To solve the given task, change the Cartesian coordinates into spherical coordinates. Rewrite the integrand and the solid hemisphere limits according to these new variables. Evaluate the final triple integral using standard techniques.
Explanation:To solve the given problem, the first step is to convert the Cartesian coordinates into spherical coordinates. In spherical coordinates, a point in three dimensions is represented by three parameters: ρ (rho) for the radial distance, θ (theta) for the azimuthal angle, and φ (phi) for the polar angle.
Here, given x² + y² + z² ≤ 4, z ≥ 0, the given region corresponds to a solid hemisphere centered at the origin and radius 2 in spherical coordinates, goes as 0 ≤ ρ ≤ 2, 0 ≤ θ ≤ 2π, and 0 ≤ φ ≤ π/2. Our spherical coordinates are x = ρsin(φ)cos(θ), y = ρsin(φ)sin(θ), and z = ρcos(φ).
Then, rewrite the integrand using spherical coordinates, (9 - x² - y²) becomes (9 - ρ²sin²(φ)) and dv becomes ρ²sin(φ)dρdφdθ.
The final integral that needs to be evaluated will read: ∫₀^2 ∫₀^π/2 ∫₀^2π (9 - ρ²sin²(φ))ρ²sin(φ) dθ dφ dρ. This is a triple integral in spherical coordinates that you can now solve using standard integration techniques.
Learn more about Spherical Coordinates here:https://brainly.com/question/31745830
#SPJ3
Find the area of the shaded region. The units are in millimeters.
Use 3.14 for π. Round your answer to two decimal places.
John predicts the basketball team will score 68 points. If the basketball team actually scores 49 points, what is John's percent error? Round your answer to the nearest tenth of a percent.
What is the value of of angle BAD:
30◦
5◦
15◦
20◦
If all of the diagonals are drawn from a vertex of a pentagon, how many triangles are formed?
A picture helps. There are 2 non-adjacent vertices, so two diagonals can be drawn, resulting in 3 triangles.
Drawing all diagonals from one vertex of a pentagon to the non-adjacent vertices creates three triangles within the pentagon.
When all the diagonals are drawn from one vertex of a pentagon, it divides the pentagon into several triangles.
A pentagon has five sides, and when selecting a single vertex, we can draw diagonals to the non-adjacent vertices. In this case, there are three non-adjacent vertices, which means we can draw three diagonals, thus creating three triangles inside the pentagon.
This is because each diagonal, combined with two sides of the pentagon, forms a triangle.
The manager at an ice cream shop keeps track of the number of deluxe cones and regular cones sold each day and the total money received. On Saturday, a total of 95 cones were sold, and the money collected was $628 . If deluxe cones are sold for $8 and regular cones are sold for $5 , how many deluxe cones and regular cones were sold?
Final answer:
By forming a system of equations and applying the elimination method, we conclude that the ice cream shop sold 51 deluxe cones and 44 regular cones on Saturday.
Explanation:
How to Solve for the Number of Deluxe and Regular Cones Sold
To solve for the number of deluxe cones and regular cones sold at the ice cream shop, we need to set up a system of equations. We are given that a total of 95 cones were sold and the total money collected was $628. Deluxe cones are sold for $8 and regular cones for $5. Let's define our variables:
x = number of deluxe conesy = number of regular conesNow, we can establish our equations:
The total number of cones sold is 95: x + y = 95The total money collected is $628: 8x + 5y = 628To solve the system, we can use the substitution or elimination method. Let's use elimination for this example.
Multiply the first equation by 5:
5x + 5y = 475Now subtract this from the second equation:
(8x + 5y) - (5x + 5y) = 628 - 4753x = 153x = 51With 51 deluxe cones sold, we can substitute that into the first equation to find y:
51 + y = 95y = 95 - 51y = 44So, 51 deluxe cones and 44 regular cones were sold on Saturday.
Which of the following statements is true? A 6.75 < 6.759 < 6.751 < 6.85 B 5.55 < 5.559 < 5.65 < 5.69 C 4.11 < 4.12 < 4.17 < 4.15 D 7.42 < 7.41 < 7.40 < 7.39
Suppose that a particular artillery piece has a range r = 8320 yards . find its range in miles. use the facts that 1mile=5280ft and 3ft=1yard.
The range in miles is 4.72 mile
What do you mean by 1 mile?mile, any of various units of distance, such as the statute mile of 5,280 feet (1.609 km). It originated from the Roman mille passus, or “thousand paces,” which measured 5,000 Roman feet.
Given : range= 8320 yards
1 mile= 5280 feet
3 feet= 1 yard
Now,
8320yards X 3ft = 24,960
Now, divide that by 5,280ft in a mile
= 24960/5280
= 4.72 miles
Hence, range is 4.72 miles.
