Question

cleary construction has a hose attached to their insulation blower that permits them to reach anywhere in a circular area of 6180 ft². find the dimensions of the largest square room with 14-ft ceilings in which they could reach all corners with the hose while leaving the blower centrally located. assume that the blower sits on the floor. the dimensions of the largest square room is \boxed{} by \boxed{} (type an integer or decimal rounded to two decimal places as needed.)

Explanation:

Step1: Find the radius of the circular area

The formula for the area of a circle is \(A = \pi r^{2}\). Given \(A=6180\ ft^{2}\), we can solve for \(r\):

$$r=\sqrt{\frac{A}{\pi}}=\sqrt{\frac{6180}{\pi}}$$

$$r=\sqrt{\frac{6180}{3.14159}}\approx\sqrt{1967.15}\approx 44.36\ ft$$

Step2: Use the Pythagorean theorem for the square - room diagonal

Let the side length of the square room be \(x\). The distance from the center of the square to a corner (using the Pythagorean theorem for a right - triangle formed by half of the square's sides) is \(d=\sqrt{(\frac{x}{2})^{2}+(\frac{x}{2})^{2}+14^{2}}\) (the \(14\) comes from the height of the ceiling). Since \(d = r\approx44.36\ ft\), we have:

$$d=\sqrt{\frac{x^{2}}{2}+196}$$

Substitute \(d = 44.36\) into the equation:

$$44.36=\sqrt{\frac{x^{2}}{2}+196}$$

Square both sides:

$$44.36^{2}=\frac{x^{2}}{2}+196$$

$$1967.81=\frac{x^{2}}{2}+196$$

$$ \frac{x^{2}}{2}=1967.81 - 196=1771.81$$

$$x^{2}=3543.62$$

$$x=\sqrt{3543.62}\approx59.53\ ft$$

Answer:

The dimensions of the largest square room is \(59.53\) by \(59.53\)