Question

(a) Write the area A of an equilateral triangle as a function of the side length s. (b) Find the (instantaneous) rate of change of the area A with respect to a side s. (c) Evaluate the rate of change of A at s " 2 and s " 10. (d) If \(s\) is measured in inches and \(A\) is measured in square inches, what units would be appropriate for \(d A / d s ?\)

Short answer

(a) \(A = s^2 \sqrt{3} / 4\) (b) \(d A / d s = s \sqrt{3} / 2\) (c) \(dA/ds= \sqrt{3}\) for \(s=2\) and \(dA/ds= 5\sqrt{3}\) for \(s=10\) (d) The units for \(d A / d s\) would be inches.

Step-by-step solution

Write the formula for the area of an equilateral triangle

The formula for finding an area \(A\) of an equilateral triangle of side length \(s\) is \(A = s^2 \sqrt{3} / 4\). This is the answer to (a).

Differentiate the area function

To find the rate of change of the area with respect to the side length, differentiate the area function with respect to \(s\), we have \(d A / d s = s \sqrt{3} / 2\). This gives us the rate of change of the area with respect to the side length \(s\). This is the answer to (b).

Evaluate the rate of change at specific values

Substitute \(s=2\) and \(s=10\) into the expression found in step 2. We get \(dA/ds= \sqrt{3}\) for \(s=2\) and \(dA/ds= 5\sqrt{3}\) for \(s=10\). This gives us the answer to (c).

Determine the units for the derivative

The area is measured in square inches, while the length is measured in inches. Hence the rate of change of the area with respect to the side length has units of square inches per inch, or simply, inches. This is the answer to (d).