Learn more about miles here:
https://brainly.com/question/23858115
#SPJ2
Final answer:
To convert the artillery piece's range of 8320 yards to miles, you multiply by 3 to change yards to feet and then divide by 5280 to convert feet to miles, resulting in a range of approximately 4.727 miles.
Explanation:
To find the artillery piece's range in miles, starting with its range in yards, we need to use unit conversion factors. Knowing that 1 mile equals 5280 feet and 3 feet equals 1 yard, we can set up a conversion to calculate the distance in miles.
Steps to Convert Yards to Miles:
Convert the given distance from yards to feet by multiplying the number of yards by 3 (since there are 3 feet in a yard).Next, convert the distance from feet to miles by dividing the number of feet by 5280 (since there are 5280 feet in a mile).For an artillery range of 8320 yards:
8320 yards is equivalent to 8320 yards * 3 feet/yard = 24960 feetThen 24960 feet / 5280 feet/mile gives us the range in miles.When you divide 24960 by 5280, you get 4.727 miles (rounded to three decimal places).
The function g(x) = 3(x - 3)2 + 2 will translate 3 units to the right. True or False?
A section of land has an area of 1 square mile and contains 640 acres. determine the number of square meters in an acre
An acre, a unit of area commonly used in the Imperial and U.S. customary systems, is approximately 4047 square meters. This is calculable by using the known conversions between acres, square miles, and square meters.
Explanation:To determine the number of square meters in an acre, it's helpful to understand the relationship between these units. A square mile is equivalent to 640 acres. Therefore, the area of one acre is 1/640 of a square mile. However, these are both Imperial measurements, and we want to convert to a metric measurement - square meters.
To make this conversion, we need to identify the conversion factor between square miles and square meters. There are 2,589,988.11 square meters in a square mile.
Our starting point is that 1 acre = 1/640 square mile. Next, we substitute the number of square meters in a square mile:
1 acre = 1/640 x 2,589,988.11 square meters = 4046.86 square meters.
Therefore, there are approximately 4047 square meters in an acre.
Learn more about Converting Measurements here:https://brainly.com/question/34081145
#SPJ6
which property of congruence would prove that BD ≈ BD?
the reflexive property of congruence
the symmetric property of congruence
the transitive property of congruence
None of the choices are correct.
Select the correct inequality for the graph below: A solid line passing through points (1, 2) and (2, 5) has shading below. y < 3x − 1 y ≤ 3x − 1 y ≥ 3x − 1 y > 3x − 1
Here is your answer:
Solving the equation:
[tex] (5-2)\div(2-1)= 3 [/tex][tex] \frac{y - y1}{(x - x1) } [/tex][tex] y-5=3(x-2) [/tex][tex] y= 3x- 6+ 5 [/tex]" [tex] y= 3x-1 [/tex] " or option B.Hope this helps!
Step 1
Find the equation of the line that passes through points [tex](1, 2)[/tex] and [tex](2, 5)[/tex]
Find the slope of the line
The formula to calculate the slope is equal to
[tex]m=\frac{y2-y1}{x2-x1}[/tex]
substitute the values
[tex]m=\frac{5-2}{2-1}[/tex]
[tex]m=\frac{3}{1}[/tex]
[tex]m=3[/tex]
Find the equation of the line
The equation of the line into slope-point form is equal to
[tex]y-y1=m(x-x1)[/tex]
we have
[tex]m=3[/tex]
[tex](1, 2)[/tex]
substitutes
[tex]y-2=3(x-1)[/tex]
[tex]y=3x-3+2[/tex]
[tex]y=3x-1[/tex]
Step 2
Find the equation of the inequality
we know that
The solution is the shaded area below the solid line
therefore
the inequality is
[tex]y\leq 3x-1[/tex]
the answer is
[tex]y\leq 3x-1[/tex]
see the attached figure to better understand the problem
The formula can be used to find the velocity v in feet per second of an object that has fallen h feet. Find the velocity of an object that has fallen 23 feet. Round your answer to the nearest hundredth.
2(8+h) simplify using distrubtive property
Find the work done by a force fâ =1.8i^+2.2j^n as it acts on an object moving from the origin to the point râ =56i^+31j^m.
The work done by the force is found by taking the dot product of the force vector and the position vector. This is achieved by multiplying corresponding components of the vectors and then adding the results together.
Explanation:To find the work done by the force, we use the formula Work = Force . Displacement, where the '.' signifies a dot product. In this case, the force vector is fâ = 1.8i^ + 2.2j^ and the position vector, râ = 56i^ +31j^. The dot product between force fâ and the displacement (position vector) râ is calculated by multiplying corresponding i and j components and then adding them together. Therefore, Work = (1.8i^.56i^) + (2.2j^.31j^). Perform the multiplication for both components and then add the results to get the total work done by the force.
