Problem 50 - University Physics with Modern Physics

Question

An acre has a length of one furlong (\\(\frac{1}{8}\\) mi) 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) 640 acres, (b) 43,560 square feet, (c) 325,829 gallons.

Step 1: Calculate Furlong in Feet

A furlong is \(\frac{1}{8}\) of a mile. Since there are 5280 feet in a mile, a furlong is \[ 5280 \times \frac{1}{8} = 660 \text{ feet}. \]

Step 2: Calculate Acre Dimensions in Feet

The width of an acre is one-tenth of its length. So, the width is \[ 660 \times \frac{1}{10} = 66 \text{ feet}. \] Thus, the dimensions of an acre are 660 feet by 66 feet.

Step 3: Calculate Area of an Acre in Square Feet

Now we calculate the area of an acre in square feet:\[ 660 \times 66 = 43,560 \text{ square feet}. \]

Step 4: Calculate Acres in a Square Mile

To find how many acres are in a square mile, calculate the total square feet in a square mile and divide it by the square feet in an acre. So,\[ 5280 \times 5280 = 27,878,400 \text{ square feet in a square mile}. \] Then divide by the area of an acre:\[ \frac{27,878,400}{43,560} \approx 640 \text{ acres}. \]

Step 5: Calculate Gallons in an Acre-Foot

The volume of water in an acre-foot is \[ 43,560 \text{ cubic feet} \] since it covers 1 acre to a depth of 1 foot. Convert cubic feet to gallons (1 cubic foot = 7.48 gallons):\[ 43,560 \times 7.48 \approx 325,829 \text{ gallons}. \]

Key Concepts

Furlong

A furlong is a unit of measurement that you might not be familiar with, but it's quite interesting! Specifically, a furlong is equal to \(\frac{1}{8}\) of a mile. Understanding how many feet are in a furlong can help in various calculations, like determining the dimensions of an acre.
To calculate how many feet are in a furlong, you start with the fact that there are 5280 feet in a mile. By multiplying this by \(\frac{1}{8}\), you find that:

• 1 furlong = 660 feet.

So, if you imagine a mile as a line, a furlong would be a small section of that line, specifically one-eighth of it. This unit is particularly useful in the context of farming and land measurement, as it gives a more manageable scale for measuring land like acres.

Square Mile

A square mile is another unit of area that expresses the total space within a mile-long and a mile-wide square. To put it into perspective, think of it as a large square field where each side measures one mile in length.
To understand how much land this represents in terms of feet, consider that:

• 1 mile = 5280 feet
• 1 square mile is therefore \(5280 \times 5280 = 27,878,400\) square feet.

Using these conversions, a square mile can cover quite a lot of space! This is why it's commonly used in geography and land planning. To better appreciate this scale, remember that there are approximately 640 acres in a square mile. This shows how vast a square mile truly is!

Square Feet

When we talk about measuring smaller areas, square feet come in handy. A square foot is a unit of area that is a square that has sides each of which is 1 foot in length. It's a widely used unit in construction, architecture, and real estate.
To grasp how square feet relate to larger land measurements:

• 1 acre is 43,560 square feet.
• This makes calculations of land area more understandable.

The conversion between acres and square feet allows for easy measurement of land. For example, knowing that 1 square mile equals 27,878,400 square feet, we can figure out how many acres fit into a square mile simply by dividing the total square feet by the square feet in an acre.

Cubic Feet

When considering volume, cubic feet is the unit to use. A cubic foot is a cube where each side is 1 foot in length, and it measures the space inside a container or area.
This concept becomes particularly relevant when considering water measurement. For example, an acre-foot, which is used in measuring water volume on land:

• Is the amount of water needed to cover 1 acre of land to a depth of 1 foot.
• This equals 43,560 cubic feet.

To convert cubic feet to gallons, a common unit for liquid measurement, use the conversion that 1 cubic foot equals 7.48 gallons. So, an acre-foot of water is equivalent to approximately 325,829 gallons. Cubic feet are helpful for various applications, from water management to storage calculations.