Question

skill #2 unit conversions
you cannot have the same unit show up twice in the numerator or in the denominator. so elim choices and see what is left
joe is ordering water for his swimming pool. he determines the volume of his pool to be about 3240 cubic feet. there are approximately 7.5 gallons of water in 1 cubic foot. a truck load holds 6000 gallons of water.
which expression would allow joe to correctly calculate the number of truck loads of water he needs to fill his pool?
(1) (\frac{3240 \text{ft}^3}{1 \text{pool}} cdot \frac{1 \text{ft}^3}{7.5 \text{gal}} cdot \frac{6000 \text{gal}}{1 \text{truck load}})
(2) (\frac{3240 \text{ft}^3}{1 \text{pool}} cdot \frac{1 \text{ft}^3}{7.5 \text{gal}} cdot \frac{1 \text{truck load}}{6000 \text{gal}})
(3) (\frac{3240 \text{ft}^3}{1 \text{pool}} cdot \frac{7.5 \text{gal}}{1 \text{ft}^3} cdot \frac{6000 \text{gal}}{1 \text{truck load}})
(4) (\frac{3240 \text{ft}^3}{1 \text{pool}} cdot \frac{7.5 \text{gal}}{1 \text{ft}^3} cdot \frac{1 \text{truck load}}{6000 \text{gal}})

Explanation:

Step1: Understand the goal

We need to find the number of truck loads by converting the pool's volume from cubic feet to gallons, then to truck loads. The conversion factors are: 1 cubic foot ≈ 7.5 gallons, and 1 truck load = 6000 gallons. The pool volume is 3240 cubic feet.

Step2: Analyze unit conversion logic

To convert cubic feet to gallons, we use the factor $\frac{7.5\ \text{gal}}{1\ \text{ft}^3}$ (since 7.5 gallons per 1 cubic foot). Then, to convert gallons to truck loads, we use $\frac{1\ \text{truck load}}{6000\ \text{gal}}$ (since 1 truck load is 6000 gallons). The pool volume is $3240\ \text{ft}^3$ per pool.

Step3: Check each option

• Option (1): $\frac{3240\ \text{ft}^3}{1\ \text{pool}} \cdot \frac{1\ \text{ft}^3}{7.5\ \text{gal}}$ – This is incorrect because it would be converting cubic feet to cubic feet (wrong direction for gallons).
• Option (2): $\frac{3240\ \text{ft}^3}{1\ \text{pool}} \cdot \frac{1\ \text{ft}^3}{7.5\ \text{gal}} \cdot \frac{1\ \text{truck load}}{6000\ \text{gal}}$ – Incorrect, same as (1) for the first conversion.
• Option (3): $\frac{3240\ \text{ft}^3}{1\ \text{pool}} \cdot \frac{7.5\ \text{gal}}{1\ \text{ft}^3} \cdot \frac{6000\ \text{gal}}{1\ \text{truck load}}$ – Incorrect, this would convert to gallons squared over truck load (wrong units).
• Option (4): $\frac{3240\ \text{ft}^3}{1\ \text{pool}} \cdot \frac{7.5\ \text{gal}}{1\ \text{ft}^3} \cdot \frac{1\ \text{truck load}}{6000\ \text{gal}}$ – Let's check units: $\text{ft}^3$ cancels with $\text{ft}^3$, $\text{gal}$ cancels with $\text{gal}$, leaving $\frac{\text{truck load}}{\text{pool}}$, which is what we want. The first conversion (cubic feet to gallons) is correct ($7.5\ \text{gal}/1\ \text{ft}^3$), then gallons to truck loads ($1\ \text{truck load}/6000\ \text{gal}$).

