PSAT Math - Solid Geometry

A right pyramid with a square base has a height that is twice the length of one edge of the base. If the height of the pyramid is 6 meters, find the volume of the pyramid.

Explanation

If the height, which is twice the length of the base edges, measures 6 meters, then each base edge must measure 3 meters.

Since the base is a square, the area of the base is 3 x 3 = 9.

Therefore the volume of the right pyramid is V = (1/3) x area of the base x height = 1/3(9)(6) = 18.

A rectangular prism has a volume of 144 and a surface area of 192. If the shortest edge is 3, what is the length of the longest diagonal through the prism?

Explanation

The volume of a rectangular prism is $length\cdot width\cdot height$ .

We are told that the shortest edge is 3. Let us call this the height.

We now have , or .

Now we replace variables by known values:

$192=2lw+6l+6w = 2(48) + 6(\frac{48}{w})+6w$

Now we have:

$96 = \frac{288}{w}+6w \Rightarrow 96w = 288 + 6w^2 \Rightarrow 16w=48+w^2$

$\Rightarrow w^2-16w+48=0\Rightarrow (w-4)(w-12)=0$

We have thus determined that the other two edges of the rectangular prism will be 4 and 12. We now need to find the longest diagonal. This is equal to:

$\sqrt{3^2+4^2+12^2}=\sqrt{9+16+144} = \sqrt{169} = 13$

If you do not remember how to find this directly, you can also do it in steps. You first find the diagonal across one of the sides (in the plane), by using the Pythagorean Theorem. For example, we choose the side with edges 3 and 4. This diagonal will be:

$\sqrt{3^2+4^2}=\sqrt{25}=5$

We then use a plane with one side given by the diagonal we just found (length 5) and the other given by the distance of the 3rd edge (length 12).

This diagonal is then $\sqrt{5^2+12^2}=\sqrt{169}=13$ .

A certain cube has a side length of 25 m. How many square tiles, each with an area of 5 m2, are needed to fully cover the surface of the cube?

100

200

500

750

1000

Explanation

A cube with a side length of 25m has a surface area of:

25m * 25m * 6 = 3,750 m2

(The surface area of a cube is equal to the area of one face of the cube multiplied by 6 sides. In other words, if the side of a cube is s, then the surface area of the cube is 6_s_2.)

Each square tile has an area of 5 m2.

Therefore, the total number of square tiles needed to fully cover the surface of the cube is:

3,750m2/5m2 = 750

Note: the volume of a cube with side length s is equal to _s_3. Therefore, if asked how many mini-cubes with side length n are needed to fill the original cube, the answer would be:

s3/n3

A regular tetrahedron has an edge length of . What is its volume?

$\frac{32}{3\sqrt{2}}$

$\frac{32}{\sqrt{2}}$

$\frac{64}{3\sqrt{2}}$

$\frac{32}{3}$

Explanation

The volume of a tetrahedron is found with the equation $V=\frac{a^3}{6\sqrt{2}}$ , where represents the length of an edge of the tetrahedron.

Plug in 4 for the edge length and reduce as much as possible to find the answer:

$V=\frac{4^3}{6\sqrt{2}}$

$V=\frac{64}{6\sqrt{2}}$

$V=\frac{32}{3\sqrt{2}}$

The volume of the tetrahedron is $\frac{32}{3\sqrt{2}}$ .

A regular tetrahedron has an edge length of . What is its volume?

$\frac{32}{3\sqrt{2}}$

$\frac{32}{\sqrt{2}}$

$\frac{64}{3\sqrt{2}}$

$\frac{32}{3}$

Explanation

The volume of a tetrahedron is found with the equation $V=\frac{a^3}{6\sqrt{2}}$ , where represents the length of an edge of the tetrahedron.

Plug in 4 for the edge length and reduce as much as possible to find the answer:

$V=\frac{4^3}{6\sqrt{2}}$

$V=\frac{64}{6\sqrt{2}}$

$V=\frac{32}{3\sqrt{2}}$

The volume of the tetrahedron is $\frac{32}{3\sqrt{2}}$ .

What is the approximate volume of a square pyramid with one edge of the base measuring $2.7\textup{inches}$ and a height equal to the diagonal of its base?

