtain their food. They must wait in line for fifteen minutes before they can even walk there from their classes. They most likely have 10 minutes to eat their food. Many of them don't even try.
You've worked as a teacher, so you know what it's like for students to go a whole day without eating. Here at Fremont, the school day lasts eight hours.
Here,
Kozol notices the stress on the 3,300 kids at fremont during lunch, when the entire meal is provided in a single 30-minute period. One teacher estimates that the lengthy distance to the cafeteria and the lengthy queues for food provide children a 10-minute window during which to eat. As a result, many students frequently skip meals entirely .
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Given question is incomplete. Completed question is here;-
Unlike the staggered luncheon sessions I observed at Walton High, lunch was served in a single sitting to the students in this school. "It's physically impossible to feed 3,300 kids at once," the teacher said. "The line for kids to get their food is very long and the entire period lasts only 30 minutes. It takes them 15 minutes just to walk there from their classes and get through the line. They get 10 minutes probably to eat their meals. A lot of them don't try. You've been a teacher, so you can imagine what it does to students when they have no food to eat for an entire day. The schoolday here at Fremont is eight hours long."
From Kozol, Jonathan. The Shame of the Nation: The Restoration of Apartheid Schooling in America. New York: Crown, 2005. Print. The passage appears on page 176.
what's 33 ÷ 33 × 33 - 33 + 33
Answer:
33
Step-by-step explanation:
first we divide 33 by 33 then we have 1 now we have times 33 which is 33 now 33-33 is 0 not we add 33 to 0 now we have 33
solve the following quadratic equation 4x^2-31x-33=12
The length of a rectangle i 7 inche le than twice the width, w, of the rectangle. Find the dimenion if the area i 30 quare meter
The dimensions of the rectangle are, length = 5 inch and width = 6 inch.
What is rectangle?
A rectangle in Euclidean plane geometry is a quadrilateral with four right angles. It can alternatively be explained in terms of an equiangular quadrilateral—a term that refers to a quadrilateral whose angles are all equal—or a parallelogram with a right angle. A square is an irregular shape with four equal sides.
Let, the length of a rectangle is 7 inches less than twice the width, w, of the rectangle.
Suppose,
width = w = x, then,
length = l = 2x - 7
Since the area of the rectangle is 30 square inches.
We know that.
Area of rectangle = length x width
30 = (2x - 7) x (x)
30 = 2x^2 - 7x
2x^2 - 7x - 30 = 0
2x^2 - 12x + 5x - 30 = 0
2x(x - 6) + 5(x - 6) = 0
(2x + 5)(x - 6) = 0
⇒ 2x + 5 = 0, x - 6 = 0
⇒ x = -5/2, x = 6
Since length is not negative.
Therefore, w = 6.
So, l = 2(6) - 7 = 12 - 7 = 5
Hence, the length is 5 inches and the width is 6 inches.
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help needed asap!!!
find the equation of the line in point-slope form with the given slope passing through the given point
Option (a) is correct, i.e., the equation of the line in point-slope form with given slope -2/3 passing through the given point (2,-3) is y+3 = -2/3 (x-2).
What is point-slope form equation of a line?
Point slope form is used to represent a straight line using its slope and a point on the line.
The equation of a line in point-slope form whose slope is 'm' and which passes through a point (x₁ ,y₁) is
y - y₁ = m (x-x₁).
Here we have slope m= -2/3 and the point (x₁ ,y₁) = (2,-3).
Therefore, the point-slope form of the equation is
y- (-3) = - 2/3 (x-2)
⇒ y + 3= -2/3 (x-2).
⇒ Option(a) is correct.
Therefore, equation of the line in point- form slope is y+3 = -2/3 (x-2).
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Sketch a number line plot your age as a point on your number line
What would you consider as the starting point on a number line? Why?
The starting point of the number line drawn using the age is zero
What is a Number line?A number line is an image of numbers plotted on a straight line, either horizontally or vertically. We can compare numbers and execute simple arithmetic operations on them easily by writing the numbers down on a number line. The starting point of a number line is often regarded as zero (0). The numbers on the left of zero are all negative, whereas the numbers on the right of zero are all positive.
The starting point of the number line drawn using the age is zero and it is why because every human life starts at the age of months/years.
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The set of data in the table below represents a linear function.
X
Y
2
-15
5
-9
7
-5
10
1
13
7
Which is an equation for this function?
