To calculate the volume of a solid bounded by two functions, set up an integral in polar coordinates. The volume is found by integrating the difference of the two volumes described by the functions (50 - sqrt(x²+y²) - sqrt(x²+y²)) over r, θ, and z within their appropriate limits.
Explanation:The subject of the question is mathematics, more specifically, integral calculus in spherical or polar coordinates. Calculating the volume of a solid bounded by two functions can be done by taking the difference of the two function volumes.
The integrals, using polar coordinates (r and θ), are set up as follows:
Volume = ∫∫∫(50 - sqrt(x²+y²) - sqrt(x²+y²)) r dr dθ dz,
where the limits for r are [0, sqrt(50)], for θ are [0, 2π], and for z are [0, 50]. Also, note that 'x²+y²' is replaced with 'r²' because in polar coordinates, 'r' represents the square root of x^2 plus y^2.
This integral, when solved, gives the volume of the ice cream cone-shaped solid.
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The volume of the ice cream shaped solid can be found by converting the limits of the cone to polar coordinates and performing a double integral. The limits of the cone are represented by the expressions z = r and z = sqrt(50 - r^2). The volume integral will be ∫ _(theta=0 to 2*pi) ∫ _(r=0 to sqrt(50)) [(sqrt(50 - r^2)) - r] r dr dθ.
Explanation:To evaluate the volume of the solid resembling an ice cream cone, we can use the method of double integration in polar coordinates. Because we're dealing with a cone, polar coordinates are suitable as they're related to circular symmetry.
The limits of the cone are given by the functions z = sqrt(x^2 + y^2) and z = sqrt(50 - x^2 - y^2). In polar coordinates these become z = r and z = sqrt(50 - r^2).
To evaluate the volume of the cone, we need to subtract the volume of the lower cone (given by z = r) from the volume of the upper cone (given by z = sqrt(50 - r^2)). We convert the values to polar coordinates and use double integration as follows:
V = ∫ _(theta=0 to 2*pi) ∫ _(r=0 to sqrt(50)) [(sqrt(50 - r^2)) - r] r dr dθ
This integral, once solved, will give the volume of the ice cream shaped solid.
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A bathtub is draining at a constant rate. After 2 minutes, it holds 30 gallons of water. Three minutes later, it holds 12 gallons of water. Write an equation that represents the number y of gallons of water in the tub after x minutes.
3 minutes - 2 minutes = 1 minute
30 gallons - 12 gallons = 18 gallons
it is draining at 18 gallons per minute
equation would be: y = -18x
What is the value of 2x to the 4th degree + 5x when x=6
Finley makes 1 batch of her favorite shade of orange paint by mixing 5 liters of yellow paint with 3 liters of red paint. How many batches of orange paint can Finley make if she has 15 liters of red paint?
You have 900-grams of an an unknown radioactive substance that has been determined to decay according to D ( t ) = 900 e − 0.002415 ⋅ t D ( t ) = 900 e - 0.002415 ⋅ t where t t is in years. How long before half of the initial amount has decayed?
It will take __ years for half of the initial amount to decay. (Round to 1 decimal place)
clark wants to figure out how many pens to order for the office. there are 36 workers, and he need to order one pen for each worker. he know that for every eight people who prefer red , there are four who prefer blue. How many blue pens should he order?
8 like red 4 like blue
8+4 = 12
36/12=3
4*3 = 12 blue pens
How much will you pay the cashier for a candy bar that originally costs 79 cents and is now marked 15% off? Sales tax is 5.5%
What is 0.10221 in scientific notation
What is the logarithmic form of 144=12^2
The logarithmic form of 144=12^2 is log12(144) = 2.
Explanation:The logarithmic form of 144=12^2 is log12(144) = 2.
The logarithm is the power to which a base number must be raised to equal a given number. In this case, the base is 12, and we are trying to find the power that results in 144.
Since 12^2 = 144, the logarithmic form is log12(144) = 2.
the logarithmic form of 144=12^2 is b: [tex]\( \log_{12}(144) = 2 \)[/tex].
The correct option is (b).
The logarithmic form of the expression [tex]\( 12^2 = 144 \)[/tex] translates an exponential equation into a logarithmic equation. The general form of converting from exponential to logarithmic form is:
[tex]\[ b^y = x \] is equivalent to[/tex]
[tex]\\ \[ \log_b(x) = y \][/tex]
where:
- [tex]\( b \)[/tex] is the base of the exponential expression,
-[tex]\( y \)[/tex] is the exponent,
-[tex]\( x \)[/tex] is the result of the exponential expression.
Applying this to the given equation[tex]\( 12^2 = 144 \)[/tex], we have:
-[tex]\( b = 12 \)[/tex],
- [tex]\( y = 2 \)[/tex],
- [tex]\( x = 144 \)[/tex].
