Mathematics 20-3

Surface area and volume measure different things: surface area is the total area of an object's faces in square units, what siding or paint covers, and volume is the space inside in cubic units, so a crate 2 by 1 by 0.5 m has 7 m² of surface and 1 m³ of volume.

Each face's area multiplies two lengths, so surface area comes out in square units; volume multiplies three, so it comes out in cubic units.

Volume is the space an object takes up, in cubic units, and capacity is the most a container can hold, in millilitres, litres or gallons: a tank 40 by 25 by 30 cm has a volume of 30 000 cm³ and holds 30 L.

A solid block of wood has a volume and no capacity; a bucket has both, the space its plastic fills and what it holds.

Objects with the same volume can need very different amounts of material: a 4 ft cube and a box 8 by 4 by 2 ft both hold 64 ft³, but the cube has 96 ft² of surface and the flat box 112 ft².

Volume counts the space and surface area counts the skin, and the shape decides how much skin wraps a given space.

The SI units tie volume to capacity: 1 cm³ holds 1 mL, 1000 cm³ holds 1 L, and 1 m³ holds 1000 L, so a 1.2 m³ water tank holds 1200 L.

A cube 10 cm on each side, 1000 cm³, holds exactly one litre; a cube 1 m on each side holds a thousand of them.

A net lays an object's faces out flat, so its surface area is the total of the net's shapes: a tent shaped like a triangular prism 3 m long, with ends 2.4 m wide and 1.6 m high and sloping sides 2.0 m wide, takes 3.84 + 12 + 7.2 = 23.04 m² of fabric with its floor.

Unfolded, the tent is two triangles for the ends, two rectangles for the sloping sides and one rectangle for the floor.

Angles of elevation and depression are both measured at the observer from the horizontal, elevation looking up and depression looking down, and the angle of depression from a cliff top to a boat equals the angle of elevation from the boat to the cliff top.

Each observer draws a horizontal line through the eye and measures from it to the line of sight.

A scale representation is used wherever the real object is too big or too small to draw or build at its own size: house plans, maps, road and site drawings, model trains and cars, sewing patterns, and enlarged drawings of small machine parts.

Reductions shrink the large: a house onto a sheet of paper, a city onto a map, a ship into a display case.

A 3-D object can be shown on flat paper in several ways, each for its own question: views give exact flat faces, an isometric drawing shows three faces at once with true lengths, and a perspective drawing shows how the object looks to the eye.

A machinist cutting a part needs its views; a catalogue showing a part's shape uses an isometric drawing; an artist or architect showing a room as a visitor sees it uses perspective.

A description becomes a sketch before any ratio: 'from the top of a 40 m fire tower, smoke is seen at an angle of depression of 8°' is a right triangle with the tower as the vertical leg, the ground distance as the horizontal leg, and 8° between the horizontal and the line of sight at the top.

Draw the horizontal through the lookout's eye, the line of sight down to the smoke, and mark 8° between them; the right angle is at the tower's foot.

A scale compares a drawn length with the real length in the same unit: 1:50 means 1 cm on the plan is 50 cm in the room, and a scale written as '1 cm = 2 m' is 1:200, because 2 m is 200 cm.

Both sides share a unit, so the ratio itself has no unit and works with any ruler: 1 mm drawn is 50 mm built.

A strategy is a plan chosen from a puzzle's structure and checked by using it: in a 3 by 3 magic square of the numbers 1 to 9, every row, column and diagonal must add to 15, because the nine numbers total 45 and the three rows share it equally.

The structure gives the first fact before any guessing: 1 + 2 + ...

A budget's income line is the money that actually arrives: a $3300 gross monthly salary that deposits $2650 gives a $2650 budget, and regular tips or a part-time job's deposits are added to it.

Deductions come off before the money reaches the account, so planning from the gross spends $650 a month that never arrives.

Simple interest is paid on the principal only, I = Prt: $2000 at 4% a year for 3 years earns 2000 × 0.04 × 3 = $240, the same $80 every year.

04; t is the time in years.

Guess and check is a strategy when each try is recorded and steers the next: two numbers that add to 25 and multiply to 154 are found by trying 10 and 15, product 150, too small, then moving closer, 11 and 14, product 154.

Write each try in a table with its result, and let the result decide the next try: with the sum fixed, numbers closer together have a larger product, so a product that is too small means move inward.

Income paid on another schedule is changed to the budget's month: a biweekly deposit of $1223 is 1223 × 26 ÷ 12 = $2650 a month, not 1223 × 2 = $2446.

Biweekly means every two weeks, 26 deposits a year, so most months have two and two months have three.

A formula is applied by putting each known value in for its letter, with the units made the same first: the finance charge on a $1200 purchase repaid in 12 payments of $110 is 12 × 110 − 1200 = $120.

Read what each letter stands for before substituting: in finance charge = total of payments − amount financed, the total of payments is the number of payments times each payment.

Slope measures how steep something is wherever a surface or a line rises or falls: wheelchair ramps, roofs, road grades, drain pipes, conveyor belts, skateboard parks and ski hills.

In each the question is how much it rises or falls for each step across.

Each trade uses the formulas of its work: a concrete finisher V = lwh for the cubic metres to order, a painter surface area for the litres, a plumber slope for the drop of a drain, an HVAC technician a room's volume for its air, and a loans officer I = Prt.

Knowing which formula a job uses, and what its answer becomes, is half the skill: the finisher's volume becomes a truck order, the painter's area becomes cans, the plumber's slope becomes the height to set each pipe hanger.

Slope is rise over run, the vertical change divided by the horizontal change: a ramp rising 0.6 m over a run of 7.2 m has a slope of 0.6 ÷ 7.2 = 1/12.

The run is measured level along the ground, not along the ramp's surface.

A formula already solved for the wanted value is used directly: a drain that must drop 1/4 in for every foot of run drops 0.25 × 40 = 10 in over a 40 ft run.

The plumbing rule gives drop = slope × length, with the slope in inches per foot and the length in feet, so the answer comes out in inches.

Each graph answers its own question: bars compare categories, a line shows change over time, a circle shows shares of one whole, and a histogram shows how measurements spread.

Start from the question the graph must answer, then pick: which crew logged the most hours is bars, how sales moved through the year is a line, where a job's money went is a circle, how long repairs usually take is a histogram.

Bars are for separate categories and a line is for a quantity changing over time, so joining the sales of pizza to the sales of tacos with a line invents a trend that does not exist.

A line between points says the values in between exist and change smoothly, which is true for months and hours and false for products, crews or provinces.

A histogram groups measurements into equal intervals and its bars touch, while a bar graph's bars stand apart because its categories are separate.

Repair times, heights and delivery times are measured on a continuous scale, so the intervals 10 to 19, 20 to 29 and so on sit side by side with no gaps.

A circle graph needs parts of one whole that add to 100%: a job's costs fit, while five products' sales in five different months do not.

Every sector is a share of the same total, so the shares must make up that total exactly.

A circle graph's sectors are each share of the total times 360°: a $4800 job's costs of $1800, $1440, $960 and $600 make sectors of 135°, 108°, 72° and 45°.

375 × 360° = 135°.