using differential determine the appropriate increase in the surface area of a sphere if the radius increases from 4 cm to 4.05 cm.
a = 4 pi r^2
da = 8 pi r dr
You have dr = 0.05 and r=4, so
da = 8 pi * 2 * 0.05 = 0.8 pi cm^2
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To determine the appropriate increase in the surface area of a sphere when the radius increases from 4 cm to 4.05 cm using differentiation, we can make use of the formula for the surface area of a sphere:
Surface Area (A) = 4πr^2
Where:
A = Surface Area
r = Radius
To find the increase in surface area (∆A), we need to find the derivative of the surface area formula with respect to the radius (dr/dA). Then, we can evaluate this derivative at the initial radius of 4 cm to get the increase in the surface area.
Step 1: Differentiate the surface area formula with respect to the radius:
dA/dr = d(4πr^2)/dr
dA/dr = 8πr
Step 2: Evaluate the derivative at the initial radius of 4 cm:
dA/dr = 8π(4)
dA/dr = 32π
So, the derivative of the surface area with respect to the radius is 32π.
Step 3: Calculate the increase in surface area when the radius increases from 4 cm to 4.05 cm:
∆A = dA/dr * ∆r
∆A = 32π * (4.05 - 4)
∆A = 32π * 0.05
∆A ≈ 5.03 cm^2
Therefore, the appropriate increase in the surface area of the sphere is approximately 5.03 cm^2 when the radius increases from 4 cm to 4.05 cm.
To determine the appropriate increase in the surface area of a sphere when the radius increases from 4 cm to 4.05 cm using differentials, you can apply the concept of derivatives.
The formula for the surface area of a sphere is given by:
A = 4πr²
Where A represents the surface area and r represents the radius.
To find the change in surface area, we need to find the derivative of the surface area equation with respect to the radius (dr):
dA/dr = d(4πr²)/dr
To simplify the differentiation, we can bring the constant 4π outside:
dA/dr = 8πr
Now, we substitute the initial radius (4 cm) into the derivative equation:
dA/dr = 8π(4)
dA/dr = 32π
This value represents the rate of change of the surface area with respect to the radius.
To find the appropriate increase in the surface area when the radius increases from 4 cm to 4.05 cm, we need to multiply the rate of change by the difference in radius:
ΔA = (dA/dr) * Δr
Where ΔA represents the change in surface area and Δr is the difference in radius.
Substituting the values:
ΔA = 32π * (4.05 - 4)
ΔA = 32π * 0.05
ΔA ≈ 5.03 cm²
Therefore, the appropriate increase in the surface area of the sphere is approximately 5.03 cm² when the radius increases from 4 cm to 4.05 cm.