If `a=b^(3)`, and `b=(sqrt(5))/(c )`, what is the value of a when `c = (1)/(3)` ?

Options

A. `0.42`
B. `1.89`
C. `60.37`
D. `301.87`

Explanation

To find the value of \( a \) given that \( a = b^3 \) and \( b = \frac{\sqrt{5}}{c} \), we will follow these steps:

### Step 1: Substitute the value of \( c \)
We know that \( c = \frac{1}{3} \). We will substitute this value into the equation for \( b \).

\[ b = \frac{\sqrt{5}}{c} = \frac{\sqrt{5}}{\frac{1}{3}} = \sqrt{5} \times 3 = 3\sqrt{5} \]

### Step 2: Calculate \( a \)
Now that we have the value of \( b \), we can find \( a \) using the equation \( a = b^3 \).

\[ a = (3\sqrt{5})^3 \]

### Step 3: Expand \( (3\sqrt{5})^3 \)
To calculate \( (3\sqrt{5})^3 \), we can use the property of exponents:

\[ (3\sqrt{5})^3 = 3^3 \cdot (\sqrt{5})^3 \]

Calculating each part:

\[ 3^3 = 27 \]
\[ (\sqrt{5})^3 = 5^{3/2} = 5 \cdot \sqrt{5} \]

### Step 4: Combine the results
Now we can combine these results:

\[ a = 27 \cdot (5\sqrt{5}) = 135\sqrt{5} \]

### Step 5: Approximate the value of \( a \)
To find a numerical approximation, we can use the value of \( \sqrt{5} \approx 2.236 \):

\[ a \approx 135 \cdot 2.236 \approx 301.86 \]

Thus, the value of \( a \) is approximately \( 301.86 \).

### Final Answer:
The value of \( a \) is \( 135\sqrt{5} \) or approximately \( 301.86 \).

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