A meteorologist estimates how long a passing storm will last by using the function `t(d)=0.08d^((3)/(2))`, where d is the diameter of the storm, in miles, and t is the time, in hours. If the storm lasts 16.2 minutes, find its diameter, in miles.

Answer

To solve the problem, we will follow these steps:

### Step 1: Convert minutes to hours
The time the storm lasts is given as 16.2 minutes. To convert this into hours, we divide by 60 (since there are 60 minutes in an hour).

\[ t = \frac{16.2}{60} \]

### Step 2: Set up the equation
Using the function provided, we can set up the equation:

\[ t(d) = 0.08d^{\frac{3}{2}} \]

Substituting \( t \) from Step 1 into the equation:

\[ \frac{16.2}{60} = 0.08d^{\frac{3}{2}} \]

### Step 3: Simplify the left side
Calculating \( \frac{16.2}{60} \):

\[ \frac{16.2}{60} = 0.27 \]

So now the equation becomes:

\[ 0.27 = 0.08d^{\frac{3}{2}} \]

### Step 4: Solve for \( d^{\frac{3}{2}} \)
To isolate \( d^{\frac{3}{2}} \), divide both sides by 0.08:

\[ d^{\frac{3}{2}} = \frac{0.27}{0.08} \]

Calculating the right side:

\[ d^{\frac{3}{2}} = 3.375 \]

### Step 5: Solve for \( d \)
To find \( d \), we need to raise both sides to the power of \( \frac{2}{3} \):

\[ d = (3.375)^{\frac{2}{3}} \]

Calculating this gives:

\[ d = 2.25 \]

### Conclusion
The diameter of the storm is \( 2.25 \) miles.

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