We are given two similar right cylinders, Cylinder A and Cylinder B. Their volumes are $300\pi$ and $2700\pi$, respectively. The radius of Cylinder A is 4, and we are tasked with finding the value of $t - d$, where $d\pi$ and $t\pi$ are the surface areas of Cylinder A and Cylinder B, respectively.

Step 1: Use the similarity ratio for the cylinders

Since the cylinders are similar, the ratio of their volumes is proportional to the cube of the ratio of their corresponding dimensions (such as their radii or heights). The volume of a cylinder is given by:

$V = \pi r^2 h$

Thus, the ratio of the volumes of Cylinder B and Cylinder A is:

$\frac{V_B}{V_A} = \frac{2700\pi}{300\pi} = 9$

This tells us that the ratio of their dimensions (including radius and height) is the cube root of 9:

$\frac{r_B}{r_A} = \frac{h_B}{h_A} = \sqrt[3]{9} = 3$

Thus, the radius and height of Cylinder B are 3 times those of Cylinder A.

Step 2: Calculate the radius of Cylinder B

Given that the radius of Cylinder A is $r_A = 4$, the radius of Cylinder B is:

$r_B = 3 \times r_A = 3 \times 4 = 12$

Step 3: Express the surface areas of Cylinder A and Cylinder B

The surface area $S$ of a cylinder is given by:

$S = 2\pi rh + 2\pi r^2$

For Cylinder A, the surface area is:

$S_A = 2\pi r_A h_A + 2\pi r_A^2 = d\pi$

For Cylinder B, the surface area is:

$S_B = 2\pi r_B h_B + 2\pi r_B^2 = t\pi$

Step 4: Relate the surface areas using the similarity ratio

Since the cylinders are similar, the surface area ratio is proportional to the square of the ratio of their corresponding dimensions. Thus, the ratio of their surface areas is:

$\frac{S_B}{S_A} = \left( \frac{r_B}{r_A} \right)^2 = 3^2 = 9$

So, the surface area of Cylinder B is 9 times that of Cylinder A:

$S_B = 9 \times S_A$

Step 5: Calculate $t - d$

From the relation $S_B = 9 \times S_A$, we have:

$t\pi = 9 \times d\pi$

Dividing both sides by $\pi$:

$t = 9d$

Thus:

$t - d = 9d - d = 8d$

Step 6: Find the surface area of Cylinder A

We already know that the radius of Cylinder A is $r_A = 4$, and its volume is $V_A = 300\pi$. Using the volume formula for a cylinder, we can find the height of Cylinder A:

$V_A = \pi r_A^2 h_A$ $300\pi = \pi (4)^2 h_A$ $300 = 16 h_A$ $h_A = \frac{300}{16} = 18.75$

Now, using the surface area formula for Cylinder A:

$S_A = 2\pi r_A h_A + 2\pi r_A^2$ $S_A = 2\pi (4)(18.75) + 2\pi (4)^2$ $S_A = 150\pi + 32\pi = 182\pi$

Thus, $d = 182$.

Step 7: Calculate $t - d$

Since $t = 9d$, we have:

$t = 9 \times 182 = 1638$

Finally:

$t - d = 1638 - 182 = 1456$

Thus, the value of $t - d$ is $\boxed{1456}$.

Would you like more details or have any further questions?

Here are 5 related questions:

1. How do you find the surface area of a cylinder given only its height and radius?
2. What is the relationship between the volume and surface area of similar figures?
3. Can you explain how to apply the concept of geometric similarity in 3D shapes?
4. How would the problem change if the cylinders weren't similar?
5. What is the cube root rule in relation to volumes of similar objects?

Tip: Always check the proportional relationships (volume, area) in similar figures—these follow predictable scaling laws!