Default Unit Worksheet

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Using Units in Maple

Units conversions and manipulations can be done most efficiently at the top-level without using the Standard or Natural Units environments, but in general you can more easily perform the computations in the other environments.

For information on inline computations, see the example worksheets for the Simple Units environment, the Standard Units environment, or the Natural Units environment. Each example in this worksheet is also in the other worksheets to show how you can perform the computations in the other environments.

$restart$

Simple Examples

Add 4 feet to 3 inches.

>	$convert (4, units, feet, inches) + 3$

$51$

(1.1)

How many meters are equivalent to 4 yards?

>	$convert (4, units, yd, m)$

$\frac{2286}{625}$

(1.2)

How many liters are equivalent to 5 UK gallons?

>	$convert (5, units, gal [UK], L)$

$\frac{454609}{20000}$

(1.3)

How many liters are equivalent to 5 US gallons?

>	$convert (5, units, gal [US_liquid], L)$

$\frac{473176473}{25000000}$

(1.4)

How many US liquid gallons are equivalent to a UK gallon?

>	$convert (1, units, gal [US_liquid], gal [UK],)$

$\frac{473176473}{568261250}$

(1.5)

Unit Names and Symbols

Unit symbols and various spellings of unit names are recognized by the package.

>	$convert (32, units, km, m), convert (32, units, km, meter), convert (32, units, km, metres)$

$32000, 32000, 32000$

(2.1)

This feature is expandable so that, for example, you can specify metr (the Polish word for this unit) as an alternate spelling of meter. See the AddUnit help page for more information.

Some units can be recognized with SI or IEC prefixes.

>	$convert (32000, units, m, km), convert (32000, units, m, kilometer), convert (32000, units, m, kilometres)$

$32, 32, 32$

(2.2)

Sample Questions with Solutions

How many miles do you travel in 35 minutes moving at 55 miles per hour?

>	$distance ≔ convert (55 \cdot 35, units, mph minutes, mi)$

$distance ≔ \frac{385}{12}$

(3.1.1)

>	$evalf (distance)$

$32.08333333$

(3.1.2)

How many seconds are equivalent to 3 weeks?

>	$convert (3, units, weeks, s)$

$1814400$

(3.2.1)

How many inches are equivalent to 5 feet 4 inches?

>	$4 + convert (5, units, ft, inches)$

$64$

(3.3.1)

Convert 50 km/h to cm/s.

>	$convert (50, units, \frac{km}{h}, \frac{cm}{s})$

$\frac{12500}{9}$

(3.4.1)

>	$evalf ()$

$1388.888889$

(3.4.2)

How many seconds does it take an object, released from rest, to fall 20 meters?

From the equation $d = \frac{g_{n} t^{2}}{2}$ , we derive:

>	$elapsed_time ≔ \sqrt{2 \cdot \frac{dist}{accel}}$

$elapsed_time ≔ \sqrt{2} \sqrt{\frac{dist}{accel}}$

(3.5.1)

>	$eval (elapsed_time, [dist = 20, accel = convert (1, units, gn, \frac{m}{s^{2}})])$

$\frac{\sqrt{2} \sqrt{400000} \sqrt{196133}}{196133}$

(3.5.2)

>	$evalf ()$

$2.019619976$

(3.5.3)

What is the rest energy of an electron?

Assume the mass of an electron is $9.11 10^{- 31}$ kilograms.

>	$mass ≔ 9.11 10^{−31}$

$mass ≔ 9.110000000 10^{−31}$

(3.6.1)

Approximate the speed of light by $3.00 10^{8}$ meters per second.

>	$c ≔ 3.00 10^{8}$

$c ≔ 3.000000000 10^{8}$

(3.6.2)

Using the formula $E = m c^{2}$ , you can approximate the rest energy of the electron.

>	$E ≔ mass \cdot c^{2}$

$E ≔ 8.199000000 10^{−14}$

(3.6.3)

A more precise answer can be found by converting the known unit the electron mass (em) to joules using energy conversions.

>	$convert (1, units, em, J, energy)$

$8.187104141 10^{−14}$

(3.6.4)

Approximately what volume does 1,000,000,000 US dollars worth of gold occupy?