Answer:

(3) $\boldsymbol{\frac{3240\ \text{ft}^3}{1\ \text{pool}} \cdot \frac{7.5\ \text{gal}}{1\ \text{ft}^3} \cdot \frac{1\ \text{truck load}}{6000\ \text{gal}}}$ (Wait, no, correction: Wait, option (4) is $\frac{3240\ \text{ft}^3}{1\ \text{pool}} \cdot \frac{7.5\ \text{gal}}{1\ \text{ft}^3} \cdot \frac{1\ \text{truck load}}{6000\ \text{gal}}$? Wait, no, looking back, option (3) is $\frac{3240\ \text{ft}^3}{1\ \text{pool}} \cdot \frac{7.5\ \text{gal}}{1\ \text{ft}^3} \cdot \frac{6000\ \text{gal}}{1\ \text{truck load}}$ (wrong), option (4) is $\frac{3240\ \text{ft}^3}{1\ \text{pool}} \cdot \frac{7.5\ \text{gal}}{1\ \text{ft}^3} \cdot \frac{1\ \text{truck load}}{6000\ \text{gal}}$ (correct). Wait, maybe I mislabeled. Let's re-express:

The correct unit conversion is:

Total gallons in pool = $3240\ \text{ft}^3 \times 7.5\ \text{gal/ft}^3$

Number of truck loads = $\frac{\text{Total gallons}}{6000\ \text{gal/truck load}} = 3240 \times \frac{7.5}{6000}$ (per pool)

In fraction form, that's $\frac{3240\ \text{ft}^3}{1\ \text{pool}} \times \frac{7.5\ \text{gal}}{1\ \text{ft}^3} \times \frac{1\ \text{truck load}}{6000\ \text{gal}}$, which is option (4) (assuming option (4) is the one with $7.5\ \text{gal}/1\ \text{ft}^3$ and $1\ \text{truck load}/6000\ \text{gal}$). Wait, the original options:

(4) $\frac{3240\ \text{ft}^3}{1\ \text{pool}} \cdot \frac{7.5\ \text{gal}}{1\ \text{ft}^3} \cdot \frac{1\ \text{truck load}}{6000\ \text{gal}}$ – Yes, this is correct.

Wait, maybe I made a mistake earlier. Let's recheck:

• To convert $3240\ \text{ft}^3$ (pool volume) to gallons: multiply by $7.5\ \text{gal}/1\ \text{ft}^3$ (so $\text{ft}^3$ cancels, gives gallons).
• Then convert gallons to truck loads: divide by $6000\ \text{gal/truck load}$, which is the same as multiplying by $1\ \text{truck load}/6000\ \text{gal}$ (gallons cancel, gives truck loads per pool).

So the correct expression is $\frac{3240\ \text{ft}^3}{1\ \text{pool}} \times \frac{7.5\ \text{gal}}{1\ \text{ft}^3} \times \frac{1\ \text{truck load}}{6000\ \text{gal}}$, which is option (4) (assuming the labels: (4) is the last one with $7.5\ \text{gal}/1\ \text{ft}^3$ and $1\ \text{truck load}/6000\ \text{gal}$). Wait, the user's image:

(4) $\frac{3240\ \text{ft}^3}{1\ \text{pool}} \cdot \frac{7.5\ \text{gal}}{1\ \text{ft}^3} \cdot \frac{1\ \text{truck load}}{6000\ \text{gal}}$ – Yes, that's correct.

So the correct option is (4). Wait, but let's check the options again:

Wait, the options:

(1) $\frac{3240\ \text{ft}^3}{1\ \text{pool}} \cdot \frac{1\ \text{ft}^3}{7.5\ \text{gal}} \cdot \frac{1\ \text{truck load}}{6000\ \text{gal}}$ – wrong (ft³/gal)

(2) $\frac{3240\ \text{ft}^3}{1\ \text{pool}} \cdot \frac{1\ \text{ft}^3}{7.5\ \text{gal}} \cdot \frac{1\ \text{truck load}}{6000\ \text{gal}}$ – same as (1)

(3) $\frac{3240\ \text{ft}^3}{1\ \text{pool}} \cdot \frac{7.5\ \text{gal}}{1\ \text{ft}^3} \cdot \frac{6000\ \text{gal}}{1\ \text{truck load}}$ – wrong (gal/gal would be squared)

(4) $\frac{3240\ \text{ft}^3}{1\ \text{pool}} \cdot \frac{7.5\ \text{gal}}{1\ \text{ft}^3} \cdot \frac{1\ \text{truck load}}{6000\ \text{gal}}$ – correct (ft³ cancels, gal cancels, leaves truck load/pool)

Yes, so the correct answer is option (4).