1. ## Need HELP: Variation

The Problem
(1)
The time T needed to mill a channel varies directly as the length, width, and depth of the channel, assuming the cutting tool and material being machined stay constant.

a) Write this as an equation using a constant of proportionality;

b) A channel 30 cm long, 5.0cm wide and 4.5 cm deep took 36 minutes to mill. Use this information to determine the constant proportionality, with units.

c) How long will it take to mill a channel in the same material, 45 cm long, 6.4 cm wide, and 3.75 cm deep?

d) How long would the machining job of b) above take, if the length, width, and depth were all doubled

(2) It takes three men three hours to assemble three shelving units. How long will it take nine men to assemble twelve shelving units? Prove your answer

(3) The time taken to machine out the interior of a hollow cylinder in a shaft on a lathe varies directly as the length of the cylinder and directly as the square of the internal diameter.

a) Write this as an equation using a constant of proportionality;

b) It took 22 minutes to machine out a cylinder 14.5 cm long and 3.45 cm in diameter. Use this information to determine the constant of proportionality, with units.

c) How long would it take to machine out a cylinder 9.6 cm long, and 5.90 cm in diameter?

(4) The cutting speed of a milling cutter varies directly as the diameter of the cutter and directly as the RPM of the cutter.

a) Write this as an equation using a constant of proportionality;

b) Turing a milling cutter 8.5 cm in diamter at 6250 RPM results in a cutting speed of 2.78 m/s. Use this information to determine the constant of proportionality, with units.

c) Calculate the needed RPM to produce a recommended cutting speed of 4.25 m/s (in another material), using a 12.5 diameter cutter.

2. Originally Posted by bigstarz
The Problem
(1)
The time T needed to mill a channel varies directly as the length, width, and depth of the channel, assuming the cutting tool and material being machined stay constant.

a) Write this as an equation using a constant of proportionality;
$T=\kappa \times l \times w \times d$

b) A channel 30 cm long, 5.0cm wide and 4.5 cm deep took 36 minutes to mill. Use this information to determine the constant proportionality, with units.
Given the above data we have:

$36= \kappa \times 30 \times 5 \times 4.5$

so:

$\kappa=\frac{36}{30 \times 5 \times 4.5}\approx 0.0533 \mbox{ min/cm^3}$

c) How long will it take to mill a channel in the same material, 45 cm long, 6.4 cm wide, and 3.75 cm deep?

d) How long would the machining job of b) above take, if the length, width, and depth were all doubled
I'm sure you can now do these two parts yourself.

RonL

3. Originally Posted by bigstarz[B
(2) [/B] It takes three men three hours to assemble three shelving units. How long will it take nine men to assemble twelve shelving units? Prove your answer
If it takes three men three hours to assemble three shelving units, it will
take one man three hours to assemble one shelving unit. therefore each
man can assemble 1/3 of a shelving unit per hour.

Twelve shelving units is 36 thirds of a shelving unit, so 36 man hours of
work is required to assemble them, which is 4 hours of work by nine men.

RonL

4. Originally Posted by bigstarz
(4) The cutting speed of a milling cutter varies directly as the diameter of the cutter and directly as the RPM of the cutter.

a) Write this as an equation using a constant of proportionality;

b) Turing a milling cutter 8.5 cm in diamter at 6250 RPM results in a cutting speed of 2.78 m/s. Use this information to determine the constant of proportionality, with units.
$\kappa=\frac{\mbox{speed}}{\mbox{diameter}}$
Thus,
$\kappa=\frac{\frac{2\pi r}{t}}{2\pi r}=\frac{1}{t}$

Notes:
Where "t" is time for revolution.
To find the speed you divide distance by time.
Distance in this case the circumfurence of circle.

5. Originally Posted by ThePerfectHacker
[QUOTE=bigstarz
(4) The cutting speed of a milling cutter varies directly as the diameter of the cutter and directly as the RPM of the cutter.

a) Write this as an equation using a constant of proportionality;

b) Turing a milling cutter 8.5 cm in diamter at 6250 RPM results in a cutting speed of 2.78 m/s. Use this information to determine the constant of proportionality, with units.
$\kappa=\frac{\mbox{speed}}{\mbox{diameter}}$[/QUOTE]

The cutting speed of a milling cutter varies directly as the diameter of the cutter and directly as the RPM of the cutter.

So you should have:

$a)\ speed=\kappa \times diameter \times RPM$,

then:

$
\kappa=\frac{speed}{diameter \times RPM}
$
,

and so:

$
b)\ \kappa=\frac{2.78}{8.5 \times 6250}=0.0000523\ \mbox{(m/s)/(cm/s)}
$

RonL