Answer:
y = 2x - 19
Step-by-step explanation:
Graph the points on the table and you'll be able to see the answer. That's how I went about it.
or use m = y^2 - y^1 divided by x^2 - x^1 to find the slope, then solve for the initial value from there
What is the equation of this line?
Responses
y=12x−3
y equals fraction 1 half end fraction x minus 3
y=−12x−3
y equals negative fraction 1 half end fraction x minus 3
y=−2x−3
y equals negative 2 x minus 3
y=2x−3
y equals 2 x minus 3
Number graph ranging from negative 5 to 5 on both the x and y axis. A line passes through the point begin ordered pair 0 comma negative 3 end ordered pair and the point begin ordered pair 4 comma negative 1 end ordered pair
The equation of the line in slope-intercept forms expressed as: y = y = 1x + 3
How to Write the Equation of a Line?
To wrote the equation of a line in slope-intercept form, first, find the slope (m) of the line, determine the y-intercept (b) of the line, then substitute the values into y = mx + b (equation in slope-intercept form).
Using two points on the line, (0, 3) and (4, -1):
Slope (m) = change in y / change in x = (3 - -1)/(0 - 4)
Slope (m) = 4/-4
Slope (m) = -1
The slope of the line is m = -1.
Find the y-intercept (b) by substituting one point, (x, y) = (0, 3), and m = -1 into y = mx + b and solve for b:
3 = 1(0) + b
3 = 0 + b
b = 3
Write the equation of the line by substituting b = 3 and m = -1 into y = mx + b:
y = 1x + 3
Thus, the equation in slope-intercept form of the line is expressed as: y = 1x + 3
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A chool will only accept tudent who core in the top 2 %of an admiion tet. If the normal ditribution with a mean of 70 %and a tandard deviation 5%. What i the minimum grade that a tudent mut core in order to be accepted?
Ninety-five percent of people have IQs between 70 and 130.99.7 percent of people have IQ scores between 55 and 145.With IQ scores between 55 and 145, only 0.3% of the population falls outside this range.
How likely is it that a student chosen at random will have an IQ of 115 or higher?
Since half should be over the normal of 100 (by evenness), this implies 50 - 34.13 = 15.87 (%) has an intelligence level score over 115.
Which level of understudies might have scored either higher than 90 or lower than 70?
That always prompts the same response: 95%!Since we are aware that one standard deviation equals five points, two standard deviations equal two x 5.As a result, 95% of scores fall between 70 and 90, which is 10 points above the mean of 80.
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Which ordered pair is in the solution set of y> -1/6x - 4?
A. (-8, 8)
B. (6,-5)
C. (4, -6)
D. (-2, -7)
Determine whether 8 > -191/48 is true or false :
=> True.
Other option are False.
Given,
In the question:
The set is given as:
y> -1/6x - 4
To check the which ordered pair is in the solution set of y> -1/6x - 4
Now, According to the question:
A) Substitute x = -8 , y = 8 into y> -1/6x - 4
8 > -1/ 6 x (-8) - 4
LHS = 8
RHS = -1/ 6 x (-8) - 4 = -191/48
Write in the form
8 > -191/48
Hence, Determine whether 8 > -191/48 is true or false :
=> True.
Other option are False.
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Answer:
Step-by-step explanation:
Bookwork code: J62
Copy and complete the table of values for
52 +3.
0 1
#| 3² | 9 | ¹ | 2 | ³ | 4 | 5 | 6
3
The values of y when x=-1,0,1,2,3,4,5,6 are 9,3,-1,-3,-3,-1,3,9 respectively.
Given the function is,
y = x²-5x+3
Finding the values we get,
substituting the value x=-1 we get,
y = (-1)²-5(-1)+3 = 1+5+3 = 9
substituting the value x=0 we get,
y = 0²-5*0+3 = 3
substituting the value x=1 we get,
y = 1²-5*1+3 = 1-5+3 = -1
substituting the value x=2 we get,
y = 2²-5*2+3 = 4-10+3 = -3
substituting the value x=3 we get,
y = 3²-5*3+3 = 9-15+3 = -3
substituting the value x=4 we get,
y = 4²-5*4+3 = 16-20+3 = -1
substituting the value x=5 we get,
y = 5²-5*5+3 = 25-25+3 = 3
substituting the value of x=6 we get,
y = 6²-5*6+3 = 36-30+3 = 9
Hence the values of y when x=-1,0,1,2,3,4,5,6 are 9,3,-1,-3,-3,-1,3,9 respectively.
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One hundred people came to the company's Thanksgiving dinner. There were fifty-two more vegetarians at the dinner than non-vegetarians. How many vegetarians were at the dinner?