>	$cost ≔ 1210.00 &num; US dollars per troy ounce November 18th, 2016$

$cost ≔ 1210.00$

(3.7.1)

>	$density ≔ 19.3 &num; grams per cubic centimeter$

$density ≔ 19.3$

(3.7.2)

>	$mass ≔ \frac{1000000000}{convert (cost, units, \frac{USD}{ounce [troy]}, \frac{USD}{g})}$

$mass ≔ 2.570535273 10^{7}$

(3.7.3)

>	$volume ≔ \frac{mass}{density}$

$volume ≔ 1.331883561 10^{6}$

(3.7.4)

What is the length of one side of a cube with this volume?

>	$length_side ≔ \sqrt[3]{volume}$

$length_side ≔ 110.0243351$

(3.7.5)

This is in centimeters. Convert it to meters.

>	$convert (length_side, units, cm, m)$

$1.100243351$

(3.7.6)

An Su-27 Flanker can travel at mach 1.1 at sea level. How fast is this in miles per hour, miles per second, and meters per second?

>	$convert (1.1, units, M, mph)$

$815.6003937$

(3.8.1)

>	$convert (1.1, units, M, \frac{mi}{s})$

$0.2265556649$

(3.8.2)

>	$convert (1.1, units, M, \frac{m}{s})$

$364.606$

(3.8.3)

Approximately how many meters are there in 3.5 miles?

>	$convert (3.5, units, mi, m)$

$5632.704000$

(3.9.1)

>	$round () ⟦m⟧$

$5633 ⟦m⟧$

(3.9.2)

Given 50 US gallons of water, how many 750 mL bottles could you fill?

>	$\frac{convert (50, units, gal [US_liquid], mL)}{750}$

$\frac{157725491}{625000}$

(3.10.1)

>	$floor ()$

$252$

(3.10.2)

Given nylon with a linear mass density of 20 deniers, what length of thread is used in an object weighing 12 grams?

>	$\frac{12}{convert (20, units, deniers, \frac{g}{m})} &num; meters$

$5400$

(3.11.1)

What is the volume in cubic inches of a 2 liter engine.

>	$convert (2, units, L, {inches}^{3})$

$\frac{250000000}{2048383}$

(3.12.1)

>	$evalf ()$

$122.0474882$

(3.12.2)

For a given phenomena with a frequency of 1.420,405,761 GHz, find the:

1. period,

2. number of cycles per year, and

3. number of cycles since the beginning of the earth.

First you must convert the frequency from GHz to Hz (cycles per second).

>	$frequency ≔ 1.420405761 convert (1, units, GHz, \frac{1}{s})$

$frequency ≔ 1.420405761 10^{9}$

(3.13.1)

The period is:

>	$\frac{1}{frequency}$

$7.040241792 10^{−10}$

(3.13.2)

>	$convert (, units, s, nanoseconds)$

$0.7040241792$

(3.13.3)

The number of cycles per year is:

>	$convert (frequency, units, \frac{1}{s}, \frac{1}{yr})$

$4.482363838 10^{16}$

(3.13.4)

The number of cycles since the beginning of the earth (10,000,000,000 years ago) is:

>	$frequency \cdot convert (10000000000, units, yr, s)$

$4.482363838 10^{26}$

(3.13.5)

Given inductance and capacitance, find the resistance in microohms.

The following formula relates the resistance to the inductance and capacitance.

>	$resistance ≔ \sqrt{\frac{inductance}{capacitance}}$

$resistance ≔ \sqrt{\frac{inductance}{capacitance}}$

(3.14.1)

Use an inductance of 124 nanohenries and a capacitance of 3.52 microfarads.

>	$eval (resistance, [inductance = convert (124., units, nH, uH), capacitance = 3.52])$

$0.1876892984$

(3.14.2)

>	$convert (0.1876892984, units, Ω, uOmega)$

$187689.2984$

(3.14.3)

Given a molar energy, find the mass energy in Btu's per pound.

>	$molar_energy ≔ 523.432 &semi; &num; Btu/mol(carbon)$

$molar_energy ≔ 523.432$

(3.15.1)

>	$carbon_mass ≔ 12 &num; kg/mol(carbon)$

$carbon_mass ≔ 12$

(3.15.2)

>	$mass_energy ≔ \frac{molar_energy}{convert (carbon_mass, units, kg, lb)}$

$mass_energy ≔ 19.78539678$

(3.15.3)

Given a distance function, find the speed by differentiation, speed at 2.5 seconds and at 5 blinks by evaluation, and distance traveled between 1 and 2.5 seconds by integration and by subtraction of the distance function values.

>	$distance ≔ \frac{5}{1 + 4 {&ExponentialE;}^{- 3 t}}$

$distance ≔ \frac{5}{1 + 4 {&ExponentialE;}^{- 3 t}}$

(3.16.1)

To find the speed function, differentiate the distance function.

>	$speed ≔ \frac{&DifferentialD;}{&DifferentialD; t} distance$

$speed ≔ \frac{60 {&ExponentialE;}^{- 3 t}}{{(1 + 4 {&ExponentialE;}^{- 3 t})}^{2}}$

(3.16.2)

To find the speed at 2.5 seconds, evaluate the speed function at t=2.5.

>	$\binom{speed}{} \| \binom{}{t = 2.5}$

$0.03303871496$

(3.16.3)

To find the speed at 5 blinks, evaluate the speed function at t=5. $⟦blink⟧$ , converted to seconds.

>	$dt ≔ convert (5., units, blink, s)$

$dt ≔ 4.320000000$

(3.16.4)

>	$\binom{speed}{} \| \binom{}{t = dt}$

$0.0001411518555$

(3.16.5)

By using a definite integral, you can determine the distance traveled from the speed function.

>	$\int_{1}^{2.5} speed &DifferentialD; t$

$0.8193365798$

(3.16.6)

The distance traveled can also be calculated directly from the distance function.

>	$\binom{distance}{} \|\binom{}{t = 2.5} - \binom{distance}{}\| \binom{}{t = 1}$

$4.988962733 - \frac{5}{1 + 4 {&ExponentialE;}^{−3}}$

(3.16.7)

>	$evalf ()$

$0.819336584$

(3.16.8)

Find the minimum and maximum length of 1.2 yards, 1 meter, 3.2 feet, and 0.6 fathoms.

>	$values ≔ op (zip ((x, y) \mapsto convert (x, units, y, m), [1.2, 1, 3.2, 0.6], [yd, m, ft, fathom]))$

$values ≔ 1.097280000, 1, 0.9753600000, 1.097280000$

(3.17.1)

>	$\min (values) &num; meters$

$0.9753600000$

(3.17.2)

>	$\max (values) &num; meters$

$1.097280000$

(3.17.3)

Given a torque of 3 newton meters, how much energy is required to move a lever through 10 degrees?

The energy required is the product of the torque and the angle in radians.

>	$energy ≔ torque \cdot angle$

$energy ≔ torque angle$

(3.18.1)

>	$eval (energy, [torque = 3, angle = convert (10, units, \deg, radians)])$

$\frac{π}{6}$

(3.18.2)

>	$evalf () &num; joules$

$0.5235987758$

(3.18.3)

The Hyper-X can travel at speeds up to 7200 miles per hour. How long would it take to circle the earth at maximum speed (assuming it could carry sufficient fuel)? How far does it travel in a 10 second flight at maximum speed?

To find the time to circle the earth, divide the distance by the speed.

The meter was originally defined as 1/10,000,000 th the distance from the North Pole to the Equator on the meridian passing through Paris. Therefore, 40,000 kilometers is a good approximation of the circumference of the earth.

>	$\frac{40000}{convert (7200, units, mph, \frac{km}{s})} &num; seconds$

$\frac{156250000}{12573}$

(3.19.1)

>	$convert (, units, s, h) &semi; &num; hours$

$\frac{390625}{113157}$

(3.19.2)

>	$evalf () &num; hours$

$3.452062179$

(3.19.3)

To find the distance traveled, multiply the speed by the time.

