if a cylinder with a height equal to its diameter has a volume of 20 cubic inches. How can you fine the volume of a sphere whose radius is equal to that of a cylinder
you need 2 formulas to do this
volumeof cylinder: pi*r^2*h
volume of sphere: 4/3 * Pi * r^3
you wanna find out the cylinder's height since you have the diameter already. plug in; if height is equal to diameter then radius is half the height, therefore solve in terms of height. r=h/2
V(c)= pir^2h
20= pir^2h
20= pi* (h/2)^2 * h
20= pi * (h^2/4) * h
20= pi * h^3/4
80= pi* h^3
80/pi= h^3
h= cube root of (80/pi)
h= 2.95 inches
r= 1.475 inches
plug this into volume of sphere
V= 4/3 Pi r^3
V= 4/3 pi * (1.475^3)
V= 4/3 pi * (3.21)
V= 4.28 pi
V= 13.446 inches cubed or 13.45 inches cubed
To find the volume of a sphere whose radius is equal to the diameter of a cylinder with a given volume, you can follow these steps:
1. Determine the diameter of the cylinder: Since the height of the cylinder is equal to its diameter, we can assume that the diameter is equal to "d".
2. Calculate the radius of the cylinder: The radius is half the diameter, so the radius "r" of the cylinder is equal to "d/2".
3. Use the given volume of the cylinder: According to the problem, the volume of the cylinder is 20 cubic inches.
4. Use the formula for the volume of a cylinder: The volume of a cylinder is given by the formula V = π * r^2 * h, where "V" is the volume, "π" is a constant approximately equal to 3.14, "r" is the radius, and "h" is the height. In this case, since the height is equal to the diameter, we can substitute "h" with "d".
5. Solve for the radius of the cylinder: Now we need to find the radius "r" of the cylinder in terms of "d". Substituting h = d into the volume formula, we have 20 = π * r^2 * d.
6. Substitute the value of the radius "r" into the equation: Using the relationship r = d/2, we substitute it into the equation and get 20 = π * (d/2)^2 * d.
7. Simplify the equation: Squaring the fraction (d/2)^2 gives (1/4)d^2, so the equation becomes 20 = π * (1/4)d^2 * d.
8. Solve for the diameter "d": Rearranging the equation, we have (1/4)πd^3 = 20.
9. Solve for "d": Divide both sides of the equation by (1/4)π, which simplifies to 4/π. Then solve for "d" by taking the cube root of both sides: d = (20 * 4/π)^(1/3).
10. Calculate the radius of the sphere: Since the radius of the sphere is equal to the diameter of the cylinder, we can substitute the value of "d" into the equation to find the radius of the sphere.
11. Calculate the volume of the sphere: Once you have the radius of the sphere, you can use the formula for the volume of a sphere, which is V = (4/3)πr^3, to calculate the volume.
By following these steps, you should be able to find the volume of the sphere whose radius is equal to the diameter of the given cylinder.