A and B

A and B Plug in the answers to your on-screen calculator. When cubed, choices (A) and (B) are less than 2 cubic feet.The other three choices for the edge of the box produce volumes over 2 cubic feet

A,B,C, D, and E

If the diagonal were exactly 10√2, then the side of the cube would be 10√2.Because the diagonal is less than 10 , each side is less than 10.Therefore, the volume must be less than 10³, or 1,000. Any value less than 1,000 is correct.

84

Use the formula for the volume of a cube to find the length of each side: V = s³, so 64 = s³, and s = 4.To find the surface area of a cube, find the area of each face of the cube and multiply by 6:Each side is 4, so each face has an area of 4 × 4 = 16, and the total surface area of the cube is 16 × 6 = 84.

It’s a geometry problem with variables in the answer choices, so draw the figure and set up your scratch paper to Plug In. Plug in an easy number like r = 5; label the radius 5 and the height 15, which is 1.5 times your diameter of 10.The surface area of a cylinder is made up of 3 smaller areas: 2 identical circular bases on top and bottom, and a rectangle that’s the height of the cylinder on one side and the circumference of the base on the other. If r = 5, then the area of each base is 25π, or 50π for the 2 of them.The rectangle is 15 × 10π = 150π, so the total surface area is 200π, your target answer. Plug 5 in for r in the answers, and only choice (C) matches your target answer of 200π.

505

To find the number of cylinders that can be made from one block of steel, divide the volume of the block by the volume of a cylinder. Start by converting the dimensions of the block into inches:Each steel block is 24 inches by 144 inches by 180 inches, so the volume is 24 × 144 × 180 = 622,080.The formula for the volume of a cylinder is V = πr²h, so the volume of each cylinder is π × 7² × 8 = 1230.88. Finally, 622,080 ÷ 1230.88 = 505.395; the problem asked for complete cylinders, so the correct answer is 505.

A,B, D, and F

First, find the dimensions of the cylinder.Because the cylinder’s height is 20 and its volume is 980π, and V = πr²h, 980π = π (r²)(20), and r = 7.The diameter of the cylinder is 14.Because the end of the cylinder is a circle with a diameter of 14, the largest box that could fit in the cylinder would have a square base with a diagonal of 14. Using the Pythagorean theorem, you can find that the length and the width of the largest possible box equal 14 divided by √2, or approximately 9.90.Therefore, the box’s length and width must each be less than 10, and its height may be up to 20.Choices (A), (B), (D), and (F) match these criteria and work as the dimensions of the box.

¹⁄₃

First, find the angle measures. Since AC and AB are radii of the circle, the triangle they form along with BC must be isosceles. Let the small angles, ∠ACB and ∠ABC, be x, which makes ∠BAC equal to 4x; now 4x + x + x = 180, so x = 30 and ∠BAC must be 120˚. At this point, you’re essentially done:Though there’s other information in the problem about diameters and heights and so on, it’s all unnecessary. Since ∠BAC represents \(\frac{120}{360}\), or ¹⁄₃, of the circular base, the shaded represents the same fraction of the entire cylinder.

Try plugging in a value for the depth, 2 feet. Note that the radii are half the given diameters.Therefore, the volume of water held by Marty’s pool is V = πr²h = π(6)²(2) = 72π and the volume of water held by Rusty’s pool is V = πr²h = π (9)²(2) = 162π. Dividing 162π by 72π yields 2.25.

First, eliminate choices (A) and (B) because the volume must increase when the side of the square base increases. Next, set up a proportion using the square base of the prism:\(\frac{13,000}{25^{2}}=\frac{x}{75^{5}}\). Finally, cross-multiply and solve for x to get choice (D).

A cube has 6 identical faces, each with an area of s², so the surface area of a cube is 6s²; the volume of a cube is s³. Quantity A is \(\frac{6\times 4^{2}}{4^{3}}=\frac{6}{4}\), and Quantity B is \(\frac{6\times 4^{2}}{4^{3}}=\frac{6}{4}\). Quantity A is greater.