If the statement “All sets of barbells in Kmart are on sale” is false, then which of the following statements must be true?
Denmark is a major manufacturer of energy-producing wind turbines. Suppose both the following statements are true:
1. Some wind turbines in Ohio are not from Denmark, but are American-built.
2. All of the wind turbines in Lorain County are indeed from Denmark.
From these two statements, which of the following can be concluded?
Consider the line y = x and the line y = 0, both of which go through the origin. They form a 45° angle in the first quadrant. The bisector of this angle is therefore a line with y = mx for its equation. So what’s the value of m? It is tempting to think that it’s ½, but this is not the case.
A grasshopper jumps from its position at (x,y) over the point (a,b) to the point that is diametrically opposite to (x,y), i.e., the three points are collinear, and the grasshopper’s new position is just as far from (a,b) as it was before it jumped. What are the coordinates of this new point, in terms of a, b, x, and y?
When the contents of a cylinder are poured into a second cylinder whose radius is 2 inches greater, the height reached in the second cylinder is one half of that reached in the first. Find to the nearest tenth of an inch the radii of the two cylinders.
The number of cubic feet in the volume of a cube is the same as the number of square inches in its surface area. The length of the edge in feet is:
Imagine a rectangular solid that is made up of smaller unit cubes. The solid measures x by y by z. Now imagine building a shell of unit cubes around the entire solid.
First question: what is an equation for the number of unit cubes in the shell, in terms of x, y, and z?
Second question: for what x, y, and z is the number of cubes in the rectangular solid equal to the number of cubes in the shell?
Of two identical barrels, one is half full and one is two-thirds full. One quarter of the liquid in the second barrel is poured into the first. The first barrel now contains 25 more gallons of liquid than the second. Find the capacity in gallons of one of the barrels.
Water is poured at a constant rate into a one-liter cube near its corner. The water rises to the halfway mark, stops for a moment, and then continues to rise at exactly half its previous rate. The reason is that inside the cube is a hollow cylindrical can that has been attached to the bottom. What are the height and radius of the can? (Recall that a one-liter cube measures 10 cm on each edge.)
In a can of tennis balls that is exactly three balls high, which is greater, the volume of the balls or the volume of the air around the balls? (Disregard the thickness of the balls.)