Question

7.

1. vic estimated the area of the following circle to be $18\frac{1}{2}$ square units. do you agree?

· if so, explain why.
· if not, explain why not and give your own estimate for the
area of the circle.
vics work:
each square on the grid has an area of $\frac{1}{2}$ square unit. i counted
24 squares inside the polygon. i estimate that there are 13 squares
in the region between the circle and the polygon. there is a total
of 37 squares because $24 + 13 = 37$. i estimate that the area of the
circle is $18\frac{1}{2}$ square units because $37\left(\frac{1}{2}\
ight)=18\frac{1}{2}$.

Explanation:

Step1: Identify circle parameters

From the grid, the center is at $(2, 2)$, and the radius $r = 1.5$ units.

Step2: Calculate actual circle area

Use the circle area formula $A=\pi r^2$.
$A=\pi (1.5)^2 = 2.25\pi \approx 7.07$ square units.

Step3: Analyze Vic's error

Vic overcounted partial squares; the 13 estimated partial squares are too many, as most partial squares are less than half a unit.

Step4: Estimate area via grid

Count full squares: ~12 full squares. Count partial squares: ~8 partial squares (each ~0.5 unit). Total estimated squares: $12 + 8(0.5) = 16$. Area: $16\times\frac{1}{2}=8$ square units (close to the calculated value).

Answer:

I do not agree with Vic's estimate. Vic overestimated the number of partial squares between the circle and the polygon, leading to an inflated total. Using the circle area formula with radius $1.5$ units, the actual area is approximately $7.07$ square units. A reasonable grid-based estimate is around 8 square units, which is far lower than Vic's $18\frac{1}{2}$ square units.