>	$10 convert (7200, units, mph, \frac{m}{s}) &semi; &num; meters$

$\frac{804672}{25}$

(3.19.4)

>	$evalf ()$

$32186.88000$

(3.19.5)

>	$convert (, units, m, km) &num; kilometers$

$32.18688000$

(3.19.6)

Given 1032 UK gallons of oil, how many cylindrical cans with a height of 1.2 feet and diameter of 0.9 feet could you fill?

The volume of a cylinder is $h π {(\frac{d}{2})}^{2}$ .

>	$\frac{convert (1032, units, gal [UK], {ft}^{3})}{1.2 \cdot π \cdot {(\frac{0.9}{2})}^{2}} &semi; &num; unitless number$

$217.0284617$

(3.20.1)

Given an power gain from 332 microwatts to 23 milliwatts, what is the gain in decibels? What would the decibel gain be if the increase were a voltage increase?

A gain is a quotient of the final value divided by the initial value.

>	$gain ≔ \frac{convert (23, units, mW, uW)}{332}$

$gain ≔ \frac{5750}{83}$

(3.21.1)

To determine the decibel gain, use the formula:

>	$10 log10 (gain) &semi; &num; decibels$

$\frac{10 \ln (\frac{5750}{83})}{\ln (10)}$

(3.21.2)

>	$evalf () &semi; &num; decibels$

$18.40589752$

(3.21.3)

Power is proportional to the square of the voltage. Therefore, the decibel increase corresponding to the voltage gain should be a factor of 2 times that of the power gain.

>	$voltage_gain ≔ \frac{convert (23, units, mV, uV)}{332}$

$voltage_gain ≔ \frac{5750}{83}$

(3.21.4)

To determine the decibel gain, use the formula:

>	$20 log10 (voltage_gain) &semi; &num; decibels$

$\frac{20 \ln (\frac{5750}{83})}{\ln (10)}$

(3.21.5)

>	$evalf () &semi; &num; decibels$

$36.81179504$

(3.21.6)

Given an initial velocity of 2.4 meters per second and an otherwise unspecified uniform acceleration, what is the generic formula for position? For starting at the origin and acceleration being gravity in the opposite direction to the initial velocity? After 0.4 seconds?

The general formula for velocity in a uniform acceleration is:

>	$v ≔ v0 + a \cdot t$

$v ≔ a t + v0$

(3.22.1)

Substitute the given starting velocity.

>	$vs ≔ \binom{v}{} \| \binom{}{v0 = 2.4 ⟦\frac{m}{s}⟧}$

$vs ≔ a t + 2.4 ⟦\frac{m}{s}⟧$

(3.22.2)

Integrating this over time answers the first question.

>	$x ≔ x0 + \int_{0}^{t0} vs &DifferentialD; t$

$x ≔ x0 + (0.5000000000 a {t0}^{2} {(⟦\frac{s}{m}⟧)}^{2} + 2.400000000 t0 ⟦\frac{s}{m}⟧) ⟦\frac{m^{2}}{s^{2}}⟧$

(3.22.3)

Starting at the origin and with the specified gravity:

>	$gravity ≔ evalf (ScientificConstants :- Constant (g, units)) &semi;$

$gravity ≔ 9.80665 ⟦\frac{m}{s^{2}}⟧$

(3.22.4)

>	$\binom{x}{} \| \binom{}{[x0 = 0 ⟦m⟧, a = - gravity]}$

$(- 4.903325000 {t0}^{2} ⟦\frac{m}{s^{2}}⟧ {(⟦\frac{s}{m}⟧)}^{2} + 2.400000000 t0 ⟦\frac{s}{m}⟧) ⟦\frac{m^{2}}{s^{2}}⟧$

(3.22.5)

After 0.4 seconds, this is the location:

>	$\binom{}{} \| \binom{}{t0 = 0.4 ⟦s⟧}$

$(- 0.7845320000 ⟦\frac{m}{s^{2}}⟧ {(⟦\frac{s}{m}⟧)}^{2} {⟦s⟧}^{2} + 0.9600000000 ⟦s⟧ ⟦\frac{s}{m}⟧) ⟦\frac{m^{2}}{s^{2}}⟧$

(3.22.6)

>	$combine (, units)$

$0.1754680000 ⟦m⟧$

(3.22.7)

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