Surface Area Of A Sphere

The formula to calculate the surface area of a sphere is given by , where is the surface area and is the radius of the sphere.

Given: and , we need to find the surface area in .

First, convert the radius from centimeters to meters: .

Substituting the values into the formula:

Simplify this expression:

Cancel the common factor of in the numerator and denominator:

Now, compute the multiplication and division:

Therefore, the correct surface area is , which matches the first option.

To find the surface area of a sphere, we use the formula:

S = 4πr²

where S is the surface area and r is the radius of the sphere.

Given that the diameter of the sphere is 28 cm, we can find the radius by dividing the diameter by 2:

r = 28 cm / 2 = 14 cm

Now, substitute the radius into the surface area formula:

S = 4π(14 cm)²

S = 4π(196 cm²)

S = 784π cm²

Approximating π as 3.14, we calculate:

S ≈ 784 × 3.14 cm²

S ≈ 2461.76 cm²

Rounding this to the nearest whole number, we get:

S ≈ 2464 cm²

Therefore, the surface area of the sphere is 2464 cm².

To determine the rate of change of the surface area of a spherical balloon when its volume is increasing at a specific rate, we can use the known formulas for the volume and surface area of a sphere. The volume of a sphere with radius is given by the formula:

Given that , we need to find when . The surface area of a sphere is given by:

To solve this, we'll differentiate both and with respect to time :

First, solve for using the volume differential equation:

Now, substitute into the surface area differential equation:

The correct rate of change of its surface area when the radius is 2 cm is therefore .

The total surface area of a hemisphere is given by:

Differentiate with respect to :

At , calculate :

Substitute into :

Thus, the rate of change of the total surface area with respect to the radius is: