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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?

Short Answer

Expert verified
(a) 640 acres, (b) 43,560 square feet, (c) 325,829 gallons.

Step by step solution

01

Calculate Furlong in Feet

A furlong is \(\frac{1}{8}\) of a mile. Since there are 5280 feet in a mile, a furlong is \[ 5280 \times \frac{1}{8} = 660 \text{ feet}. \]
02

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.
03

Calculate Area of an Acre in Square Feet

Now we calculate the area of an acre in square feet:\[ 660 \times 66 = 43,560 \text{ square feet}. \]
04

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:\[ \frac{27,878,400}{43,560} \approx 640 \text{ acres}. \]
05

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):\[ 43,560 \times 7.48 \approx 325,829 \text{ gallons}. \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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