Question

what is the diameter of circle t?
sector area \\( \frac { 64 } { 5 } \pi \\)
the diameter is \\( \square \\).
(type an integer or a decimal.)

Explanation:

Step1: Recall the sector - area formula

The formula for the area of a sector of a circle is \(A=\frac{\theta}{360^{\circ}}\times\pi r^{2}\), where \(\theta\) is the central angle of the sector and \(r\) is the radius of the circle. Given \(\theta = 72^{\circ}\) and \(A=\frac{64}{5}\pi\).
Substitute the values into the formula: \(\frac{64}{5}\pi=\frac{72^{\circ}}{360^{\circ}}\times\pi r^{2}\).

Step2: Simplify the equation

First, simplify \(\frac{72}{360}=\frac{1}{5}\). The equation becomes \(\frac{64}{5}\pi=\frac{1}{5}\pi r^{2}\).
Multiply both sides of the equation by \(5\) to get rid of the denominators: \(64\pi=\pi r^{2}\).
Divide both sides by \(\pi\): \(r^{2}=64\).

Step3: Solve for \(r\)

Take the square root of both sides. Since \(r>0\) (radius is a non - negative quantity), \(r = 8\).

Step4: Calculate the diameter

The diameter \(d\) of a circle is related to the radius by the formula \(d = 2r\). Substitute \(r = 8\) into the formula: \(d=2\times8\).

Answer:

\(16\)