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 $90^{\circ}\text{F}$.

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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.