An underground gasoline tank at 47 ◦F can
hold 1100 gallons of gasoline.
If the driver of a tanker truck fills the underground tank on a day when the temperature
is 90 ◦F, how much (according to his gauge
on the truck) can he pour in? Assume that
the temperature of the gasoline cools to 47◦F
upon entering the tank and its thermal coefficient of expansion is 0.00096 (◦C)−1
.
Answer in units of gal.

To solve this problem, we need to calculate the volume change due to temperature reduction from \(90^{\circ}\text{F}\) to \(47^{\circ}\text{F}\). This involves the concept of thermal expansion for liquids.

Here’s the step-by-step solution:

### Step 1: Convert Temperatures to Celsius
First, let's convert both temperatures from Fahrenheit to Celsius, since the coefficient of thermal expansion is given in \((^{\circ}\text{C})^{-1}\).

Using the formula:
\[
T(\text{°C}) = \frac{5}{9}(T(\text{°F}) - 32)
\]
we get:
1. \(90^{\circ}\text{F}\):
   \[
   T_1 = \frac{5}{9}(90 - 32) = \frac{5}{9} \times 58 = 32.22^{\circ}\text{C}
   \]
2. \(47^{\circ}\text{F}\):
   \[
   T_2 = \frac{5}{9}(47 - 32) = \frac{5}{9} \times 15 = 8.33^{\circ}\text{C}
   \]

### Step 2: Calculate the Temperature Change in Celsius
\[
\Delta T = T_1 - T_2 = 32.22 - 8.33 = 23.89^{\circ}\text{C}
\]

### Step 3: Apply the Thermal Expansion Formula
The volume of gasoline at \(90^{\circ}\text{F}\) (or \(32.22^{\circ}\text{C}\)) is what we need to find. The underground tank's volume capacity at \(47^{\circ}\text{F}\) (or \(8.33^{\circ}\text{C}\)) is given as 1100 gallons. 

For thermal expansion:
\[
V = V_0 (1 + \beta \Delta T)
\]
where:
- \(V\) is the volume at \(90^{\circ}\text{F}\).
- \(V_0 = 1100 \, \text{gallons}\) (volume at \(47^{\circ}\text{F}\)),
- \(\beta = 0.00096 \, (^{\circ}\text{C})^{-1}\),
- \(\Delta T = 23.89^{\circ}\text{C}\).

### Step 4: Calculate \(V\)
\[
V = 1100 \times (1 + 0.00096 \times 23.89)
\]
\[
V = 1100 \times (1 + 0.0229)
\]
\[
V = 1100 \times 1.0229 = 1125.19 \, \text{gallons}
\]

### Conclusion
The driver can pour **1125.19 gallons** of gasoline (as measured on the truck’s gauge) into the underground tank when the gasoline is at \(90^{\circ}\text{F}\).

---

Would you like further details or have any questions?

Here are some related questions to expand your understanding:
1. How would the result change if the thermal coefficient of expansion were different?
2. What would happen if the underground tank were filled with a liquid with a higher thermal expansion coefficient?
3. How would the answer change if the initial and final temperatures were given in Kelvin?
4. What assumptions are we making about the cooling process of the gasoline as it enters the tank?
5. How would a change in tank material affect the calculation?

**Tip:** Remember that thermal expansion coefficients are temperature-dependent and may vary with changes in environmental conditions.

An underground gasoline tank at 47 ◦F can hold 1100 gallons of gasoline. If the driver of a tanker truck fills the underground tank on a day when the temperature is 90 ◦F, how much (according to his gauge on the truck) can he pour in? Assume that the temperature of the gasoline cools to 47◦F upon entering the tank and its thermal coefficient of expansion is 0.00096 (◦C)−1. Answer in units of gal.

Explore how to calculate the volume of gasoline that can be poured into an underground tank when the temperature changes from 90°F to 47°F. This solution involves concepts of thermal expansion and temperature conversion. The maximum volume that can be poured is 1125.19 gallons.

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