Prisms - SAT Math

Since the volume is the product of length, width, and height, and each of these three dimensions are integers, it is important to know the factors of the volume. 70 = (2)(5)(7). This implies that each of these factors (and only these factors with the exception of 1) will show up in the three dimensions exactly once. This creates precisely the following five possibilities:

2, 5, 7

SA = 2((2)(5)+(2)(7)+(5)(7)) = 118

1, 7, 10

SA = 2((1)(7)+(1)(10)+(7)(10)) = 174

1, 5, 14

SA = 2((1)(5)+(1)(14)+(5)(14)) = 178

1, 2, 35

SA = 2((1)(2)+(1)(35)+(2)(35)) = 214

1, 1, 70

SA = 2((1)(1)+(1)(70)+(1)(70)) = 282

Let's call the side lengths of the box l, w, and h. We are told that l, w, and h must all be integer lengths greater than one. We are also told that the volume of the box is 182 cubic units.

Since the volume of a rectangular box is the product of its side lengths, this means that lwh must equal 182.

(l)(w)(h) = 182.

In order to determine possible values of l, w, and h, it would help us to figure out the factors of 182. We want to express 182 as a product of three integers each greater than 1.

Let's factor 182. Because 182 is even, it is divisible by 2.

182 = 2(91).

91 is equal to the product of 7 and 13.

Thus, 182 = 2(7)(13).

This means that the lengths of the box must be 2, 7, and 13 units.

In order to find the surface area, we can use the following formula:

surface area = 2lw + 2lh + 2hw.

surface area = 2(2)(7) + 2(2)(13) + 2(7)(13)

= 28 + 52 + 182

= 262 square units.

The answer is 262.

There are six faces to a right, rectangular prism. Based on our dimensions, we know that we must have a face that is 3 x 5, a face that is 5 x 20 and a face that is 3 x 20. To think this through, imagine that the front face is 3 x 5, the right side is 5 x 20, and the top is 3 x 20. Now, each of these sides has a matching side opposite (the left has the right, the top has the bottom, the front has the back).

Therefore, we know we have the following areas for the faces of our prism:

2 * 3 * 5 = 30

2 * 5 * 20 = 200

2 * 3 * 20 = 120

Add these to get the total surface area:

30 + 200 + 120 = 350

There are six faces to a right, rectangular prism. Based on our dimensions, we know that we must have a face that is 12.4 x 2.3, a face that is 2.3 x 33 and a face that is 33 x 12.4. To think this through, imagine that the front face is 12.4 x 2.3, the left side is 2.3 x 33, and the top is 33 x 12.4. Now, each of these sides has a matching side opposite (the left has the right, the top has the bottom, the front has the back).

Therefore, we know we have the following areas for the faces of our prism:

2 * 12.4 * 2.3 = 57.04

2 * 2.3 * 33 = 151.8

2 * 12.4 * 33 = 818.4

Add these to get the total surface area:

57.04 + 151.8 + 818.4 = 1027.24

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

Dim1: x

Dim2: 2 * Dim1 = 2x

Dim3: 2 * Dim2 = 2 * 2x = 4x

More directly stated, therefore, our dimensions are: x, 2x, and 4x. Therefore, the volume is x * 2x * 4x = 216, which simplifies to 8x3 = 216 or x3 = 27. Solving for x, we find x = 3. Therefore, our dimensions are:

x = 3

2x = 6

4x = 12

Or: 3 x 6 x 12

Now, to find the surface area, we must consider that this means that our prism has sides of the following dimensions: 3 x 6, 6 x 12, and 3 x 12. Since each side has a "matching" side opposite it, we know that we have the following values for the areas of the faces:

2 * 3 * 6 = 36

2 * 6 * 12 = 144

2 * 3 * 12 = 72

The total surface area therefore equals: 36 + 144 + 72 = 252 square units.

Converting squared units is not difficult, though you have to be careful not to make a simple mistake. It is tempting to think you can merely divide the initial value (30,096) by 12, as though you were converting from inches to feet.

Begin by thinking this through as follows. In the case of a single dimension, we know that:

1 ft = 12 in or 1 in = (1/12) ft

Now, think the case of a square with dimensions 1 ft x 1 ft. This square has the following dimensions in inches: 12 in x 12 in. The area is therefore 12 * 12 = 144 in2. This holds for all two-dimensional conversions. Therefore, the two dimensional conversion equation is:

1 ft2 = 144 in2 or 1 in2 = (1/144) ft2

Based on this, we can convert our value 30,096 in2 thus: 30,096/144 = 209 ft2.

Converting squared units is not difficult, though you have to be careful not to make a simple mistake. It is tempting to think you can merely multiply the initial value (24) by 36, as though you were converting from yards to inches.