$9.28\textup{ cubic inches}$

$27.84\textup{ cubic inches}$

$3.82\textup{ cubic inches}$

$14.58\textup{ cubic inches}$

$5.4\textup{ cubic inches}$

Explanation

If one edge of the base is 2.7 inches, and the height of the pyramid is equal to the diagonal of the base, we can find the height using the edge. The height is going to be equal to $\sqrt{2.7^2+2.7^2}=\sqrt{14.58}$ . The area of the base is . Volume of a pyramid is equal to $\left ( \frac{1}{3} \right )\left ( \textup{area of the base} \right )\left ( \textup{height} \right )$

This leaves us with

$v=(7.29)(\sqrt{14.58})\left ( \frac{1}{3} \right )\approx9.28$

A certain cube has a side length of 25 m. How many square tiles, each with an area of 5 m2, are needed to fully cover the surface of the cube?

100

200

500

750

1000

Explanation

A cube with a side length of 25m has a surface area of:

25m * 25m * 6 = 3,750 m2

(The surface area of a cube is equal to the area of one face of the cube multiplied by 6 sides. In other words, if the side of a cube is s, then the surface area of the cube is 6_s_2.)

Each square tile has an area of 5 m2.

Therefore, the total number of square tiles needed to fully cover the surface of the cube is:

3,750m2/5m2 = 750

Note: the volume of a cube with side length s is equal to _s_3. Therefore, if asked how many mini-cubes with side length n are needed to fill the original cube, the answer would be:

s3/n3

A rectangular prism has a volume of 144 and a surface area of 192. If the shortest edge is 3, what is the length of the longest diagonal through the prism?

Explanation

The volume of a rectangular prism is $length\cdot width\cdot height$ .

We are told that the shortest edge is 3. Let us call this the height.

We now have , or .

Now we replace variables by known values:

$192=2lw+6l+6w = 2(48) + 6(\frac{48}{w})+6w$

Now we have:

$96 = \frac{288}{w}+6w \Rightarrow 96w = 288 + 6w^2 \Rightarrow 16w=48+w^2$

$\Rightarrow w^2-16w+48=0\Rightarrow (w-4)(w-12)=0$

We have thus determined that the other two edges of the rectangular prism will be 4 and 12. We now need to find the longest diagonal. This is equal to:

$\sqrt{3^2+4^2+12^2}=\sqrt{9+16+144} = \sqrt{169} = 13$

$\sqrt{3^2+4^2}=\sqrt{25}=5$

We then use a plane with one side given by the diagonal we just found (length 5) and the other given by the distance of the 3rd edge (length 12).

This diagonal is then $\sqrt{5^2+12^2}=\sqrt{169}=13$ .

A regular tetrahedron has an edge length of . What is its volume?

$\frac{32}{3\sqrt{2}}$

$\frac{32}{\sqrt{2}}$

$\frac{64}{3\sqrt{2}}$

$\frac{32}{3}$

Explanation

The volume of a tetrahedron is found with the equation $V=\frac{a^3}{6\sqrt{2}}$ , where represents the length of an edge of the tetrahedron.

Plug in 4 for the edge length and reduce as much as possible to find the answer:

$V=\frac{4^3}{6\sqrt{2}}$

$V=\frac{64}{6\sqrt{2}}$

$V=\frac{32}{3\sqrt{2}}$

The volume of the tetrahedron is $\frac{32}{3\sqrt{2}}$ .

A certain cube has a side length of 25 m. How many square tiles, each with an area of 5 m2, are needed to fully cover the surface of the cube?

100

200

500

750

1000

Explanation

A cube with a side length of 25m has a surface area of:

25m * 25m * 6 = 3,750 m2

(The surface area of a cube is equal to the area of one face of the cube multiplied by 6 sides. In other words, if the side of a cube is s, then the surface area of the cube is 6_s_2.)

Each square tile has an area of 5 m2.

Therefore, the total number of square tiles needed to fully cover the surface of the cube is:

3,750m2/5m2 = 750

Note: the volume of a cube with side length s is equal to _s_3. Therefore, if asked how many mini-cubes with side length n are needed to fill the original cube, the answer would be:

s3/n3

Solid Geometry

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