Two similar cylindrical jugs have heights 5 cm and 8 cm, respectively. If the capacity of the smaller jug is 64 `cm^(3)`, what is the capacity (correct to 2 decimal places, in `cm^(3)`) of the larger jug?

Options

A. 262.14
B. `102.40`
C. 294.54
D. 163.84

Answer

To solve the problem of finding the capacity of the larger jug, we can follow these steps:

### Step 1: Understand the relationship between the heights and volumes of similar jugs
Since the two jugs are similar, the ratio of their volumes is equal to the cube of the ratio of their heights.

### Step 2: Identify the heights of the jugs
The height of the smaller jug is 5 cm and the height of the larger jug is 8 cm.

### Step 3: Write the ratio of the heights
The ratio of the heights of the smaller jug to the larger jug is:
\[ \text{Ratio of heights} = \frac{5}{8} \]

### Step 4: Calculate the ratio of the volumes
Since the volumes are proportional to the cube of the heights, we calculate the ratio of the volumes:
\[ \text{Ratio of volumes} = \left(\frac{5}{8}\right)^3 = \frac{5^3}{8^3} = \frac{125}{512} \]

### Step 5: Set up the equation for the volumes
Let \( V_s \) be the volume of the smaller jug and \( V_l \) be the volume of the larger jug. We know that:
\[ \frac{V_s}{V_l} = \frac{125}{512} \]
Given that \( V_s = 64 \, \text{cm}^3 \), we can set up the equation:
\[ \frac{64}{V_l} = \frac{125}{512} \]

### Step 6: Cross-multiply to solve for \( V_l \)
Cross-multiplying gives:
\[ 64 \times 512 = 125 \times V_l \]
\[ 32768 = 125 \times V_l \]

### Step 7: Solve for \( V_l \)
Now, divide both sides by 125:
\[ V_l = \frac{32768}{125} \]
Calculating this gives:
\[ V_l = 262.144 \, \text{cm}^3 \]

### Step 8: Round the result to two decimal places
Rounding \( 262.144 \) to two decimal places gives:
\[ V_l \approx 262.14 \, \text{cm}^3 \]

Thus, the capacity of the larger jug is **262.14 cm³**.

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