Question

Use differentials to approximate the increase in the area of a soap bubble when its radius increases from 4 inches to 4.035 inches.

Answer: ______square inches

Answer 1

the increase in area of the soap bubble when its radius increases from 4 inches to 4.035 inches is approximately 3.52 square inches.

Explanation:

Let's start by using the formula for the surface area of a sphere to find the area of the soap bubble:

A = 4πr^2

where A is the surface area and r is the radius.

We want to find the increase in area when the radius increases from 4 inches to 4.035 inches. Let's call the initial radius r1 and the final radius r2:

r1 = 4 inches

r2 = 4.035 inches

We can find the increase in area by taking the difference between the areas at the two radii:

ΔA = A2 - A1

where A1 is the area at r1 and A2 is the area at r2.

To use differentials, we start by finding the differential of A with respect to r:

dA/dr = 8πr

This tells us how much the area changes for a small change in the radius.

We can use this differential to approximate the change in area between r1 and r2:

ΔA ≈ dA/dr * Δr

where Δr = r2 - r1 is the change in radius.

Plugging in our values, we get:

ΔA ≈ (8πr) * (4.035 - 4)

ΔA ≈ 8π(4) * 0.035

ΔA ≈ 3.52 square inches

Therefore, the increase in area of the soap bubble when its radius increases from 4 inches to 4.035 inches is approximately 3.52 square inches.

Answer 2

The formula for the surface area of a sphere is A = 4πr^2, where r is the radius of the sphere.

To approximate the increase in the area of a soap bubble when its radius increases from 4 inches to 4.035 inches, we can use differentials.

We first find the differential of the surface area formula with respect to the radius:

dA = 8πr dr

Substituting r = 4 inches and dr = 0.035 inches, we get:

dA = 8π(4)(0.035) = 3.5395 square inches (rounded to four decimal places)

Therefore, the approximate increase in the area of the soap bubble is 3.5395 square inches when the radius increases from 4 inches to 4.035 inches