The edge of a cube was found to have a length of 50 cm with a possible error in measurement of 0.1 cm. Based on the measurement, you determine that the volume is 125,000 cm3. Use tangent line approximation to estimate the percentage error in volume.

v = s^3

dv = 3 s^2 ds
at s = 50 v = 125,000
dv = 3 (2500) ds = 7500(.1) = 750

100 (750/125,000) = .6 %

Thanks doll

To estimate the percentage error in the volume of the cube using tangent line approximation, we can use the formula:

Percentage error = (2 × error in length × tangent of slope) × 100

First, let's calculate the slope of the tangent line to the volume equation, which is the derivative of the volume function with respect to the edge length.

Given that the volume (V) of a cube is given by V = (edge length)^3, we have:

dV/dL = 3 × (edge length)^2

Substituting the given edge length of 50 cm into the derivative equation, we get:

dV/dL = 3 × (50 cm)^2
= 3 × 2500 cm^2
= 7500 cm^2

Next, let's calculate the error in length. The possible error in measurement is given as 0.1 cm.

Using the formula for percentage error:

Percentage error = (2 × error in length × tangent of slope) × 100

Substituting the values:

Percentage error = (2 × 0.1 cm × 7500 cm^2) × 100
= (2 × 0.1 cm × 7500 cm^2) × 100
= (2 × 0.1 cm × 7500 cm^2) × 100
= 1500 cm × 100
= 150,000 cm

Therefore, the estimated percentage error in volume is 150,000 %.

To estimate the percentage error in volume using the tangent line approximation method, we need to calculate the derivative of the volume function with respect to the edge length, and then use this derivative to find the approximate change in volume for a given change in edge length.

Step 1: Find the derivative of the volume function with respect to the edge length.
The volume V of a cube is given by V = s^3, where s is the length of the edge.
Differentiating the volume function with respect to s, we get: dV/ds = 3s^2.

Step 2: Calculate the absolute error in the edge length.
The absolute error in the edge length is given as 0.1 cm.

Step 3: Use the tangent line approximation formula to estimate the change in volume.
The general formula for the tangent line approximation is: ΔV ≈ dV/ds * Δs,
where ΔV represents the change in volume, dV/ds is the derivative of the volume function, and Δs is the change in edge length.

In this case, we want to estimate the change in volume (ΔV) for the given absolute error in the edge length (Δs = 0.1 cm), so we have:
ΔV ≈ dV/ds * Δs
≈ (3s^2) * (0.1)
≈ 0.3s^2

Step 4: Calculate the approximate change in volume.
Given that the length of the edge (s) is 50 cm, we can substitute this value into our approximation formula:
ΔV ≈ 0.3s^2
≈ 0.3*(50^2)
≈ 0.3*2500
≈ 750 cm³

Step 5: Calculate the percentage error in volume.
The estimated change in volume (ΔV) is 750 cm³. The original volume (V) of the cube is given as 125,000 cm³.
The percentage error in volume can be calculated by dividing the absolute change in volume (ΔV) by the original volume (V) and then multiplying by 100:
Percentage Error = (ΔV / V) * 100
= (750 / 125,000) * 100
= 0.6%

Therefore, the estimated percentage error in volume is approximately 0.6%.