Worksheet titled "Related Rates Worksheet 2" featuring six word problems related to calculus, including scenarios like a balloon inflating, a ladder sliding, and a cone-shaped tank filling.

1. The ratio of the lengths of the sides of the two similar triangles is 3:5. Since the area of a triangle is proportional to the square of its side lengths, the ratio of the areas is \(3^2 : 5^2 = 9 : 25\). The area of the smaller triangle is 18 square units, so let the area of the larger triangle be \(A\). Then:
\[
\frac{9}{25} = \frac{18}{A}
\]
Solving for \(A\):
\[
9A = 18 \times 25 = 450 \implies A = \frac{450}{9} = 50
\]
The area of the larger triangle is 50 square units.

2. The ratio of the radii of the two similar circles is 2:3. The area of a circle is proportional to the square of its radius, so the ratio of the areas is \(2^2 : 3^2 = 4 : 9\). The area of the smaller circle is 12 square units, so let the area of the larger circle be \(A\). Then:
\[
\frac{4}{9} = \frac{12}{A}
\]
Solving for \(A\):
\[
4A = 12 \times 9 = 108 \implies A = \frac{108}{4} = 27
\]
The area of the larger circle is 27 square units.

3. The ratio of the surface areas of two similar cubes is 4:9. Since surface area is proportional to the square of the side length, the ratio of the side lengths is \(\sqrt{4} : \sqrt{9} = 2 : 3\). The volume of a cube is proportional to the cube of its side length, so the ratio of the volumes is \(2^3 : 3^3 = 8 : 27\). The volume of the smaller cube is 16 cubic units, so let the volume of the larger cube be \(V\). Then:
\[
\frac{8}{27} = \frac{16}{V}
\]
Solving for \(V\):
\[
8V = 16 \times 27 = 432 \implies V = \frac{432}{8} = 54
\]
The volume of the larger cube is 54 cubic units.

4. The ratio of the areas of two similar squares is 1:4. Since area is proportional to the square of the side length, the ratio of the side lengths is \(\sqrt{1} : \sqrt{4} = 1 : 2\). The perimeter of a square is proportional to its side length, so the ratio of the perimeters is also 1:2. The perimeter of the smaller square is 12 units, so let the perimeter of the larger square be \(P\). Then:
\[
\frac{1}{2} = \frac{12}{P}
\]
Solving for \(P\):
\[
P = 12 \times 2 = 24
\]
The perimeter of the larger square is 24 units.

5. The ratio of the volumes of two similar spheres is 8:27. Since volume is proportional to the cube of the radius, the ratio of the radii is \(\sqrt[3]{8} : \sqrt[3]{27} = 2 : 3\). The surface area of a sphere is proportional to the square of the radius, so the ratio of the surface areas is \(2^2 : 3^2 = 4 : 9\). The surface area of the smaller sphere is 16 square units, so let the surface area of the larger sphere be \(S\). Then:
\[
\frac{4}{9} = \frac{16}{S}
\]
Solving for \(S\):
\[
4S = 16 \times 9 = 144 \implies S = \frac{144}{4} = 36
\]
The surface area of the larger sphere is 36 square units.

6. The ratio of the volumes of two similar cylinders is 1:8. Since volume is proportional to the cube of the radius (assuming height is proportional), the ratio of the radii is \(\sqrt[3]{1} : \sqrt[3]{8} = 1 : 2\). The lateral surface area of a cylinder is proportional to the radius (since \(2\pi r h\), and height is proportional to radius), so the ratio of the lateral surface areas is 1:2. The lateral surface area of the smaller cylinder is 10 square units, so let the lateral surface area of the larger cylinder be \(L\). Then:
\[
\frac{1}{2} = \frac{10}{L}
\]
Solving for \(L\):
\[
L = 10 \times 2 = 20
\]
The lateral surface area of the larger cylinder is 20 square units.

7. The ratio of the volumes of two similar cones is 1:27. Since volume is proportional to the cube of the radius (assuming height is proportional), the ratio of the radii is \(\sqrt[3]{1} : \sqrt[3]{27} = 1 : 3\). The lateral surface area of a cone is proportional to the radius (since \(\pi r l\), and slant height is proportional to radius), so the ratio of the lateral surface areas is 1:3. The lateral surface area of the smaller cone is 12 square units, so let the lateral surface area of the larger cone be \(L\). Then:
\[
\frac{1}{3} = \frac{12}{L}
\]
Solving for \(L\):
\[
L = 12 \times 3 = 36
\]
The lateral surface area of the larger cone is 36 square units.

8. The ratio of the volumes of two similar pyramids is 1:64. Since volume is proportional to the cube of the height (assuming base area is proportional to height squared), the ratio of the heights is \(\sqrt[3]{1} : \sqrt[3]{64} = 1 : 4\). The base area of a pyramid is proportional to the square of the height (since base area is proportional to height squared), so the ratio of the base areas is \(1^2 : 4^2 = 1 : 16\). The base area of the smaller pyramid is 5 square units, so let the base area of the larger pyramid be \(B\). Then:
\[
\frac{1}{16} = \frac{5}{B}
\]
Solving for \(B\):
\[
B = 5 \times 16 = 80
\]
The base area of the larger pyramid is 80 square units.

9. The ratio of the volumes of two similar prisms is 1:125. Since volume is proportional to the cube of the side length, the ratio of the side lengths is \(\sqrt[3]{1} : \sqrt[3]{125} = 1 : 5\). The surface area of a prism is proportional to the square of the side length, so the ratio of the surface areas is \(1^2 : 5^2 = 1 : 25\). The surface area of the smaller prism is 20 square units, so let the surface area of the larger prism be \(S\). Then:
\[
\frac{1}{25} = \frac{20}{S}
\]
Solving for \(S\):
\[
S = 20 \times 25 = 500
\]
The surface area of the larger prism is 500 square units.

10. The ratio of the volumes of two similar spheres is 1:27. Since volume is proportional to the cube of the radius, the ratio of the radii is \(\sqrt[3]{1} : \sqrt[3]{27} = 1 : 3\). The diameter is twice the radius, so the ratio of the diameters is also 1:3. The diameter of the smaller sphere is 4 units, so let the diameter of the larger sphere be \(D\). Then:
\[
\frac{1}{3} = \frac{4}{D}
\]
Solving for \(D\):
\[
D = 4 \times 3 = 12
\]
The diameter of the larger sphere is 12 units.