The hypotenuse of a right-angled triangle (T) measures 35 cm. T's shortest side is 25% shorter than its 2nd longest side. If T's 2nd longest side and shortest side are equal to the height and radius respectively of a right circular cylinder (C), then what is C's approximate volume?

Options

A. 11250
B. 22500
C. 45000
D. 38800

Answer

To solve the problem step-by-step, we will break down the information given and apply the relevant mathematical concepts.

### Step 1: Understand the triangle dimensions
We know that the hypotenuse (H) of the right-angled triangle T is 35 cm. Let’s denote the second longest side as \( x \) and the shortest side as \( y \).

### Step 2: Relate the sides of the triangle
According to the problem, the shortest side \( y \) is 25% shorter than the second longest side \( x \). This can be expressed mathematically as:
\[ y = x - 0.25x = 0.75x \]
Thus, we have:
\[ y = \frac{3}{4}x \]

### Step 3: Apply the Pythagorean theorem
In a right-angled triangle, the relationship between the sides is given by:
\[ H^2 = x^2 + y^2 \]
Substituting the values we have:
\[ 35^2 = x^2 + \left(\frac{3}{4}x\right)^2 \]
Calculating \( 35^2 \):
\[ 1225 = x^2 + \frac{9}{16}x^2 \]

### Step 4: Combine the terms
We can combine the terms on the right-hand side:
\[ 1225 = x^2 + \frac{9}{16}x^2 = \left(1 + \frac{9}{16}\right)x^2 = \left(\frac{16}{16} + \frac{9}{16}\right)x^2 = \frac{25}{16}x^2 \]

### Step 5: Solve for \( x^2 \)
Now, we can solve for \( x^2 \):
\[ 1225 = \frac{25}{16}x^2 \]
Multiplying both sides by \( \frac{16}{25} \):
\[ x^2 = 1225 \times \frac{16}{25} \]
Calculating \( 1225 \div 25 = 49 \):
\[ x^2 = 49 \times 16 = 784 \]

### Step 6: Find \( x \)
Taking the square root of both sides:
\[ x = \sqrt{784} = 28 \text{ cm} \]

### Step 7: Find the shortest side \( y \)
Using the relation \( y = \frac{3}{4}x \):
\[ y = \frac{3}{4} \times 28 = 21 \text{ cm} \]

### Step 8: Calculate the volume of the cylinder
The height \( h \) of the cylinder is equal to the second longest side \( x \) and the radius \( r \) is equal to the shortest side \( y \):
- Height \( h = 28 \text{ cm} \)
- Radius \( r = 21 \text{ cm} \)

The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
Using \( \pi \approx \frac{22}{7} \):
\[ V = \frac{22}{7} \times (21)^2 \times 28 \]

### Step 9: Calculate \( r^2 \)
Calculating \( r^2 \):
\[ (21)^2 = 441 \]
Now substituting back:
\[ V = \frac{22}{7} \times 441 \times 28 \]

### Step 10: Simplify the volume calculation
Calculating \( \frac{22 \times 441 \times 28}{7} \):
First, simplify \( \frac{441}{7} = 63 \):
\[ V = 22 \times 63 \times 28 \]

Calculating \( 22 \times 63 = 1386 \):
Now, calculating \( 1386 \times 28 \):
\[ 1386 \times 28 = 38708 \]

### Final Step: Approximate the volume
The approximate volume of the cylinder \( C \) is:
\[ V \approx 38708 \text{ cm}^3 \]