Question

if mario plans to develop a circular race track 1 mi in circumference on a square plot of land, then what is the minimum number of acres that he needs? (one acre is equal to 43,560 ft².)

Explanation:

Step1: Convert miles to feet

Since 1 mile = 5280 feet, the circumference of the circular race - track $C = 5280$ feet.

Step2: Find the radius of the circular race - track

The formula for the circumference of a circle is $C = 2\pi r$. Given $C = 5280$ feet, we can solve for $r$: $r=\frac{C}{2\pi}=\frac{5280}{2\pi}=\frac{2640}{\pi}$ feet.

Step3: Find the side - length of the square plot

The diameter of the circular race - track $d = 2r=\frac{5280}{\pi}$ feet. The side - length of the square plot of land that can enclose the circular race - track is equal to the diameter of the circle. So, the area of the square plot $A_{square}=d^{2}=(\frac{5280}{\pi})^{2}$ square feet.

Step4: Calculate the area in square feet

$A_{square}=\frac{5280^{2}}{\pi^{2}}=\frac{27878400}{\pi^{2}}\approx\frac{27878400}{9.8696}\approx2824467.7$ square feet.

Step5: Convert square feet to acres

Since 1 acre = 43560 square feet, the number of acres $n=\frac{A_{square}}{43560}=\frac{2824467.7}{43560}\approx64.84$ acres.

Answer:

Approximately 64.84 acres