Let \( d \) be the distance to the beach in miles. The time taken to drive to the beach at a speed of 60 miles per hour can be expressed as:
\[ t_1 = \frac{d}{60} \]
If you drove at a speed of 40 miles per hour, the time taken would be:
\[ t_2 = \frac{d}{40} \]
According to the problem, the difference in time between driving at 60 miles per hour and driving at 40 miles per hour is 45 minutes. We need to convert 45 minutes into hours:
\[ 45 \text{ minutes} = \frac{45}{60} \text{ hours} = \frac{3}{4} \text{ hours} \]
This gives us the equation:
\[ t_2 - t_1 = \frac{3}{4} \]
Substituting the expressions for \( t_1 \) and \( t_2 \):
\[ \frac{d}{40} - \frac{d}{60} = \frac{3}{4} \]
To solve this equation, we will first find a common denominator for the left-hand side. The common denominator of 40 and 60 is 120. Thus, we rewrite the fractions:
\[ \frac{d}{40} = \frac{3d}{120} \]
\[ \frac{d}{60} = \frac{2d}{120} \]
Now substitute these into the equation:
\[ \frac{3d}{120} - \frac{2d}{120} = \frac{3}{4} \]
This simplifies to:
\[ \frac{3d - 2d}{120} = \frac{3}{4} \]
\[ \frac{d}{120} = \frac{3}{4} \]
Now, we can solve for \( d \) by multiplying both sides by 120:
\[ d = 120 \cdot \frac{3}{4} = 120 \cdot 0.75 = 90 \]
Thus, the distance driven to the beach is:
\[ \boxed{90} \text{ miles} \]
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