Solid Geometry - PSAT Math

The tetrahedron has four faces, each of which is an equilateral triangle with sidelength 7. Each face has area

The total surface area is four times this, or about .

Rounded, this is 84.9.

We are looking for the height of the pyramid.

The base is an equilateral triangle with sidelength 4, so its area can be calculated as follows:

The height of a pyramid can be calculated using the fomula

We set and and solve for :

The height of the pyramid is . The base is an equilateral triangle with sidelength 4, so its area can be calculated as follows:

The volume of a pyramid can be calculated using the fomula

The volume of a tetrahedron is found with the equation , 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:

The volume of the tetrahedron is .

The diagonal from the top back left corner to the bottom front right corner will be the hypotenuse of a right triangle. The sides of the triangle will be the height of the box and the diagonal through the middle of one of the rectangular faces. We will be able to solve for the length using the Pythagorean Theorem.

To calculate the length of the hypotenuse, we first must find the length of the rectangular diagonal using the sides of the rectangle. This diagonal will be the hypotenuse of a right triangle with sides 7 and 4. Solve for the diagonal length using the Pythagorean Theorem.

Now we can return to our first triangle. We are given the height, 4, and now have the length of the rectangular diagonal. Use these values to solve for the length of the diagonal that connects the top back left corner and the bottom front right corner.

The problem is simple, but be careful. The units are not equal. First convert the last dimension into inches. There are 12 inches per foot. Therefore, the prism's dimensions really are: 4 in x 12 in x 24 in.

From this point, things are relatively easy. The distance from corner to corner in a three-dimensional prism like this can be found by using a variation on the Pythagorean Theorem that merely adds one dimension. That is, _d_2 = _x_2 + _y_2 + _z_2, or d = √(_x_2 + _y_2 + _z_2)

For our data, this would be:

d = √(42 + 122 + 242) = √(16 + 144 + 576) = √(736) = √(2 * 2 * 2 * 2 * 2 * 23) = 4√(46)

Based on our prompt, we can say that the prism has dimensions that can be represented as:

Dim1: x

Dim2: 3 * Dim1 = 3x

Dim3: 5 * Dim1 = 5x

More directly stated, therefore, our dimensions are: x, 3x, and 5x. Therefore, the volume is x * 3x * 5x = 120, which simplifies to 15x3 = 120 or x3 = 8. Solving for x, we find x = 2. Therefore, our dimensions are:

x = 2

3x = 6

5x = 10

Or: 2 x 6 x 10

The height of the submerged part of the cylinder is 27cm. 2πrh + πr2 is equal to 270π + 25π = 295π

The equation for the volume of the cylinder is πr2h. When the radius doubles (r becomes 2r) you get π(2r)2h = 4πr2h. So when the radius doubles, the volume quadruples, giving a new volume of 80.

First, let's represent our dimensions. We know the bottom could be represented as being x by x. The height is said to be three times one of these dimensions, so let's call it 3_x_. Based on this, we know the dimensions of the prism are x, x, and 3_x_. Now, the volume of a right rectangular prism is found by multiplying together its three dimensions. Therefore, if we know the overall volume is 375 in3, we can say:

375 = x * x * 3_x_ or 375 = 3_x_3

Simplifying, we first divide by 3: 125 = _x_3. Taking the cube root of both sides, we find that x = 5.

Now, be careful. The dimensions are not 5, 5, 5. They are (recall) x, x, and 3_x_. If x = 5, this means the dimensions are 5, 5, and 15.

At this point, things are beginning to progress to the end of the problem. The distance from corner to corner in a three-dimensional prism like this can be found by using a variation on the Pythagorean Theorem that merely adds one dimension. That is, _d_2 = _x_2 + _y_2 + _z_2, or d = √(_x_2 + _y_2 + _z_2)

For our data, this would be: d = √(52 + 52 + 152) = √(25 + 25 + 225) = √(275) = √(5 * 5 * 11) = 5√(11) in

There are two things to be careful with here. First, we must solve for the radius of the base. Secondly, note that the height is given in feet, not inches. Notice that all the answers are in cubic inches. Therefore, it will be easiest to convert all of our units to inches.

First, solve for the radius, recalling that C = 2_πr_, or, for our values 77_π_ = 2_πr_. Solving for r, we get r = 77/2 or r = 38.5.

The height, in inches, is 24.

The basic form for the volume of a cone is: V = (1 / 3)πr_2_h

For our values this would be:

V = (1/3)π * 38.52 * 24 = 8 * 1482.25_π_ = 11,858π in3

Since the diagonal of the cube is a line segment that goes through the center of the cube (and also the circumscribed sphere), it is clear that the diagonal of the cube is also the diameter of the sphere. Since the radius = 1, the diameter = 2.

The first thing necessary is to determine the dimensions of the cube. This can be done using the volume formula for cubes: V = _s_3, where s is the length of the cube. For our data, this is:

_s_3 = 512, or (taking the cube root of both sides), s = 8.

The distance from corner to corner of the cube will be equal to the distance between (0,0,0) and (8,8,8). The distance formula for three dimensions is very similar to that of 2 dimensions (and hence like the Pythagorean Theorem):

d = √( (_x_1 – _x_2)2 + (_y_1 – _y_2)2 + (_z_1 – _z_2)2)

Or for our simpler case:

d = √( (x)2 + (y)2 + (z)2) = √( (s)2 + (s)2 + (s)2) = √( (8)2 + (8)2 + (8)2) = √( 64 + 64 + 64) = √(64 * 3) = 8√(3)

The formula for the volume of a sphere is

Therefore,

Dividing both sides by , leaves us with . Taking the cube root, we find , meaning our diameter .

We know that the cube has a volume of 27 cubic inches, so each side of the cube must be ∛27=3 inches. Since the cube is inscribed inside the sphere, the diameter of the sphere is the diagonal length of the cube, so the radius of the sphere is half of the diagonal length of the cube. To find the diagonal length of the cube, we use the distance formula d=√(32+32+32 )=√(3*32 )=3√3, and then divide the result by 2 to find the radius of the sphere, (3√3)/2.

S = 4π(r2)

100π = 4π(r2)

100 = 4r2

25 = r2

5 = r

Use the equation for the volume of a sphere to find the radius.

So, the radius of the sphere is 3

Given radius , the surface area of a sphere is given by the formula

Set and solve for :

meters.

Given the radius , the volume of a sphere is given by the formula

We find the inner radius by setting :

meters.