How many circles fits in the rectangle?

[megslove1212]

A rectangle has dimensions 18 ft by 72 ft. How many circles (placed side by side with sides touching) with area 354.4690049 can be placed inside of the rectangle? What is the measure of the area inside the rectangle but not inside any of the circles?

Help!? Thanks!

 

[Denis]

A rectangle has dimensions 18 ft by 72 ft. How many circles (placed side by side with sides touching) with area 354.4690049 can be placed inside of the rectangle? What is the measure of the area inside the rectangle but not inside any of the circles?

That area means circles have diameter of ~21 feet, so will not fit in a rectangle width 18.
Where did you get that?

 

[megslove1212]

That area means circles have diameter of ~21 feet, so will not fit in a rectangle width 18.
Where did you get that?

My geometry teacher gave it to me for bonus. I was pretty sure I had it but I just wanted a second opinion before I gave him the answer. thanks!
p.s. my math teacher isn't very smart...

 

[Denis]

Well, do you agree that a circle with diameter of approximately 21 feet
cannot be inserted in a rectangle of width 18?

 

[megslove1212]

Yes, I do. and Thankyou

 

[soroban]

Hello, megslove1212!

I assume there is a typo.
I'll take a guess at correcting it.

A rectangle has dimensions 18 ft by 72 ft.
How many non-overlapping circles with area 254.4690049 can be placed inside of the rectangle?
What is the measure of the area inside the rectangle but not inside any of the circles?

\(\displaystyle \text{The area of a circle is: }\:\pi r^2 \:=\:254.4690049 \quad\Rightarrow\quad r^2 \:=\:80.99999999 \quad\Rightarrow\quad r \:=\:9\)

\(\displaystyle \text{The diameter is 18 feet.}\)

\(\displaystyle \text{Exactly }f\!our\text{ circles can be placed in the rectangle.}\)

. . \(\displaystyle \begin{array}{c}\_\_\_\_\_\_\_\_\_\_\_ \\ [-2mm]|\!\!\bigcirc\!\!\bigcirc\!\!\bigcirc\!\!\bigcirc\!| \\ [-3mm] \_\_\_\_\_\_\_\_\_\_\_ \end{array}\)


\(\displaystyle \text{The area of the rectangle is: }\:18 \times 72 \:=\:1296\text{ ft}^2\)

\(\displaystyle \text{The area of the circles is: }\:4 \times 254.4690049 \:\approx\:1017.876\text{ ft}^2\)


\(\displaystyle \text{Therefore, the area between the rectangle and the circles is: }\:1296 \,-\, 1017.876\:=\:278.124\text{ ft}^2\)