Question

Consider a circle whose radius is greater than 9 and whose area is given by \(A=\pi\left(x^{2}-18 x+81\right)\). (Use \(\pi \approx 3.14\) ) If the area of the circle is 12.56 square meters, what is the value of \(x ?\)

Short answer

The value of \(x\) is 9.

Step-by-step solution

Setup

The area \(A\) of the circle is given by \(A = \pi(x^2 - 18x + 81)\), with \(\pi \approx 3.14\). The area of the circle is also given to be 12.56 square meters. Thus, one can set up the equation \(3.14(x^2 - 18x + 81) = 12.56\).

Simplify Equation

Divide both sides by 3.14; This gives \(x^2 - 18x + 81=4\).

Rearrange the Equation

Rearrange the equation so it is set equal to zero, which helps to solve for \(x\). Thus, \(x^2 - 18x + 81 - 4=0\), which simplifies to \(x^2 - 18x + 77= 0\).

Solve the Equation

Factor the quadratic equation to solve for \(x\). This yields \((x - 9)^2=0\). So, \(x - 9 = 0\), meaning \(x=9\).