Learn more about Work Done here:https://brainly.com/question/35917320
#SPJ2
Helpp! -- A piece of wire 5 inches long is to be cut in two pieces. One piece is x inches long and is to be bent into the shape of a square. The other piece is to be bent in the shape of a circle. Find an expression for the total area made up by the square and the circle as a function of x.
The expression for the total area made up by the square and the circle as a function of x is f(x) = x^2/16 + (25 - 10x + x^2)/(4*Pi) square inches.
Explanation:To solve this problem, you need to know the formulas for the areas of a square (Area = side^2) and a circle (Area = pi * radius^2). If a piece of the wire with length x inches is to be bent into the shape of a square, then each side of the square will be x/4 inches long (since a square has 4 equal sides). Therefore, the area of the square = (x/4)^2 = x^2/16 sq.inches.
The remaining piece of the wire is 5-x inches long and is to be bent into the shape of a circle. The circumference of the circle is 5-x inches (since it uses the remaining wire), therefore the radius of the circle is (5-x)/(2*Pi) inches. Therefore, the area of the circle = Pi*((5-x)/(2*Pi))^2 = (25 - 10x + x^2)/(4*Pi) sq.inches.
The total area made up by the square and the circle is the sum of their areas, therefore the function representing the total area f(x) = x^2/16 + (25 - 10x + x^2)/(4*Pi) sq.inches.
Learn more about Area Function here:https://brainly.com/question/35160023
#SPJ11
someone please help me to find the answer to this geography problem please explain
James wants to tile his floor using tiles in the shape of a trapezoid. To make the pattern a little more interesting he has decided to cut the tiles in half along the median. The top base of each tile is 15 inches in length and the bottom base is 21 inches. How long of a cut will John need to make so that he cuts the tiles along the median?
Answer:18 inches
Step-by-step explanation:
The answer would be 18 inches.
What is the median of a trapezoid?The median of a trapezoid is the segment that connects the midpoints of the non-parallel sides.
The length of the median is the average of the length of the bases.
The top base of each tile is 15 inches in length and the bottom base is 21 inches,
Add the top base and bottom base,
So, 15+21=36
Now, divide that by 2
⇒ 18
Hence, the answer would be 18 inches.
Learn morea about trapezoid here
https://brainly.com/question/8643562
#SPJ2
Evaluate the surface integral s f · ds for the given vector field f and the oriented surface s. in other words, find the flux of f across s. for closed surfaces, use the positive (outward) orientation. f(x, y, z) = x i + y j + 10 k s is the boundary of the region enclosed by the cylinder x2 + z2 = 1 and the planes y = 0 and x + y = 2
A set of data with a mean of 45 and a standard deviation of 8.3 is normally distributed. Find the value that is +1 standard deviation away from the mean.
Answer:
53.3
Step-by-step explanation:
Add one standard deviation to the mean:
45 +1(8.3) = 53.3
Answer:
53.3
Step-by-step explanation:
+1 standard deviation away from the mean is 45+1(8.3)
A pizzeria owner wants to know which pizza topping is least liked by her customers so she can take it off the menu. She used four different methods to find this information.
Method 1: The owner asked every third customer to rate all the pizza toppings in order of preference.
Method 2: The owner gave all customers a toll-free telephone number and asked them to phone in their topping preferences.
Method 3: The owner asked the preferences of every other teenager who entered the pizzeria.
Method 4: The owner reviewed the pizzeria’s complaint cards and assessed all complaints related to pizza toppings.
Which method is most likely to give a valid generalization
The valid method is the owner asked every third customer to rate all the pizza toppings in order of preference. The correct option is A.
What is feedback?information that is utilized as a foundation for improvement, such as data on how people react to a product or how well someone performs a task.
I suggest method 1 because it is unbiased and systematic. The second method requires a lot of the initiative of customers and is likely to have extreme cases only (liked very much or disliked very much).
The third is biased towards teenagers, which may not be the only category of customers who ordered pizzas. Again, the fourth requires initiative from the customer, so biased towards customers who had something to say.
Therefore, the valid method is for the owner to ask every third customer to rate all the pizza toppings in order of preference. The correct option is A.
To know more about feedback follow
https://brainly.com/question/25658249
#SPJ2
The pressure,p, of a gas varies inversely with its volume,v. Pressure is measured in units of pa. Suppose that a particular amount of a gas is initially at a pressure of 104 pa at a volume of 108 L. If the volume is expanded to 432 L, what will the new pressure be
A rectangle is 6 centimeters longer than is wide and has an area of 280 square centimeters if l represent the length,in centimeters of the rectangle solving which of the following formulas will give the correct value for l?
4 students equally share 3/4 of a pizza. How much of the pizza does each student get?