So, the logarithmic form is:
[tex]\[ \log_{12}(144) = 2 \][/tex]
complete question given below:
What is the logarithmic form of 12² = 144?
a.log212=144
b.log12144=2
c.log1442=12
d.log2144=12
An aluminum recycling center pays $0.08 per pound for cans. How much will Mark receive for 21.5 pounds of cans?
Jessica purchased an $84 suit. The sales tax is 6.5%. What is the total price of the suit including tax?
write 4.51 in expanded form
The expanded form of 4.52 is 4 + 0.5 + 0.01
Given.
4.51
We need to find the expanded form.
What is expanded form?It is a form of writing a given number by splitting it based on the place value.
Example:
56743
We can write it as:
50000 + 6000 + 700 + 40 + 3
Find the expanded form of 4.51
4 is in ones place
So,
4 - 4
5 is in tenths place
So,
0.5 - 0.5
1 is in hundredths place
So,
.01 - .01
The expanded form is:
4 + 0.5 + 0.01
Thus the expanded form of 4.52 is 4 + 0.5 + 0.01
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Find each percentage decrease. round to the nearest percentage from 16 bagels to 0 bagels
To solve this problem:
The formula would be: New Value – Old Value)/Old Value] x 100 or in the form of variable it would be: [(B-A)/A] x 100 = Answer.
So plugging in the information given in our problem it would be:
0 – 16 / 16 x 100 = percent of change
0/16 x 100 = 0%
The answer is 0%
Which are properties of rotations? Check all that apply.
1 Rotations are always counterclockwise.
2 Rotations preserve size.
3 Rotations of 360° map a figure to itself.
4 Lines connecting corresponding points on the pre-image and image are parallel.
5 Lines connecting corresponding points on the pre-image and image have equal length.
6 Lines connecting the center of rotation to the pre-image and the corresponding point on the image have equal length.
Answer:- The correct properties of rotation are as follows:-
2.Rotations preserve size.
3. Rotations of 360° map a figure to itself.
6. Lines connecting the center of rotation to the pre-image and the corresponding point on the image have equal length.
Explanation:-
Rotation transformation is a rigid transformation which create the image of a figure with same shape and size by rotating it by some degrees about the the center of rotation.1. False ; Rotations are always counterclockwise.→ it is not a property of rotation, ∵rotation can be clockwise or counterclockwise both.
2.True; Rotations preserve size. [by definition of rotation]
3.True; Rotations of 360° map a figure to itself. [It completes one turn and get back to the original position ]
4.False; Lines connecting corresponding points on the pre-image and image are parallel. [They can be intersect]
5.False ; Lines connecting corresponding points on the pre-image and image have equal length.[The point which is the center of rotation in a figure is nearest to its image rather than the other points. ]
6. True ; Lines connecting the center of rotation to the pre-image and the corresponding point on the image have equal length. [∵ all the points of the figure is rotated about the center of the point ,so the distance of the center of rotation to the pre-image and the corresponding point on the image have equal length]
Rotations have several properties, including preserving size, mapping a figure to itself when rotated by 360°, and having lines connecting the centre of rotation to the pre-image and image equal in length.
Properties of rotations:
Rotations are not always counterclockwise.Rotations preserve size.Rotations of 360° map a figure to itself.Lines connecting corresponding points on the pre-image and image are parallel.Lines connecting corresponding points on the preimage and image have equal lengths.Lines connecting the centre of rotation to the pre-image and the corresponding point on the image have equal lengths.What is the relationship between 0.04 and 0.004?
Manuel and Sonja are shopping for school supplies. Manuel is buying 5 notebooks and 3 pens at a cost of $21. Sonja is buying 6 notebooks and 5 pens at a cost of $28. The first equation, representing Manuel’s purchase, is mc015-1.jpg. What is the second equation needed to solve the system?
Answer:
Second equation is 6x + 5y = 28
Step-by-step explanation:
Let x is the cost of notebooks and y is the cost of pens.
Now we know Manuel is buying 5 notebooks and 3 pens at a cost of $21.
To represent this statement equation is 5x + 3y = 21
Similarly we will form the second equation to solve the system of equations.
Cost of 6 notebooks = 6x
Cost of 5 pens = 5y
Total cost of 6 notebooks and 5 pens = $28
So 6x + 5y = 28 will be the second equation.
By solving this system of equations we can find the cost of notebook as well as cost of pen purchased by Manuel.
Find the instantaneous rate of change of w with respect to z if w=7/3z^2
The instantaneous rate of change of 'w' with respect to 'z', for the function w = 7/3z², is 14/3*z.
Explanation:The question is asking for the instantaneous rate of change of 'w' with respect to 'z'. To compute this, we need to take the derivative of w with respect to z, for the given function w = 7/3z². Using the power rule, which says that the derivative of z^n is n*z^(n-1), our derivative would be: (7/3)*2*z^(2-1) simplifying this gives 14/3*z. Hence, the instantaneous rate of change of 'w' with respect to 'z' for given function is 14/3*z.
