How many numbers are there between 200 to 300 which are perfect cubes?

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find how many numbers between 200 and 300 are perfect cubes, we can follow these steps:

### Step 1: Identify the range of perfect cubes
We need to find the smallest integer \( n \) such that \( n^3 \) is greater than or equal to 200, and the largest integer \( m \) such that \( m^3 \) is less than or equal to 300.

### Step 2: Calculate the cube roots
- For the lower limit (200):
\[ n = \lceil \sqrt[3]{200} \rceil \]
Calculating \( \sqrt[3]{200} \):
\[ \sqrt[3]{200} \approx 5.848 \]
So, \( n = 6 \) (since we round up to the nearest whole number).

- For the upper limit (300):
\[ m = \lfloor \sqrt[3]{300} \rfloor \]
Calculating \( \sqrt[3]{300} \):
\[ \sqrt[3]{300} \approx 6.669 \]
So, \( m = 6 \) (since we round down to the nearest whole number).

### Step 3: List the perfect cubes
Now we can find the perfect cubes for \( n = 6 \):
- \( 6^3 = 216 \)

### Step 4: Check if there are any other perfect cubes
Since \( n \) and \( m \) are both 6, we only have one perfect cube in the range:
- The only perfect cube between 200 and 300 is \( 216 \).

### Step 5: Conclusion
Thus, there is **1 perfect cube** between 200 and 300.

### Summary of Steps:
1. Calculate the cube root of 200 and round up to find the smallest integer.
2. Calculate the cube root of 300 and round down to find the largest integer.
3. List the perfect cubes between these two integers.
4. Count the perfect cubes found.