The volume of a cube is (that's why you call to the third power "-cubed"), where represents the length of one of the sides. The surface area is , or more conceptually the sum of the areas of each of the six faces (top, bottom, front, back, left, right) of the cube.

So if the volume of a cube is , that means that , so .

Plugging that into the surface area formula, you have which is .

The total interior angle of a polygon with sides equals . Thus, the total interior angle of an octagon is and the total interior angle of a pentagon is . A regular polygon has the property that all interior angles are congruent, so an interior angle of a regular octagon is . Similarly, the interior angle of a pentagon is . is equivalent to .

For quicker math-by-hand, recognize that you're being asked about a ratio. If you set up the measure of one angle of a regular octagon as (the total measure divided by the number of angles) and the measure of one angle of a regular pentagon as , then notice that with a 180 term multiplied in the numerator of each portion of the ratio, the 180s can factor out. Then you're just taking the ratio of to , which nets quickly to .

The most concrete piece of given information on this problem is that the area of a rectangle is . You know that Area = Length × Width, so you can say here that . You're also told how the length and width relate to each other. The longer side (which is the one they ask you to solve) is inches longer than the shorter side, so you can call those (for the longer) and for the shorter. This then means that:

You can then expand the multiplication:

Which becomes a quadratic if you subtract from each side:

And now you have a choice: you could solve this algebraically by factoring the quadratic, but of course may not be the easiest number to quickly factor. Instead, you could test the answer choices to see which potential multiplies with to yield a product of .

If you start in the middle with , you'll see that would be . Before you calculate, first check to see whether you will indeed get a units digit of (otherwise why do the math?). You will, but when you do do that math you'll see that is , which is too small since you need a larger number in .

So then assess the remaining larger answers. If the answer were E, the sides would be 20×16 which will not end in a , so that's out. If it were D, then your sides would be , which does end in a , and which does yield .

Had you wished to factor the quadratic, you would find that factors to , again yielding as the answer for the longer side.

The sum of the interior angles of an n-sided polygon can be calculated as:

Which you can test for yourself: a triangle (3 sides) has 180 degrees: .

A rectangle has 360 degrees.

So in this case, where , you'd calculate as:

.

The most concrete piece of given information on this problem is that the area of a rectangle is . You know that Area = Length × Width, so you can say here that . You're also told how the length and width relate to each other. The longer side (which is the one they ask you to solve) is inches longer than the shorter side, so you can call those (for the longer) and for the shorter. This then means that:

You can then expand the multiplication:

Which becomes a quadratic if you subtract from each side:

And now you have a choice: you could solve this algebraically by factoring the quadratic, but of course may not be the easiest number to quickly factor. Instead, you could test the answer choices to see which potential multiplies with to yield a product of .

If you start in the middle with , you'll see that would be . Before you calculate, first check to see whether you will indeed get a units digit of (otherwise why do the math?). You will, but when you do do that math you'll see that is , which is too small since you need a larger number in .

So then assess the remaining larger answers. If the answer were E, the sides would be 20×16 which will not end in a , so that's out. If it were D, then your sides would be , which does end in a , and which does yield .

Had you wished to factor the quadratic, you would find that factors to , again yielding as the answer for the longer side.

The volume of a cube is (that's why you call to the third power "-cubed"), where represents the length of one of the sides. The surface area is , or more conceptually the sum of the areas of each of the six faces (top, bottom, front, back, left, right) of the cube.

So if the volume of a cube is , that means that , so .

Plugging that into the surface area formula, you have which is .

The sum of the interior angles of an n-sided polygon can be calculated as:

Which you can test for yourself: a triangle (3 sides) has 180 degrees: .

A rectangle has 360 degrees.

So in this case, where , you'd calculate as:

.

The total interior angle of a polygon with sides equals . Thus, the total interior angle of an octagon is and the total interior angle of a pentagon is . A regular polygon has the property that all interior angles are congruent, so an interior angle of a regular octagon is . Similarly, the interior angle of a pentagon is . is equivalent to .

For quicker math-by-hand, recognize that you're being asked about a ratio. If you set up the measure of one angle of a regular octagon as (the total measure divided by the number of angles) and the measure of one angle of a regular pentagon as , then notice that with a 180 term multiplied in the numerator of each portion of the ratio, the 180s can factor out. Then you're just taking the ratio of to , which nets quickly to .

This problem asks you to find the surface area for sides of the box, since the top side will not have area. You should then determine the dimensions of each side that you'll be using.

For the left and right sides, the measurement will be square inches, and since you'll have two of those sides you'll multiply by to have square inches of sides.

The front and back will measure square inches, and since you'll have two of those sides you should multiply by to have square inches of front/back.

Then you'll need to account for the bottom, which measures square inches.

So your total calculation is square inches.

The key to solving this problem is in dividing the given figure into two: a rectangle on the left and a right triangle on the right:

If you do so, you should recognize something familiar with the right triangle: the hypotenuse has a length of and one side has a length of 30, meaning that this triangle will fit the side ratio. You then know that the bottom side of the triangle must measure .

With that, you can fill in figures for the bottom of the quadrilateral. The bottom of the rectangle will measure , symmetrical to the top, and the bottom of the triangle will also measure , meaning that the entire bottom side of the quadrilateral measures . Therefore the perimeter is .