Answer:
52????? It's in the question.
Step-by-step explanation:
∠A and angle ∠B are supplementary angles. If angle A=(5x-22) and m∠B=(x+28)
what are the mesaurements for ∠A?
The corresponding measurements for angles A and B are 123° and 57°
How to find the measurements of the angles?Supplementary angles are angles that sum up to 180 degrees e.g angle 140° and angle 40° are supplementary angles because the sum of 140° and 40° is equal to 180°
Given that: angle A= (5x-22) and m∠B= (x+28), and ∠A and angle ∠B are supplementary angles. Thus:
A + B = 180°
(5x-22) + (x+28) = 180°
5x-22 + x+28 = 180°
6x + 6 = 180
6x = 180 -6
6x = 174
x = 174/6 = 29°
∠A = (5x-22) put x = 29°
∠A = 5(29) -22
= 145-22
= 123°
∠B = 180 -123 = 57°
Therefore, the measurements for angles A and B are 123° and 57° respectively
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Answer:The corresponding measurements for angles A and B are 123° and 57°
The corresponding measurements for angles A and B are 123° and 57°
How to find the measurements of the angles?
Supplementary angles are angles that sum up to 180 degrees e.g angle 140° and angle 40° are supplementary angles because the sum of 140° and 40° is equal to 180°
Given that: angle A= (5x-22) and m∠B= (x+28), and ∠A and angle ∠B are supplementary angles. Thus:
A + B = 180°
(5x-22) + (x+28) = 180°
5x-22 + x+28 = 180°
6x + 6 = 180
6x = 180 -6
6x = 174
x = 174/6 = 29°
∠A = (5x-22) put x = 29°
∠A = 5(29) -22
= 145-22
= 123°
∠B = 180 -123 = 57°
Therefore, the measurements for angles A and B are 123° and 57° respectively
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2^18 times 2^9 what is the missing exponent
The value of the exponent given as 2^18 times 2^9 is 2^27.
What is an exponent?The exponent is the number of times that a number is multiplied by itself. It should be noted that the power is an expression which shows the multiplication for the same number. For example, in 6⁴ , 4 is the exponent and 6⁴ is called 6 raise to the power of 4.
In this case, 2^18 times 2^9 will be illustrated thus:
= 2^18 times 2^9
= 2^(18 + 9)
= 2^27
The value is 2^27.
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A rectangular box has a square base. The combined length of a side of the square base, and the height is 20 in. Let x be the length of a side of the base of the box.
a. Write a polynomial function in factored form modeling the volume V of the box.
b. What is the maximum possible volume of the box?
c. Explain how you found the maximum possible volume.
Answer:
Step-by-step explanation:
V(x) = x3 + 2x2 - 11x - 12
To factor, you want 3 numbers that multiply to give -12 (the constant term) and add to give 2 (coefficient of the x2 term). The answer is 4, -3, and 1: (4)(-3)(1) = -12, 4 + (-3) + 1 = 2. So the volume polynomial factors to:
V(x) = (x+4)(x-3)(x+1)
Now the volume of a rectangular box is:
V = Length*Width*Height = (x+4)(x-3)(x+1)
So let's pick height = (x+1) = 4.5 (you could pick any one of the three factors).
x+1 = 4.5
x = 3.5
Length = (x+4) = 3.5+4 = 7.5
Width = (x-3) = 3.5 - 3 = 0.5
The box's dimensions are 7.5 m by 0.5 m by 4.5 m
I need help with please!!!!
Answer: -5/2
Step-by-step explanation:
Using slope formula which is [tex]\frac{y_2-y_1}{x_2-x_1}[/tex] you can substitute in the numbers which leads you to getting [tex]\frac{-1-4}{-3-(-5)}[/tex] which equals to -5/2
natalie is paid 42 more per hour than travis. when natalie works for 40 hours and travis works for 35 hours, they earn $755 between them. how much does each earn per hour?
Using Linear equation we get , Travis and Natalie's combined hourly wages are $135 and $137.
Combined Natalie earns $2 more an hour than Travis.
Natalie works 40 hours
Travis works 35 hours
for his or her work Salary difference = $755
we have to find out what percentage each of them contributes per hour.
