You are standing outside in the sunshine. An airplane at 3000 feet is going to fly directly over your head in a few minutes. It is traveling 200 mph. You don’t notice it until you hear it, and the instant you hear it you spot it at x degrees above the horizon. However, the sound you hear came from the plane somewhat earlier, when it was 20° above the horizon. Take the speed of sound at 1100 ft/sec, find x, the angle at which you looked up and saw the plane.
Find Sine, Given Tangent
Let tan x = (2ab)/(a2 – b2), where a > b > 0 and 0 < x < 90. Draw a right triangle with angle x and label two sides of the triangle to show tan x. Now find an expression for sin x.
Slope of Angle Bisector
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.
Find Edge of Cube Given Diagonal
The diagonal of a cube is 20 cm. How long is its edge?
Tetrahedron in Cube
Four of the 8 vertices of a cube are vertices of a regular tetrahedron. Find the ratio of the surface area of the cube to the surface area of the tetrahedron.
Square and Triangles Area
In this diagram, not drawn to scale, figures I and III are equilateral triangular regions with areas of 32√3 and 8√3 square inches, respectively. Figure II is a square region with area 32 square inches. Let the length of segment AD be decreased by 12.5% of itself, while the lengths of AB and CD remain unchanged. What’s the percent decrease in the area of the square?
Trapezoid, Find Angle Measure
Here is a nice trapezoid, ABCD, with AB parallel to CD. Diagonal BD is drawn, and BD = AD. ∠DCB = 110° and ∠CBD = 30°. What is ∠ADB?
Triangular Garden Bed, Bandstand
A triangular garden bed is going to be installed near the bandstand in the Milan town square. The high school math club has designed it. It is going to have an obtuse angle, and the three sides of the triangle are whole numbers (measured in yards). The two short sides of the triangle are 3 yards and 4 yards. How long is the longest side?
One Sequence to Continue
What is the next term in this sequence: 2, 9, 28, 65, 126, 217, _____?
Log with Radical Base
Find x if , (note that is the base of the logarithm).