1. The Volume Of A Sphere Is 2100 cm Cube And Its Radius Is xcm. Find The Value Of x And Hence The Curved Surface Area Of The Sphere(take pie=22/7). 2. A Hemispherical Bolw Of Diameter 30cm Has The Same Volume As A Cylinder Of Height 22.5cm. Find The Radius Of The Cylinder(take pie=22/7).
4π/3 x^3 = 2100
x^3 = 1575/π
a = 4πx^2 = ?
2π/3 r^3 = 2π/3*15^3 = 2250π
for the cylinder, then
πr^2 45/2 = 2250π
r = 10
Vary good to me
1. To find the value of x, we need to use the formula for the volume of a sphere:
V = (4/3) * π * r^3
We are given that the volume V is 2100 cm^3. Therefore, we can write the equation as:
2100 = (4/3) * (22/7) * x^3
Simplifying the equation, we get:
(4/3) * (22/7) * x^3 = 2100
Multiply both sides by (3/4) * (7/22) to isolate x^3:
x^3 = 2100 * (3/4) * (7/22)
x^3 = 2250
Taking the cube root of both sides, we find:
x = ∛2250
Using a calculator, we can evaluate the cube root to approximately:
x ≈ 13.7724 cm
Now, to find the curved surface area of the sphere, we can use the formula:
A = 4 * π * r^2
Substituting the value of x we found:
A = 4 * (22/7) * (13.7724)^2
Simplifying, we get:
A ≈ 3018.4996 cm^2
Therefore, the curved surface area of the sphere is approximately 3018.4996 cm^2.
2. To find the radius of the cylinder, we first need to find its volume using the formula:
V = π * r^2 * h
We are given that the height h is 22.5 cm. The volume of the cylinder is the same as the volume of the hemisphere, which we can calculate as:
V = (2/3) * π * r^3
Equating the volumes of the cylinder and the hemisphere, we get:
π * r^2 * 22.5 = (2/3) * π * r^3
Cancelling π from both sides, we get:
r^2 * 22.5 = (2/3) * r^3
Dividing both sides by r^2 to isolate r, we have:
22.5 = (2/3) * r
Multiplying both sides by (3/2), we find:
r = (3/2) * 22.5
Simplifying, we get:
r = 33.75 cm
Therefore, the radius of the cylinder is 33.75 cm.
1. The volume of a sphere can be calculated using the formula V = (4/3)πr^3, where V is the volume and r is the radius of the sphere.
Given that the volume of the sphere is 2100 cm^3, we can write the equation as follows:
2100 = (4/3)πr^3
To find the value of x (radius), we need to solve for r in the above equation. Let's go step by step:
Step 1: Divide both sides of the equation by (4/3)π to isolate r^3.
(4/3)πr^3 = 2100
Step 2: Divide both sides of the equation by (4/3)π to solve for r^3:
r^3 = (2100) / [(4/3)π]
Step 3: Simplify the right side of the equation:
r^3 = 2100 / (4/3)*(22/7)
Step 4: Evaluate the expression on the right side:
r^3 = 225
Step 5: Find the cube root of both sides to solve for r:
r = ∛225
r ≈ 6.30 cm
So, the value of x (radius) is approximately 6.30 cm.
To find the curved surface area of the sphere, we use the formula A = 4πr^2, where A is the curved surface area and r is the radius.
Substituting the value of x we found earlier,
A = 4(22/7)(6.30)^2
A = (4/7)(22)(6.30)^2
A ≈ 498.52 cm^2
Therefore, the curved surface area of the sphere is approximately 498.52 cm^2.
2. The volume of a hemisphere can be calculated using the formula V = (2/3)πr^3, where V is the volume and r is the radius of the hemisphere.
Given that the hemisphere has the same volume as a cylinder, we can equate their volumes and solve for r.
The volume of the hemisphere is given by (2/3)πr^3, and the volume of the cylinder is given by πr^2h, where h is the height of the cylinder.
Let's set up the equation:
(2/3)πr^3 = πr^2h
Given that the diameter of the hemisphere is 30 cm, the radius would be half of that, i.e., r = (30/2) cm = 15 cm.
Now, given that the height of the cylinder is 22.5 cm, we can substitute the values into the equation:
(2/3)(22/7)(15^3) = (22/7)(15^2)(22.5)
Let's solve for h:
(2/3)(22/7)(15^3) = (22/7)(15^2)(22.5)
[(2/3)(22/7)(15^3)] / [(22/7)(15^2)] = h
Canceling out common factors:
[2/(3*15)] * 15 = h
Simplifying:
2/3 * 15 = h
h = 10 cm
Therefore, the radius of the cylinder is 15 cm.