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How do you find the volume of water in a hemispherical bowl of radius A and depth H using integration?
Slice the bowl horizontally into circles, then integrate the area of the circles. The area of each circle is (pi * r^2). The height of each slice is dh. The 1st (bottom) circle is r=0. The r^2 of each circle-slice is (2*A*h-h^2), where A is the spherical radius, and h is the variable height of any given slice. At the top of the water level, (r^2=2*A*H-H^2). Integrate the area over the interval h=0->H as follows: V=pi * integral[(2*A*h - h^2) dh]; h=0->H to yield V=pi * (2*A*h^2 / 2 - h^3 / 3); h=0->H V=pi * (A*H^2 - H^3 / 3). As a check, plug the full diameter (2*A) in for H. If you did the integration correctly, you will get the full volume of the sphere, (4/3 * pi * A^3).
What else can I help you with?
A hemi-spherical vessel has internal radius 0.5 m It is initially empty Water flows in at a constant rate of 1 liter per second Find an expression for the depth of the water after t seconds?
Since the vessel is hemispherical, its volume can be given by:V=((4/3)(pi)r3)/2V=(2/3)(pi)r3where r is the radius of the vessel.Since water is flowing into the vessel at a constant rate of 1 L/s, the volume of water in the vessel is thereby increasing at a constant rate of 1 L/s.By deriving the volume equation for the vessel with respect to time, we can equate the rate of change of the volume of water to the rate of change of the radius of the surface of water:dV/dt = (2/3)(pi)(3r2)(dr/dt)You must derive implicitly, so r3 derives down to 3r2(dr/dt) since the radius is also in itself a function of time. This equation can be cleaned up:dV/dt = 2(pi)r2(dr/dt)By solving for dr/dt, we get an expression for rate of change of the radius of the surface of water.dr/dt = (dV/dt)/(2(pi)r2)From the problem, we know that dV/dt is 1 L/s, and the radius of the hemisphere is a constant 0.5 m. We can substitute these known values into the equation:dr/dt = 1/(2(pi)(0.5)2)dr/dt = 2/piThis is the rate of change of the radius of the surface of water. The rate of change is a constant, which is important. Since it is constant, you can simply multiply this rate of change by a quantity of time to find the radius of the water level at any specific time. This is analogous to multiplying a constant velocity times a quantity of time to know an object's position at that time (a rate of change times an amount of change). We know that the vessel has an overall radius of 0.5 m, so the radius of the surface of water cannot exceed 0.5m.(dr/dt)= 2/pitherefore, depth at time t, D= 2t/piThis model gives the depth of the water (D) at any given time (t). As t increases, D(t) will return larger and larger values, which is expected since the water depth will increase as more water flows in.
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What is the maximal of nodes in binary tree of depth D?
2^(d+1) - 1
What is the Froude number at the beginning of the jump if sequent depth ratio of a hydraulic jump in a rectangular channel is 16.48?
12
How many yds of concrete in 366sq ft of floor 6inch thick?
As far as I know; to equal ft. and inches (366sq. ft. and 6 inches ) if you convert this iti is 366 divided in half to give you a 12 inch depth =183 cubic ft and divide by 3 to get cubic yards = 61 cubic yards
Circumference is 929 depth 6 what is the volume?
To find the volume we must first work out what the radius is: 2*pi*radius = 929 Divide both sides by 2*pi to find the value of the radius: radius = 147.8549421 volume = pi*147.85494212*6 volume = 412071.7235 cubic units
Calculate the volume of a cylinder with a radius of 3m and a depth of 5m?
V = 141.37 m3
What is the volume of a beaker that is 1.750 wide by 2.625 deep?
The volume of the cylinder is found by multiplying the depth by the square of the radius and by 3.142. The radius of the beaker is thus 6.31 cubic meters.
How to calculate the volume of water in a cylindrical well with given diameter and height?
Volume of water = (pi) x (Radius of the well)2 x (depth of the water)
A circular pool is 3.5 meters deep and has a radius of 9 meters what is the volume?
The volume of a cylinder is the area of its base multiplied by the height. [In this case, the "height" of the cylinder is the depth of the pool.] The base of the pool is a circle with radius of 9 m. The area of a circle is the radius squared times pi, which in this case is 81*pi m^2. Multiply this by the depth of 3.5 and you get 283.5 * pi cubic meters.
What formula do you use to find the volume of an object?
The formula to find the volume of an object depends on its shape. For a cube or rectangular prism, the volume is length x width x height. For a cylinder, the volume is π x radius^2 x height. And for a sphere, the volume is 4/3 x π x radius^3.
How do you calculate the volume of a round bathtub?
A round bath is a cylinder. The volume of a cylinder = area of the base x perpendicular height. Area of the base is πr2 (pi x radius x radius). The radius is half the diameter. The diameter is the width of the circular base. The perpendicular height will be the depth of the water, whether it's up to the top or up to where you have a bath.
How do you write a formula for volume equals area x average depth?
Volume= surface area (length x width) x depth re arrange to surface area= depth= Volume/Area Area= Volume/Depth
What is the volume of a circular spa with a 12 foot diameter and a constant depth of 3.5 feet?
The volume for radius r and depth d is = pi*r2*d. So V = pi*6*6*3.5 = 395.84 cubic feet (to 2 dp).
How many gallons in a pool size 15' x 36?
Volume is a 3 dimensional measurement, so 3 dimensions must be specified (length, width, depth for a rectangular pool). If it is a circle, then only need radius or diameter, and depth to calculate volume. If volume is calculated as cubic feet, then there are approximately 7.48 gallons (US) in a cubic foot.
How do you find the height of a rectangular box given the width depth and volume?
height * width * depth = volume height = volume / (depth * width) Volume = lengthXwidthXheight V=LWH H=V/LW
How much water is in a 27' round pool that is 45 feet deep?
Volume = Area x Depth Area of a circle = Pi * R^2 Pi = 3.1415 Radius = Diameter / 2 Diameter = 27 Radius = 13.5 Depth = 45 Volume = 3.1415 x 13.5^2 x 45 = 3.1415 x 182.25 x 45 = 572.54 x 45 = 25,764 cu. ft.