To find out how much pizza each of 4 students gets when they equally share 3/4 of a pizza, we divide the amount of pizza (3/4) by the number of students (4). The result is 3/16, which means each student gets 3/16 of the pizza.
Explanation:The question asks how 4 students can equally share 3/4 of a pizza. In mathematics, to divide a quantity equally amongst a group, we perform division. So, in this scenario, we need to divide 3/4 (the pizza) by 4 (the students).
Step 1: Represent the given scenario as a division problem: 3/4 ÷ 4. Step 2: Convert the division into multiplication by the reciprocal. This changes our problem into a multiplication problem: 3/4 * 1/4. Step 3: Multiply the numerators together to get the numerator of the answer: 3 * 1 = 3. Step 4: Multiply the denominators together to get the denominator of the answer: 4 * 4 = 16.
So, the answer of 3/4 ÷ 4 is equal to 3/16. Therefore, each student gets 3/16 of the pizza.
Learn more about Division of Fractionshttps://brainly.com/question/4528098
#SPJ11
Which figure has the correct lines of symmetry drawn in?
Two garden plots are to have the same area. One is square and one is rectangular. The rectangular plot is 2 meters wide and 18 meters long. How long is one side of the of the square garden plot in meters?
area of rectangle = l x w
area of rectangle = 2*18 = 36 square meters
area of square = S^2, or s = square root(area)
sqrt(36) = 6
length of side of square = 6 meters
The side length of the square garden plot with the same area as a rectangular plot of dimensions 2 meters by 18 meters is 6 meters.
The student is asking about finding the length of a side of a square garden plot that has the same area as a rectangular garden plot measuring 2 meters wide by 18 meters long. To find the area of the rectangular plot, we multiply the length by the width: 2 meters imes 18 meters = 36 square meters.
Since the area of the square plot must equal the area of the rectangular plot, we can use the formula for the area of a square, A = s2, where s is the side length of the square. Therefore, to find the side length of the square plot, we take the square root of the area of the rectangular plot:
√(36 square meters) = 6 meters
So, the side length of the square garden plot is 6 meters.
Find the area of a square with radius 17√2 in. Round to he nearest whole number. Please help me!!
The area of the square with radius 17√2 is 1156 square inches.
What is area?The area is the amount of space within the perimeter of a 2D shape
What is the formula for the area of square?The formula for the area of square is
[tex]A = \frac{1}{2} d^{2}[/tex]
Where, d is the length of the diagonal of the square.
According to the given question.
The radius of the square is [tex]17\sqrt{2}[/tex] in.
⇒ Diameter or the length of the square = 2 × 17 √2 = 34√2
Therefore,
The area of the square
= [tex]\frac{1}{2}( 34(\sqrt{2)} )^{2}[/tex]
= [tex]\frac{2312}{2}[/tex]
=[tex]1156\ in^{2}[/tex]
Hence, the area of the square with radius 17√2 is 1156 square inches.
Find out more information about area of square here:
https://brainly.com/question/22964077
#SPJ2
The area of a rectangular patio is 5 5/8 square yards, and it's length is 1 1/2 yards. What is the patio's width in yards?
Davis Construction is building a new housing development. They begin by mapping the development out on a coordinate grid. They place one of the two swimming pools at coordinate (-16, -25) and the other at (18, 23). If they want the club house to be exactly halfway between the two pools, find the coordinate point for the clubhouse
To solve this problem, we can use the following ratio formulas to find for the coordinates of the club house:
x = [m / (m + n)] (x2 – x1) + x1
y = [m / (m + n)] (y2 – y1) + y1
Where,
m and n are the ratios (0.5 and 0.5 each since the club house are to be located exactly halfway)
x1 and y1 are the x and y coordinates of the 1st pool (-16, -25)
x2 and y2 are the x and y coordinates of the 2nd pool (18, 23)
Substituting:
x = [0.5 / (0.5 + 0.5)] (18 - -16) + -16
x = 1
y = [0.5 / (0.5 + 0.5)] (23 - -25) + -25
y = -1
Therefore the club house should be located at coordinates (1, -1).
Pick the correct description of the line 5x + 3y=15
Answer:
The correct description is [tex]y=5-\frac{5x}{3}[/tex]
Step-by-step explanation:
In this case, firts we have to organize the equation in the normal way, where y is alone in one side, this is:
[tex]5x+3y=15\\3y=15-5x\\y=\frac{15}{3}-\frac{5x}{3} \\y=5-\frac{5x}{3}[/tex]
The correct description is [tex]y=5-\frac{5x}{3}[/tex]
Now we can see that the slope of the equation is [tex]-\frac{5}{3}[/tex]
Calculus AB
lim x-->4 sqrt(x+5)-3 / 4-x
The answer is apparently -1/6
How do you solve this