The edge x of a cube is measured with an error of at most 0.5%. What is the maximum corresponding percentage error in computing the cube’s

a. surface area?

검증된 단계별 안내

First, understand that the surface area of a cube with edge length x is given by the formula: \( A = 6x^2 \).

Next, we need to find the relationship between the error in x and the error in A. Use differentials to approximate the change in A: \( dA = \frac{dA}{dx} \cdot dx \).

Calculate the derivative of the surface area with respect to x: \( \frac{dA}{dx} = 12x \).

Substitute \( dx = 0.005x \) (since the error in x is 0.5% of x) into the differential equation: \( dA = 12x \cdot 0.005x \).

Finally, express the percentage error in A as \( \frac{dA}{A} \times 100\% \) and simplify using the expressions for dA and A.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Differential Approximation

Differential approximation is a method used to estimate the change in a function's value based on the change in its input. For a function f(x), the differential df is approximately f'(x)dx, where f'(x) is the derivative and dx is the small change in x. This concept helps in estimating errors in measurements.

Surface Area of a Cube

The surface area of a cube with edge length x is given by the formula 6x^2. Understanding this formula is crucial because it allows us to relate changes in the edge length to changes in the surface area, which is necessary for calculating the percentage error in the surface area.

Percentage Error

Percentage error quantifies the accuracy of a measurement by comparing the error to the actual value, expressed as a percentage. It is calculated as (error/actual value) * 100%. In this context, it helps in determining how a small error in measuring the edge length of a cube affects the calculated surface area.

The edge x of a cube is measured with an error of at most 0.5%. What is the maximum corresponding percentage error in computing the cube’sa. surface area?