Let Travis earn in dollars per hour = x
So if he works 35 hours his earnings in dollars
= 35x
Natalie can earn in dollars per hour = x+2
So If she works 40 hours, her income will be in dollars = 40x + 80 [use distribution]
The income difference will be:
⇒ 40x + 80 -35x = 755
=> 5x = 675 => x = 135
So Travis' hourly rate = $135
Natalie's hourly rate = 135+2= $137
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Kal practices lines for 28 minutes at his acting class. This is 70% of the entire acting class. How many minutes long is his acting class? use any strategy to solve.
Using percent method we can solve the problem. Kal's acting class is 40 minutes long in total.
In which he practices lines about 28 minutes.
let "x minutes" be the total timing of kal's acting class .
Percent is always running between 0 to 100 where 0 % means nothing and 100% total about which we are considering the percent.
so, here x mintues is equal 100% timing
we have given that, kal practice lines for 28 minutes at his acting class and in percent form his timing is equal to 70% .
That's 70% time of entire acting class = 28 min
but we have to find the 100% of entire acting class i.e x minutes
Now, 70% time of entire class = 28 minutes
1% time of entire acting class = 28/70 minutes
100% time of entire acting class i.e x = ( 28/70)100
= 40 minutes .
So, the entire kal's acting class runs for 40 minutes .
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Write a linear equation that models the data in the table
X Y
-6 0
-3 2
0 4
3 6
Equation:?
to get the equation of any straight line, we simply need two points off of it, let's use the ones in red provided in the table in the picture below
[tex](\stackrel{x_1}{-3}~,~\stackrel{y_1}{2})\qquad (\stackrel{x_2}{3}~,~\stackrel{y_2}{6}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{6}-\stackrel{y1}{2}}}{\underset{run} {\underset{x_2}{3}-\underset{x_1}{(-3)}}} \implies \cfrac{4}{3 +3} \implies \cfrac{ 4 }{ 6 } \implies \cfrac{2 }{ 3 }[/tex]
[tex]\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{2}=\stackrel{m}{ \cfrac{2 }{ 3 }}(x-\stackrel{x_1}{(-3)}) \implies y -2 = \cfrac{2 }{ 3 } ( x +3) \\\\\\ y-2=\cfrac{2 }{ 3 }x+2\implies {\Large \begin{array}{llll} y=\cfrac{2 }{ 3 }x+4 \end{array}}[/tex]
a ____________ is a table in which frequencies correspond to two variables.
A contingency table is a table in which frequencies correspond to two variables.
Explanation:
In contingency table one variable is used to categorise rows and second variable is used to categories columns.In contingency table have at least two rows and at least two columns.What is frequency table?
A frequency table is just a two-column "t-chart" or table that lists all of the potential outcomes and their corresponding frequencies as seen in a sample.What does "two way frequency table" mean?
A two-way or contingency table is the name for this kind of table. A statistical table called a two-way or contingency table displays the observed number or frequency for two variables, with the rows denoting one category and the columns denoting the other. Example, the gender (male or female) row category.To know more about contingency table check the below link:
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After much tinkering, Jaylen's time machine finally works! For his first trip,
Jaylen will travel 25 years into the future. If his time machine uses 3
kilowatts of power to transport him 1 month into the future, how many
kilowatts of power will it use for his first trip?
Answer:
900kW
Step-by-step explanation:
3kW = 1 month
xkW = 25 years
First, change 25 years into months:
12 months = 1 year
Therefore,
25 years = 12 × 25
= 300 months
If
3kW = 1 month
x = 300 months
cross-multiply
Hence,
1 monthX = 300 months × 3kW
= 900kW months
Divide both sides by the coefficient of X
X = 900kW months / 1 month
= 900kW
Therefore, Jalen's first trip will take 900kW
For a project in her Geometry cla, Violet ue a mirror on the ground to meaure the height of her chool’ football goalpot. She walk a ditance of 12. 75 meter from the goalpot, then place a mirror flat on the ground, marked with an X at the center. She then walk 6. 55 more meter pat the mirror, o that when he turn around and look down at the mirror, he can ee the top of the goalpot clearly marked in the X. Her partner meaure the ditance from her eye to the ground to be 1. 75 meter. How tall i the goalpot? Round your anwer to the nearet hundredth of a meter
3.406 tall is the goalpost
given that
she walk a distance of 12.75 meters from the goalpost, then place a mirror flat on the ground, marked with an X at the center. She then walks 6. 55 more meters past the mirror
Her partner measures the distance from her eye to the ground to be 1. 75 meters.
based on the given condition [tex]\frac{1.75}{6.55}[/tex] = [tex]\frac{x}{12.75}[/tex]
x = [tex]\frac{1.75}{6.55}[/tex] ×12.75
x = 3.406
Therefore, 3.406 tall is the goalpost
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Zola took a picture of Greg standing next to a building. Greg is 5 feet tall. The picture Zola took
shows Greg as ½ inch tall. If the building is 25 feet tall in real life, how tall should the building be
in the picture? Explain.
