Iron weights 8 times the weight of oak. Find the diameter of an iron ball whose weight is equal to that of a ball of oak 18 cm in diameter.

Options

A. 4.5 cm
B. 9 cm
C. 12 cm
D. 15 cm

Answer

To find the diameter of an iron ball whose weight is equal to that of a ball of oak with a diameter of 18 cm, we can follow these steps:

### Step 1: Understand the relationship between the weights of iron and oak.
Given that iron weighs 8 times the weight of oak, we can express the weight of the iron ball in terms of the weight of the oak ball.

### Step 2: Calculate the volume of the oak ball.
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

Since the diameter of the oak ball is 18 cm, the radius \( r \) is:
\[ r = \frac{diameter}{2} = \frac{18}{2} = 9 \text{ cm} \]

Now, substituting the radius into the volume formula:
\[ V_{oak} = \frac{4}{3} \pi (9)^3 \]

### Step 3: Calculate \( 9^3 \).
Calculating \( 9^3 \):
\[ 9^3 = 729 \]

So, the volume of the oak ball becomes:
\[ V_{oak} = \frac{4}{3} \pi (729) = \frac{2916}{3} \pi = 972 \pi \text{ cm}^3 \]

### Step 4: Relate the volumes of the iron ball and the oak ball.
Since the weight of the iron ball is equal to the weight of the oak ball, we can express this relationship in terms of volume:
\[ V_{iron} = \frac{1}{8} V_{oak} \]
because the weight of iron is 8 times that of oak, meaning the volume of iron for the same weight must be \( \frac{1}{8} \) of the volume of oak.

### Step 5: Calculate the volume of the iron ball.
Substituting the volume of the oak ball:
\[ V_{iron} = \frac{1}{8} (972 \pi) = 121.5 \pi \text{ cm}^3 \]

### Step 6: Set up the equation for the volume of the iron ball.
Let \( d \) be the diameter of the iron ball. The radius \( r_{iron} \) will be \( \frac{d}{2} \). Thus, the volume of the iron ball is:
\[ V_{iron} = \frac{4}{3} \pi \left(\frac{d}{2}\right)^3 = \frac{4}{3} \pi \left(\frac{d^3}{8}\right) = \frac{1}{6} \pi d^3 \]

### Step 7: Equate the volumes.
Now we set the volume of the iron ball equal to the volume we calculated:
\[ \frac{1}{6} \pi d^3 = 121.5 \pi \]

### Step 8: Solve for \( d^3 \).
Dividing both sides by \( \pi \):
\[ \frac{1}{6} d^3 = 121.5 \]
Multiplying both sides by 6:
\[ d^3 = 729 \]

### Step 9: Find \( d \).
Taking the cube root of both sides:
\[ d = \sqrt[3]{729} = 9 \text{ cm} \]

### Final Answer:
The diameter of the iron ball is **9 cm**.

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