We want to find a solution to the initial value problem:
We can start by integrating the equation once:
Using the initial condition , we can determine the integration constant
:
Therefore, we have:
We can now integrate again:
The integration constant is determined by using
:
Finally, the solution is:
Answer:
approximately 8.5
Step-by-step explanation:
So you can use the distance formula: which is derived from the Pythagorean theorem because normally x2-x1 would kind of give the distance length between x2 and x1 except sometimes it would be negative which wouldn't make much sense right? but it doesn't matter because it's being squared so the value ultimately becomes positive, and the same thing goes for the y-value. the distance between x2 - x1 really represents the base and y2-y1 represents the height and then adding them together gives the hypotenuse squared which is why you take the square root over the entire thing to find the distance, since the hypotenuse is the shortest distance between two points
Now all you have to do is plug the values in to get
Answer:
The rotation of the rectangle, of area 12 squared units, about y-axis produces a cylinder, with a volume of 48 cubic units.
Step-by-step explanation:
Rotation about x-axis or y-axis at an angle requires some general rules. When a 2D figure is rotated about a line a 3d figure would be produced. This can be done through a process called solid of revolution.
From the question, connecting the given coordinates produces a rectangle with a length of 4 units and width 3units.
Area of the rectangle = Length × width
= 4 × 3
= 12 square units
The required rule for rotation of a figure about y-axis is: (x, y) → (-y, x).
If the given rectangle is rotated about y-axis, a cylinder would be formed.
The volume of the cylinder = V = h
where: r = 4 units and h = 3 units.
V = ×
× 3
= 48 cubic units
Therefore, the rotation of the rectangle of area 12 squared units produces a cylinder with volume of 48 cubic units.