Question

In Exercises 1-4, (a) write a formula for the function and (b) use the formula to find the indicated value of the function. the area A of a circle as a function of its diameter d; the area of a circle of diameter 4 in.

Short answer

The formula for the area A of a circle as a function of its diameter d is \(A = \pi d^2/4\). Therefore, the area of a circle with a diameter of 4 inches is \(4\pi\) square inches or approximately 12.566 square inches.

Step-by-step solution

Express Area Formula in Terms of Diameter

The traditional formula for the area of a circle is \(A = \pi r^2\) where r is the radius. However, the radius is just half of the diameter, making \(r = d/2\), where d represents the diameter. Substitute \(d/2\) in place of r in the formula for the area to get a formula for the area of a circle in terms of its diameter. Thus, the formula becomes \(A = \pi (d/2)^2 = \pi d^2/4\).

Apply the Diameter Value to the Equation

Given that the diameter d of the circle is 4 inches, substitute 4 inches for d in the formula obtained in Step 1, yielding \(A = \pi (4^2)/4\).

Simplify and Compute the Area

Solving \(A = \pi (4^2)/4\), simplifies to \(A = \pi (16)/4\), and eventually \(A = 4\pi\) square inches. Therefore, the area of a circle with a diameter of 4 inches is \(4\pi\) or approximately 12.566 square inches if you approximate \(\pi\) as 3.14159.