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Find the minimum and maximum of f(x,y,z)=x^2+y^2+z^2 subject to two constraints, x+2y+z=4 and x-y=8.
7,209,000 in the year 2050 and that this will be 24% of the population. What is the total population?
T Shirt sore keeps 7 white T-Shirts on the shelves for every 3 purple T- Shirts on the shelve how many white T-Shirts on the shelve if 15 purple T-Shirts on the shelve. Show how you get the answer in long form
G use lagrange multipliers to find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the given plane.x + 9y + 8z = 27
A restaurant requires customers to pay a 15% tip for the server. your family spends $45 on the meal.
a.how much money do you need to pay the server?
b.how much money will you spend total?
A) 45 * 0.15 = $6.75 paid to server
B 45 + 6.75 = $51.75 total
Is this equation correct? 18/48=27/72?
What is the equation of the line that passes through
(10, 4)
and is perpendicular to
5x−y=3
Find dy/dx by implicit differentiation. 3x + y = 8 + x2y2
To find dy/dx by implicit differentiation, differentiate both sides of the equation with respect to x, apply the chain rule and product rule, isolate the terms involving dy, and simplify the equation to find dy/dx.
Explanation:To find dy/dx by implicit differentiation, we need to differentiate both sides of the equation with respect to x. Using the chain rule and product rule, we can simplify the equation and solve for dy/dx. Starting with 3x + y = 8 + x^2y^2:
Step 1: Differentiate each term with respect to x.
Step 2: Apply the chain rule and product rule to simplify the equation.
Step 3: Solve for dy/dx by isolating the terms involving dy.
Step 4: Simplify the equation and rearrange to get dy/dx.
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Suppose that f(x+h)−f(x)= −6hx2−7hx−6h2x−7h2+2h3.
Find f′(x).
f′(x)=
Final answer:
To find f'(x), simplify the given expression by canceling the common factor of h, then take the limit as h approaches zero. The resulting derivative is f'(x) = -6x² -7x.
Explanation:
To find the derivative of function f(x), denoted as f'(x), we use the definition of the derivative and the given equation f(x+h) - f(x) = -6hx2 - 7hx - 6h2x - 7h2 + 2h3. The definition of the derivative is the limit of the average rate of change as the interval approaches zero, formally:
f'(x) = lim (h -> 0) [f(x+h) - f(x)] / h
In the given difference quotient, every term includes an h, allowing us to simplify by cancelling an h from each term before taking the limit as h approaches zero. This gives us:
f'(x) = lim (h -> 0) [-6x² - 7x - 6h2 - 7h + 2h2]
When h approaches zero, the terms containing h also approach zero. Thus, we are left with:
f'(x) = -6x² - 7x
How much force is required to push a 54-pound sofa across a carpeted floor?
Michael started a savings account with $300. After 4 weeks, he had $350 dollars, and after 9 weeks, he had $400. What is the rate of change of money in his savings account per week?
after 75% reduction, you purchase a new clothes dryer on sale for $200. What was the original price?
Original price was $800 for purchasing a new clothe dryer on sale for $200 after 75% reduction.
Original price is the price offered initially without applying any offer.
Discount offered is the amount of relaxation that is the difference between the marked price and selling price.
Given the following information:
Discount provided= 75%
Purchasing price after discount=200
Let the original price be x
x-75%x=200
[tex]x-\dfrac{75}{100}x=200\\\\100x-75x=200\times 100\\[/tex]
Solving the equation as follows to calculate the value of x:
[tex]25x=20000\\x=20000/25\\x=800[/tex]
Thus, original price was $800.
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Suppose x follows a continuous uniform distribution from 1 to 5. determine the conditional probability p(x >2.5 | x ≤ 4).
Final answer:
The conditional probability P(x > 2.5 | x ≤ 4) for a variable x following a uniform distribution from 1 to 5 is calculated using the conditional probability formula, resulting in a probability of 0.5.
Explanation:
The question asks to determine the conditional probability P(x > 2.5 | x ≤ 4) when x follows a continuous uniform distribution from 1 to 5. The formula for conditional probability is P(A|B) = P(A ∩ B) / P(B), where P(A ∩ B) is the probability of A and B both happening, and P(B) is the probability of B happening.
Here, event A is 'x > 2.5' and event B is 'x ≤ 4'. The probability of x being greater than 2.5 and less than or equal to 4, P(A ∩ B), is the length of the interval from 2.5 to 4, which is 1.5 units. Since x is uniformly distributed, this translates to a probability of 1.5/(5-1), because the total length of the distribution is 4 units. Event B 'x ≤ 4' has a probability of 3/(5-1) since the interval from 1 to 4 is 3 units long.
Hence, the conditional probability is P(A|B) = (1.5/4) / (3/4) = 1.5/3 = 0.5.
Evaluate. 4⋅(4+(2^6⋅2^−3))