Question

1. on pipers map, the area of the field is 16 square inches. piper says that the actual area of the field is 320 square feet. is piper correct? explain your thinking.

Explanation:

Step1: Assume a scale factor

Let's assume the scale of the map is 1 inch represents $x$ feet. Then 1 square - inch on the map represents $x^{2}$ square - feet in reality. But we don't know the scale factor here. However, if we assume a simple case of a linear scale factor, we need to convert the units properly.

Step2: Consider the relationship between inches and feet

We know that 1 foot = 12 inches. So, 1 square - foot = 144 square - inches.
If we assume the map has a scale such that the area ratio is directly related to the square of the linear - scale factor. Let's check if there is a reasonable scale.
Let's assume the linear scale factor is $k$. Then the area scale factor is $k^{2}$.
If we assume the map has a scale where 1 inch on the map represents $y$ feet in reality, then the area on the map ($A_m = 16$ square inches) and the actual area ($A_a$) are related by $A_a=A_m\times y^{2}$.
If we assume $A_a = 320$ square feet, and convert the area on the map to square - feet. 16 square inches $=\frac{16}{144}=\frac{1}{9}$ square feet.
Let the scale factor be such that if 1 inch on the map represents $s$ feet in reality, then the area relationship gives us $\frac{1}{9}\times s^{2}=320$.
$s^{2}=320\times9 = 2880$. Then $s=\sqrt{2880}\approx53.67$ feet per inch. This is an extremely large and non - standard scale for a typical map.
In a more general sense, without knowing the scale of the map, we cannot directly convert the area on the map to the actual area. But if we assume a "normal" map scale, we can say that if we consider the conversion of inches to feet for area (1 square foot = 144 square inches), 16 square inches is $\frac{16}{144}=\frac{1}{9}$ square feet. And $\frac{1}{9}$ square feet is much less than 320 square feet.

Answer:

Piper is not correct. Without knowing the scale of the map, we cannot make the conversion as Piper did. And if we consider the basic conversion of area units from square inches to square feet (1 square foot = 144 square inches), 16 square inches is a very small fraction of a square foot and is not equal to 320 square feet.