Begin by thinking this through as follows. In the case of a single dimension, we know that:

1 yd = 36 in

Now, think the case of a square with dimensions 1 yd x 1 yd. This square has the following dimensions in inches: 36 in x 36 in. The area is therefore 36 * 36 = 1296 in2. This holds for all two-dimensional conversions. Therefore, the two dimensional conversion equation is:

1 yd2 = 1296 in2

Based on this, we can convert our value 24 yd2 thus: 24 * 1296 = 31,104 in2.

The surface area of the prism can be broken into three rectangular sides and two equilateral triangular bases.

The area of the sides is given by: , so for all three sides we get .

The equilateral triangle is also an equiangular triangle by definition, so the base has congruent sides of 6 in and three angles of 60 degrees. We use a special right traingle to figure out the height of the triangle: 30 - 60 - 90. The height is the side opposite the 60 degree angle, so it becomes or 5.196.

The area for a triangle is given by and since we need two of them we get .

Therefore the total surface area is .

First we must calculate the surface area of the cube. We know that there are six surfaces and each surface has the same area:

Now we will determine the amount of paint needed

Since the front face of the prism is a square, the common sidelength - and the missing dimension - is .

There are two faces (front and back) that are squares of sidelength ; the area of each is the square of this, or .

There are four faces (left, right, top, bottom) that are rectangles of dimensions 25 and ; the area of each is the product of the two, .

The surface area is the total of their areas:

Since the top face of the prism is a square, the common sidelength - and the missing dimension - is 25.

The surface area of a rectangular prism with length , width , and height can be found using the formula

.

Setting, and solving for :

In order to figure out how many cubic centimeters can fit into the box, we need to figure out the volume of the box in terms of cubic centimeters. However, the measurements of the box are given in meters. Therefore, we need to convert these measurements to centimeters and then determine the volume of the box.

There are 100 centimeters in one meter. This means that in order to convert from meters to centimeters, we must multiply by 100.

The length of the box is 2 meters, which is equal to 2 x 100, or 200, centimeters.

The width of the box is 0.5(100) = 50 centimeters.

The height of the box is 3.2(100) = 320 centimeters.

Now that all of our measurements are in centimeters, we can calculate the volume of the box in cubic centimeters. Remember that the volume of a rectangular box (or prism) is equal to the product of the length, width, and height.

V = length x width x height

V = (200 cm)(50 cm)(320 cm) = 3,200,000 cm3

To rewrite this in scientific notation, we must move the decimal six places to the left.

V = 3.2 x 106 cm3

The answer is 3.2 x 106.

Let l be the length, w be the width, and h be the height of the prism. We are told that the length is twice the width, and that the width is twice the height. We can set up the following two equations:

l = 2_w_

w = 2_h_

Next, we are told that the surface area is equal to 252 square units. Using the formula for the surface area of the rectangular prism, we can write the following equation:

surface area = 2_lw_ + 2_lh_ + 2_wh_ = 252

We now have three equations and three unknowns. In order to solve for one of the variables, let's try to write w and l in terms of h. We know that w = 2_h_. Because l = 2_w_, we can write l as follows:

l = 2_w_ = 2(2_h_) = 4_h_

Now, let's substitute w = 2_h_ and l = 4_h_ into the equation we wrote for surface area.

2(4_h_)(2_h_) + 2(4_h_)(h) + 2(2_h_)(h) = 252

Simplify each term.

16_h_2 + 8_h_2 + 4_h_2 = 252

Combine _h_2 terms.

28_h_2 = 252

Divide both sides by 28.

_h_2 = 9

Take the square root of both sides.

h = 3.

This means that h = 3. Because w = 2_h_, the width must be 6. And because l = 2_w_, the length must be 12.

Because we now know the length, width, and height, we can find the volume of the prism, which is what the question ultimately requires us to find.

volume of a prism = l • w • h

volume = 12(6)(3)

= 216 cubic units

The answer is 216.

The volume of B is 7920 in3. The volume of A is 7722 in3. Treasure Chest B can hold more treasure.

The question tells us that the cube is split into two prisms of equal size. Therefore, we don't even need to use the formula for the volume of a prism. If the cube is split into two equal forms, then the two prisms will be equal to one-half of the cube's volume.

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)

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

First, let's represent our dimensions. We know the bottom could be represented as being x by 3_x_. The height is said to be twice the longer dimension, so let's call it 2 * 3_x_, or 6_x_. Based on this, we know the dimensions of the prism are x, 2_x_, and 6_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 13,122 in3, we can say:

13,122 = x * 3_x_ * 6_x_ or 13,122 = 18_x_3

Simplifying, we first divide by 18: 729 = _x_3. Taking the cube root of both sides, we find that x = 9.

Now, be careful. The dimensions are not 9, 9, and 9. They are (recall) x, 3_x_, and 6_x_. If x = 9, this means the dimensions are 9, 27, and 54.

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 = √(92 + 272 + 542) = √(81 + 729 + 2916) = √(3726) = √(2 * 3 * 3 * 3 * 3 * 23) = 9√(46) in.