Problem 24 - The Calculus with Analytic Geometry

Question

Use the total differential to find approximately the greatest error in calculating the area of a right triangle from the lengths of the legs if they are measured to be 6 in. and 8 in., respectively, with a possible error of \(0.1\) in. for each measurement. Also find the approximate percent error.

Step-by-Step Solution

Answer

The greatest possible error is 0.7 square inches, and the approximate percent error is 2.92%.

Step 1 - Identify the Function

The area of a right triangle can be calculated using the formula \(A = \frac{1}{2}ab\), where \(a\) and \(b\) are the lengths of the legs.

Step 2 - Write the Total Differential

To find the error in the area, use the total differential formula for \(A = \frac{1}{2}ab\): \(dA = \frac{\text{∂}A}{\text{∂}a} \text{d}a + \frac{\text{∂}A}{\text{∂}b} \text{d}b\). This simplifies to \(dA = \frac{1}{2}(b \text{d}a + a \text{d}b)\).

Step 3 - Differentiate the Function

Calculate the partial derivatives: \(\frac{\text{∂}A}{\text{∂}a} = \frac{1}{2} b\) and \(\frac{\text{∂}A}{\text{∂}b} = \frac{1}{2} a\). Therefore: \(dA = \frac{1}{2}(b \text{d}a + a \text{d}b)\)

Step 4 - Input the Values

Substitute the given values: \(a = 6\), \(b = 8\), \(\text{d}a = 0.1\), and \(\text{d}b = 0.1\). This results in \(dA = \frac{1}{2}(8 \times 0.1 + 6 \times 0.1) = \frac{1}{2}(0.8 + 0.6) = \frac{1}{2}(1.4) = 0.7\).

Step 5 - Calculate the Percentage Error

The actual area \(A\) is \(A = \frac{1}{2}(6 \times 8) = 24\text{ in}^2\). The percentage error can be found by: $\text{Percentage error} = \frac{0.7}{24} \times 100\text{%} = 2.92\text{%}.

Key Concepts

error approximation

When working with measurements, it's important to understand how potential errors can affect calculated results. Error approximation helps us estimate these effects. The total differential equation is useful here. It allows us to compute approximate errors by considering the possible deviations in the measurements of input variables. For instance, if we have small errors in measuring the legs of a triangle, the total differential helps us find the resulting error in the triangle's area.

right triangle area

The area of a right triangle is calculated using a simple formula: \(A = \frac{1}{2}ab\) where \(a\) and \(b\) are the lengths of the triangle's two legs. This formula is derived from the general formula for the area of a triangle, which is half the product of the base and the height. In a right triangle, the legs serve as the base and height. So if you have the lengths of the legs, you can easily find the area.

partial derivatives

To find the total differential, we use partial derivatives. These derivatives show how the function changes as each variable (holding others constant) changes. For area \(A = \frac{1}{2}ab\), we need the partial derivatives with respect to \(a\) and \(b\). They are derived as follows:

• The partial derivative of \(A\) with respect to \(a\) is \(\frac{\partial A}{\partial a} = \frac{1}{2} b\).
• The partial derivative of \(A\) with respect to \(b\) is \(\frac{\partial A}{\partial b} = \frac{1}{2} a\).

Combining these, the total differential becomes \(dA = \frac{1}{2} (b \, da + a \, db)\). This equation allows us to approximate the change in area \(dA\) based on small changes in \(a\) and \(b\).

percentage error

Calculating the percentage error gives insight into how significant the approximate error is relative to the actual value. In this context, we first find the approximate error in the area using:

\(dA = \frac{1}{2} (b \, da + a \, db)\).
By substituting the measured and error values: \(a = 6\), \(b = 8\), \(da = 0.1\), and \(db = 0.1\), we find:
\(dA = 0.7\) square inches.
Next, we calculate the actual area \(A\) using \(A = \frac{1}{2}(6 \, \times 8) = 24\) square inches.

The percentage error is then:
\(\text{Percentage error} = \frac{0.7}{24} \times 100\% = 2.92\%\).This tells us that the approximation error is about 2.92% of the calculated area, reflecting the relative impact of measurement inaccuracies.