1) To calculate the volume of a pyramid with a square base, use the formula:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
The base area of a square with side 7 cm is \(7^2 = 49 \, \text{cm}^2\). The height is 11 cm.
\[
V = \frac{1}{3} \times 49 \times 11 = \frac{539}{3} \approx 179.7 \, \text{cm}^3
\]
The volume is approximately 179.7 cm³.
2) The total surface area of a cone is the sum of the base area and the lateral (curved) surface area. The formula is:
\[
A = \pi r^2 + \pi r l
\]
where \(r\) is the radius and \(l\) is the slant height. Here, \(r = 2.5\) cm and \(l = 10\) cm.
\[
A = \pi (2.5)^2 + \pi (2.5)(10) = \pi (6.25 + 25) = 31.25\pi \, \text{cm}^2
\]
The total surface area is \(31.25\pi \, \text{cm}^2\).
3) The volume of a sphere is given by:
\[
V = \frac{4}{3} \pi r^3
\]
The diameter is 40 mm, so the radius \(r = 20\) mm.
\[
V = \frac{4}{3} \pi (20)^3 = \frac{4}{3} \pi (8000) = \frac{32000}{3} \pi \approx 10666.7\pi \, \text{mm}^3
\]
The volume is approximately \(10666.7\pi \, \text{mm}^3\).
4) The sphere fits exactly into a cube, so the diameter of the sphere is equal to the side of the cube, which is 18 mm. Thus, the radius \(r = 9\) mm. The surface area of a sphere is:
\[
A = 4 \pi r^2
\]
\[
A = 4 \pi (9)^2 = 4 \pi (81) = 324\pi \, \text{mm}^2
\]
Convert to cm²: \(1 \, \text{cm}^2 = 100 \, \text{mm}^2\), so:
\[
A = \frac{324\pi}{100} = 3.24\pi \, \text{cm}^2
\]
The surface area is \(3.24\pi \, \text{cm}^2\).
5) The volume of the sphere is equal to the volume of the cylinder since the metal is melted and recast. The volume of the sphere with diameter 14 cm (radius 7 cm) is:
\[
V_{\text{sphere}} = \frac{4}{3} \pi (7)^3 = \frac{4}{3} \pi (343) = \frac{1372}{3} \pi \, \text{cm}^3
\]
The cylinder has diameter 4 cm, so radius 2 cm. Let \(h\) be the height. The volume of the cylinder is:
\[
V_{\text{cylinder}} = \pi (2)^2 h = 4\pi h
\]
Set the volumes equal:
\[
4\pi h = \frac{1372}{3} \pi
\]
\[
h = \frac{1372}{3 \times 4} = \frac{1372}{12} \approx 114.3 \, \text{cm}
\]
The height of the cylinder is approximately 114.3 cm.
6) The curved surface area of a cone is given by:
\[
A = \pi r l
\]
where \(r\) is the radius and \(l\) is the slant height. Given \(A = 256\pi \, \text{cm}^2\) and \(l = 32\) cm:
\[
256\pi = \pi r (32)
\]
\[
r = \frac{256}{32} = 8 \, \text{cm}
\]
Use the Pythagorean theorem to find the height \(h\):
\[
l^2 = r^2 + h^2
\]
\[
32^2 = 8^2 + h^2
\]
\[
1024 = 64 + h^2
\]
\[
h^2 = 960
\]
\[
h = \sqrt{960} = \sqrt{64 \times 15} = 8\sqrt{15} \, \text{cm}
\]
The volume of the cone is:
\[
V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi (8)^2 (8\sqrt{15}) = \frac{1}{3} \pi (64)(8\sqrt{15}) = \frac{512}{3} \pi \sqrt{15} \, \text{cm}^3
\]
The volume is \(\frac{512}{3} \pi \sqrt{15} \, \text{cm}^3\).
Parent Tip: Review the logic above to help your child master the concept of volume of a cylinder word problems worksheet.
