ACT Preparation / Unit 14 ACT 1332 review, 2 to 3 minutes
A Silo: Cylinder Plus Hemisphere
The idea
Measuring figures comes down to a few formulas and the judgment to apply them. Areas: a rectangle is length times width, a triangle half the base times the height, a trapezoid half the height times the sum of the bases, and a circle πr², with composite and shaded regions built by adding and subtracting. Arcs and sectors are the fraction θ/360 of the circumference and the area; an inscribed angle is half its arc, and a tangent meets its radius at a right angle. Volume fills and surface area covers: a box is lwh, a cylinder πr²h, a cone a third of that, and a sphere (4/3)πr³. The circle equation shows its center and radius. Right-triangle trigonometry pairs sides with angles through sine, cosine and tangent, the unit circle extends them to every angle, and the laws of sines and cosines handle any triangle.
In practice
Worked example: a grain silo is a cylinder of radius 4 meters and height 15 meters, topped by a hemisphere of the same radius. Volume: the cylinder holds π × 16 × 15 = 240π, and the hemisphere, half a sphere, holds ½ × (4/3)π × 64 = (128/3)π, about 42.7π. Total: about 282.7π, roughly 888 cubic meters. Painting the outside, with the base on the ground: the cylinder's side is 2π × 4 × 15 = 120π, and the dome is half a sphere's surface, ½ × 4π × 16 = 32π. Total: 152π, about 478 square meters. From 30 meters away on level ground, the top of the dome, 19 meters up, is seen at an angle of elevation whose tangent is 19/30, about 32°.
Where people slip
Three habits carry forward. Decide whether a question wants a length, an area or a volume, and match the units. Read the figure for what is exposed, removed or shaded before adding or subtracting. And place every trig angle inside its triangle, measured from the horizontal. This ends the geometry; the last math unit turns to data, chance and counting.
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