Q50P - University Physics with Modern Physics

Question

An acre has a length of one furlong $\left(1/8 \mathrm{mi} \right)$and a width one-tenth of its length. (a) How many acres are in a square mile? (b) How many square feet are in an acre? See Appendix E. (c) An acre-foot is the volume of water that would cover 1 acre of flat land to a depth of 1 foot. How many gallons are in 1 acre-foot?

Step-by-Step Solution

Answer

a) The area of an acre of land is $\frac{1}{640}{\mathrm{mi}}^2$. i.e., $640$ acres to a square mile.

b) The number of square feet in an acre is $43,560{\mathrm{ft}}^2$.

c) The gallons in 1 acre-foot are $3.25\times {10}^5\mathrm{gal}$.

Concept of the length and Area

Length is a dimension of an object to describe how long the object is, and the area is length times width which defines the size of a closed region in a plane or two-dimensions. 

Identification of given data

The given data can be listed below,

• The length of an acre is, $l=\frac{1}{8} mi$.
• The width of an acre is, $w =\left(\frac{1}{10}\right)\times l =\frac{1}{80} mi$.
• The depth of an acre is, $d=1 ft$.

(a) Determination of acres in a square mile

Area of an acre can be given as,

$A=l\times w$

Here, l  is the acre’s length, and w is the width of the acre.

Substitute values in the above expression,

$\begin{array}{rcl}A& =& \left(\frac{1}{8}\mathrm{mi}\right)\times \left(\frac{1}{80}\mathrm{mi}\right)\\ & =& \frac{1}{640}{\mathrm{mi}}^2\end{array}$

Thus, an acre of the land area is$\frac{1}{640}{\mathrm{mi}}^2$, i.e., 640 acres to a square mile.

(b) Determination of square feet in an acre

Conversion of a mile into the foot is given by,

$1\mathrm{mi}=5280\mathrm{ft}$

From the above conversion, the number of square $\mathrm{ft}$ in an acre is given by,

$\begin{array}{rcl}N& =& \left(1\text{ acre }\right)\left(\frac{1{\mathrm{mi}}^2}{640\text{ acre }}\right)\times {\left(\frac{5280\mathrm{ft}}{1\mathrm{mi}}\right)}^2\\ & =& 43,560{\mathrm{ft}}^2\end{array}$

Thus, the number of square feet in an acre is $43,560{\mathrm{ft}}^2$.

(c) Determination of gallons in 1 acre-foot

Gallon in one cubic foot can be given by,

$1{\mathrm{ft}}^3=7.477\mathrm{gal}$

From the above conversion,

$\begin{array}{rcl}\text{ 1acre-foot }& =& \left(43,560{\mathrm{ft}}^3\right)\times \left(\frac{7.477\mathrm{gal}}{1{\mathrm{ft}}^3}\right)\\ & =& 3.25\times {10}^5\mathrm{gal}\end{array}$

Thus, the gallons in 1 acre-foot are $3.25\times {10}^5\mathrm{gal}$.

Final Solution

a) an acre of land comprises of${\left(\frac{1}{640}\right)}^{th}$.part of a square mile.

b) an acre of land comprises of $43,560{\mathrm{ft}}^2$

c) The number of gallons in 1 acre-foot are $3.25\times {10}^5\mathrm{gal}$.