The measurement of the edge of a cube is measured to be 18 inches, with a possible error of 0 . on inch. Use differentials to
approximate the possible error and the relative error in computing the following values. (Round your answers to three
decimal places.)
(a) the volume of the cube
possible error relative error
possible error relative error the surface area of the cube
(b)
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Ask by Murray Watkins. in the United States

Mar 31,2025

Question

The measurement of the edge of a cube is measured to be 18 inches, with a possible error of 0 . on inch. Use differentials to
approximate the possible error and the relative error in computing the following values. (Round your answers to three
decimal places.)
(a) the volume of the cube
possible error relative error
possible error relative error the surface area of the cube
(b)
Need Help? Readit

Ask by Murray Watkins. in the United States

Mar 31,2025

UpStudy AI · Accepted Answer

We are given a cube with edge length $$ x = 18 $$ inches measured with a possible error of $$ dx = 0.01 $$ inch.

### (a) Volume of the Cube

The volume of a cube is given by
$$
V = x^3.
$$
Its differential is
$$
dV = 3x^2 \, dx.
$$
Substitute $$ x = 18 $$ and $$ dx = 0.01 $$:
$$
dV = 3 \times (18)^2 \times 0.01 = 3 \times 324 \times 0.01 = 9.72 \text{ in}^3.
$$

The relative error in the volume is estimated by
$$
\frac{dV}{V} = \frac{9.72}{18^3} = \frac{9.72}{5832} \approx 0.00167.
$$

### (b) Surface Area of the Cube

The surface area of a cube is given by
$$
A = 6x^2.
$$
Its differential is
$$
dA = 12x \, dx.
$$
Substitute $$ x = 18 $$ and $$ dx = 0.01 $$:
$$
dA = 12 \times 18 \times 0.01 = 2.16 \text{ in}^2.
$$

The relative error in the surface area is estimated by
$$
\frac{dA}{A} = \frac{2.16}{6 \times (18)^2} = \frac{2.16}{6 \times 324} = \frac{2.16}{1944} \approx 0.00111.
$$

### Final Answers

**Volume of the cube:**
- Possible error: $$ \pm 9.72 \text{ in}^3 $$
- Relative error: $$ \approx \pm 0.00167 $$

**Surface area of the cube:**
- Possible error: $$ \pm 2.16 \text{ in}^2 $$
- Relative error: $$ \approx \pm 0.00111 $$
**Volume of the cube:** - Possible error: $$ \pm 9.72 \text{ in}^3 $$ - Relative error: $$ \approx \pm 0.00167 $$ **Surface area of the cube:** - Possible error: $$ \pm 2.16 \text{ in}^2 $$ - Relative error: $$ \approx \pm 0.00111 $$

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