Chem 1 - Units conversion problem

Question

In the United States, water used for irrigation is measured in acre-feet. An acre-foot of water covers an acre to a depth of exactly 1 ft. An acre is 4840 yd². An acre-foot is enough water to supply two typical households for 1.00 yr.

(a) If desalinated water costs $1,950 per acre-foot, how much does desalinated water cost per liter?

Task: Determine cost per liter of desalinated water.

Given: 1 foot-acre of desalinated water costs $1,950.

Here's how I solve this:

1) Convert foot-acre to Liters

4840 yd² × (3 ft / yd)² × 1 ft × (12in/ft × 2.54cm/in. ÷ 1dm/10cm)³ = 1.233 × 10⁶ L

2) $1,950 / 1.233 × 10^{6 =} $1.58 × 10^–3 per Liter

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Here's the Solutions Manual:

$1950/acre-ft × 1 acre/4840 yd² × 3ft/1yd × (1.094yd)³/(1m)³ × (1m)³/(10 dm)³ × (10dm)³/1L = $1.583 ×10¯³ L

My problem is that the solutions manual "solution" doesn't make any sense to me! I've been staring at this for hours! Acre-foot is a unit of VOLUME. They are converting from volume (acre-ft) to area (yd²) then to length (ft). So I'd like some help unpacking the math in each step in this conversion.

Thank you!

Answer 1

Hello Daniel,

Since volume is the product of an area unit and a length unit, think of 1 acre-ft as (1 acre)(1 ft), that is as the product of 1 acre and 1 ft. So that the solution given begins with the price per acre-foot followed by two conversion factors that will cancel the acre and the foot - resulting in the price per cubic yard.

[$1950 / (1 acre)(1 foot)] x (1 acre / 4840 yd²) x (3 ft / 1 yd) = $(5850/4840) per yd³

Next, the solution uses three conversion factors to go from yd³ to m³ to dm³ to L as follows:

[$(5850/4840) / yd³] x (1.094 yd / 1 m)³ x (1 m / 10 dm)³ x (1 dm³ / 1 L) = $1.583 x 10^-3 L

Note: for the last conversion factor, 1 L = 1 dm³

Hope this helps.