Theory: Rectangular coordinates and polar coordinates are two different ways of using two numbers to locate a point on a plane. Rectangular coordinates are in the form (x,y), where 'x' and 'y' are the horizontal and vertical distances from the origin: Polar coordinates are in the form: (r,), where 'r' is the distance from the origin to the point, and '' is the angle measured from the positive 'x' axis to the point: To convert between polar and rectangular coordinates, we make a right triangle to the point (x,y), like this:
Transcript
Theory:
Rectangular coordinates and polar coordinates are two different ways of using two numbers to locate a point on a plane.
Rectangular coordinates are in the form (x,y), where 'x' and 'y' are the horizontal and vertical distances from the origin:
Polar coordinates are in the form: (r,), where 'r' is the distance from the origin to the point, and '' is the angle measured from the positive 'x' axis to the point:
To convert between polar and rectangular coordinates, we make a right triangle to the point (x,y), like this:
1. Polar to Rectangular
From the diagram above, these formulas convert polar coordinates to rectangular coordinates:
x = r cos, y = r sin
So the polar point: (r,) can be converted to rectangular coordinates like this:
( r cos, r sin ) ( x, y )
2. Rectangular to Polar
Again, from the diagram above, these formulas convert rectangular coordinates to polar coordinates:
By the rule of Pythagoras:
Tan = y/x , so therefore:
= tan-1( y/x )
So the rectangular point: (x,y) can be converted to polar coordinates like this:
( , tan-1( y/x ) ) ( r , )
EXAMPLE PROBLEMS
A point has polar coordinates: (5, 30º). Convert to rectangular coordinates.
Example: A point has rectangular coordinates: (3, 4). Convert to polar coordinates.
Solution: r = square root of(3² + 4²) = 5, = tan-1(4/3) = 53.13º so (r,) = (5, 53.13º)
Question2010-11-7 7:41:58 PST
Solid mensuration problems help :)?
1.) An edge of a cube of ice measures 20 inches. The ice melts until it weighs half as heavy as the original size. Find the dimensions of the new ice cube. please I need the answer asap thanks in advance.
Answer
density of ice = 57.4 lb/ft^3 = 920 kg / m^3 = 0.92 g/cm^320 inches = 50.8 cmvolume of ice = 50.8 x 50.8 x 50.8volume of ice = 131.096.512 cm^3weight of ice = 131.096.512cm^3 x (0.92 g/cm^3)weight of ice = 120,608.791 gmsweight of ice = 120.609 kghalf of the wieght to determine dimension of the icenew wt. = 60.304 kg = 60304.396 gm.determine volume of new icevolume of new ice = wt of new ice / 0.92 gm/cm^2volume of new ice = 60304.396 gm. / 0.92Volume of new ice = 65,548.256 cm^3get the cube root to determine new dimension= 40.32 cm = 15.87 inches
Q uestion #8
The vertex of a cone is at the upper vertex of a cube. The base of a cone is circle which is inscribed in one of the faces of the cube (see illustration). If the edge of a cube is 10cm, find the volume of the cone.
Required: Volume,V of the coneSolution:Since the base is circular, then to find its area, we haveB=⺆D2/4=(10cm)2 / 4= 25cm2To find the volumeV = 1 Bh / 3=1/3 (25cm2)(10cm)Hence,V=261.799 cm3
Question #15
The diagonal of a cube is1 7 3 . 2 cm. Find its total surface and volume
Required: Total surface area, TVolume, VSolution:
Let D = diagonal of the cube = 173.2 cmd = diagonal of one facea = edge of the cube
Solve for d (diagonal of one face usingPythagorean Theorem)
For the total surface area:Solving for the edge of the cube
For the volume:
V = a3
= (100 cm)3
=1,000,000 cm3
A cube has one face that is equal to the total surface area of another cube. Find the ratio of their volume.
Solution:
The total surface area of a cube is 6 times the area of one face, which is the square of the length of the edge: A = 6s².The volume is the cube of the length of the edge: V = s³.
If the second cube's face = 6s², the second cube's edge is √(6s²) = s√6.
If the edge = s√6, the volume = (s√6)³ = s³·6^(3/2), or (6√6)s³.
The ratio of the volumes is (6√6)s³ / s³ = 6√6, approximately 14.7.
Find the total surface area (in m2) of a cube of side 10 m.
Total surface area of a cube = 6 × Side2
= 6 × 102 = 600 m2.
A solid metal cube of side 6 inches is placed in a rectangular tank whose length, width, and height are 1, 2, and 3 feet. What is the volume, in cubic units, of water that the tank can now hold?
note that 12 in = 1 ft. So the side of the cube is ft.
find the volume of the cube: . So the volume of the cube is of a cubic foot.
Now find the volume of the tank: . So the tank is 6 cubic feet.
Now subtract the volume of the cube from the volume of the tank: .
So the tank can now hold 5.875 cubic feet of water.