Answer
2inch tall
Explanation
let the height of the building in the picture = y
5ft -––- ½ inch
20 ft -––- y inch
Cross multiply ✖️
20 × ½ = 5 × y
10 = 5y
10/5 = y
y = 2
Write the equation of the parabola that has vertex (-5,6) and directrix y=2.
The equation of parabola is ( x + 5 )² = 16 ( y - 6 )
What is a Parabola?
A Parabola, open curve, a conic section produced by the intersection of a right circular cone and a plane parallel to an element of the cone. A parabola is a plane curve generated by a point moving so that its distance from a fixed point is equal to its distance from a fixed line
The equation of the parabola is given by
( x - h )² = 4p ( y - k )
where ( h , k ) is the vertex and ( h , k + p ) is the focus
y is the directrix and y = k - p
Given data ,
The vertex of the parabola is ( -5 , 6 )
The directrix of the parabola y = 2
From the equations of parabola , we know that
The vertex ( h , k ) = ( -5 , 6 )
h = -5
k = 6
y = k - p
y = 2
k - p = 2
Substituting the value k in the equation , we get
6 - p = 2
On Simplifying , we get
p = 4
So , now we have the values of h , k , and p
The equation of the parabola is
( x - h )² = 4p ( y - k )
Substituting the values , we get
( x - (-5) )² = 4 x 4 ( y - 6 )
( x + 5 )² = 16 ( y - 6 )
x² + 25 + 10x = 16y - 96
x² + 10x - 16y = -121
Hence , the equation of parabola is ( x + 5 )² = 16 ( y - 6 )
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a conical tank has height 3 m and radius 2 m at the base. water flows in at a rate of 2 m3/min. cone with bottom radius 2 meters and height 3 meters. how fast is the water level rising when the level is 1 m and when the level is 1.8 m? (use decimal notation. give your answers to four decimal places.)
The speed at which conical tank water will rise is 9/8[tex]\pi[/tex] m/min.
What is cone?
A cone could be a three-dimensional geometric type with a flat base and a sleek tapering apex or vertex.
Main body:
The quick way of solving this is simply to consider the horizontal cross sectional area of the tank at the 2m water depth. This is 2/3 of the way up a tank having radius increasing linearly with depth, and so the radius is 2/3 of the 2m at the top, i.e: 4/3 m. So the area is:
πr²=π(4/3)2=16/9π
Now water having this cross section filling at 2 m³ per minute would fill a cylinder having this volume and cross section each minute. A cylinder has volume equal to cross section times length, so the cylinder would have a height of:
2/(16/9π)=9/8π metres
Therefore the water at this depth is rising at a rate of 9/8π meters per minute.
The longer way around is to calculate the water volume for a given depth and differentiate. We know that the volume of a cone is a third that of a cylinder with the same dimensions, ie: 1/3πr²h . So if our tank has radius R and height H, then the radius of the surface of the water at depth h is (R/H)h , and so the water volume having this depth is:
v=1/3π(R/H)h)²h=[tex]\frac{\pi r^{2} }{3H^3} h^2[/tex]
We’re interested in rate of change with respect to time, so let’s try differentiating this formula with respect to time, remembering that h is changing with time and so we need to use the chain rule:
[tex]\frac{dv}{dt}[/tex]=[tex]\frac{\pi r^{2} }{3H^2} 3h^2[/tex][tex]\frac{dh}{dt}[/tex]=[tex]\frac{\pi r^{2} }{H^2} h^2[/tex][tex]\frac{dh}{dt}[/tex]
Notice that we can solve this for [tex]\frac{dh}{dt}[/tex] , which is the rate of water height change, i.e: the value that we want:
[tex]\frac{dh}{dt}[/tex]= [tex]\frac{H^2 }{\pi r^{2}} h^2[/tex][tex]\frac{dh}{dt}[/tex]
The rate of change of water volume dv/dt is just the rate that water flows in, so now all we need to do is slot the given numbers into the equation:
dh/dt=2(3²/π2²2²)=9/8π meters per minute
Hence the answer is 9/8[tex]\pi[/tex] m/min.
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What is the range of the data in how many teams complete the escape room this week
Please add an image or some more data.