Cylinder A Cylinder B Shape Volume 300π 2700π Cylinder A and Cylinder B are similar right cylinders. The radius of Cylinder A is 4, and its surface area is dπ . The surface area of Cylinder B is tπ. What is the value of t-d? ( Note: surface area of a cylinder is 2πrh+2πr^2)

Given that Cylinder A and Cylinder B are similar right cylinders, we can use the properties of similar shapes to solve for \( t - d \), where \( t \) is the surface area of Cylinder B and \( d \) is the surface area of Cylinder A.

### Step 1: Determine the dimensions of Cylinder A
- **Volume of Cylinder A** is \( 300\pi \).
- **Radius of Cylinder A** is 4 units.

The volume \( V \) of a right cylinder is given by the formula:
\[
V = \pi r^2 h
\]
Substitute the known values for Cylinder A:
\[
300\pi = \pi (4)^2 h
\]
\[
300\pi = 16\pi h
\]
Dividing both sides by \( 16\pi \):
\[
h = \frac{300}{16} = 18.75
\]
So, the height of Cylinder A is 18.75 units.

### Step 2: Calculate the surface area of Cylinder A (denoted as \( d \))
The surface area \( A \) of a right cylinder is given by the formula:
\[
A = 2\pi rh + 2\pi r^2
\]
Substituting \( r = 4 \) and \( h = 18.75 \) into the formula for Cylinder A:
\[
d = 2\pi (4)(18.75) + 2\pi (4)^2
\]
\[
d = 2\pi (75) + 2\pi (16)
\]
\[
d = 150\pi + 32\pi = 182\pi
\]
Thus, the surface area of Cylinder A is \( d = 182\pi \).

### Step 3: Use the properties of similar cylinders to find the surface area of Cylinder B (denoted as \( t \))
Since Cylinder A and Cylinder B are similar, the ratio of their corresponding linear dimensions (such as radius) is the same. Let's denote the radius of Cylinder B as \( r_B \) and the height as \( h_B \). The ratio of their volumes is equal to the cube of the ratio of their radii:
\[
\frac{V_B}{V_A} = \left( \frac{r_B}{r_A} \right)^3
\]
Substituting the volumes of the cylinders:
\[
\frac{2700\pi}{300\pi} = \left( \frac{r_B}{4} \right)^3
\]
\[
\frac{2700}{300} = \left( \frac{r_B}{4} \right)^3
\]
\[
9 = \left( \frac{r_B}{4} \right)^3
\]
Taking the cube root of both sides:
\[
\frac{r_B}{4} =  \sqrt[3]{9} = 3
\]
\[
r_B = 12
\]
Thus, the radius of Cylinder B is 12 units.

Since the cylinders are similar, the ratio of their surface areas is the square of the ratio of their radii:
\[
\frac{A_B}{A_A} = \left( \frac{r_B}{r_A} \right)^2 = \left( \frac{12}{4} \right)^2 = 9
\]
Therefore, the surface area of Cylinder B is:
\[
t = 9 \times d = 9 \times 182\pi = 1638\pi
\]

### Step 4: Find \( t - d \)
Now that we have \( t = 1638\pi \) and \( d = 182\pi \), we can find \( t - d \):
\[
t - d = 1638\pi - 182\pi = 1456\pi
\]

Thus, the value of \( t - d \) is \( 1456\pi \).

Would you like to see more details on any part of the solution? Here are five related questions to further explore this concept:

1. How would the height of Cylinder B be calculated using the properties of similar cylinders?
2. What happens to the surface area ratio if the cylinders are not similar?
3. Can you find the total surface area of both cylinders combined?
4. How does the volume formula for a cylinder change if the cylinder is tilted (oblique cylinder)?
5. How would you find the surface area if the cylinders were hollow?

**Tip**: When dealing with similar figures, remember that areas scale by the square of the ratio of corresponding lengths, and volumes scale by the cube.

Cylinder A and Cylinder B are similar right cylinders. The radius of Cylinder A is 4, and its surface area is dπ. The surface area of Cylinder B is tπ. What is the value of t-d?

Explore how to calculate the difference in surface areas between two similar right cylinders. This problem involves understanding the relationship between the radii and heights of the cylinders, applying key geometric formulas, and using properties of similar shapes. Ideal for students in Grades 9-12.

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