If your solution to stated problem does not match the given solution, you should self-critique per instructions at

 

http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm.

 

Your solution, attempt at solution. 

 

If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it.  This response should be given, based on the work you did in completing the assignment, before you look at the given solution.

 

062.  General rate definition.

 

 

Question:  `q001

A 'graph rectangle' is a rectangle, one of whose sides is part of the horizontal axis.

On a graph of speed in miles / hour vs. clock time in hours, we find a graph rectangle with base 3 and altitude 40.

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

Given Solution: 

A graph of y vs. x represents y on the vertical axis and x on the horizontal axis.  So a graph of speed in miles / hour vs. clock time in hours represents speed as the 'vertical' quantity and clock time as the 'horizontal' coordinate. 

On the present graph, the speed is said to be in miles/hour and the clock time is said to be in hours. 

 

Thus the altitude of the graph represents the speed in miles / hour, and the base represents the change in clock time in hours.

 

In the process of solving this problem you should have sketched a graph, with the vertical axis labeled 'speed in miles / hour' and the horizontal axis labeled 'clock time in hours'. 

When you multiply the base of the rectangle by the altitude to get the area, your result will represent the product of 40 mile / hour and 3 hours.

The units of the product are miles / hour * hours.

 

miles / hour * hours  = miles / hour * hours / 1 = miles * hours / (hour * 1) = miles * hour / hour = mile * (hours / hour) = miles

 

According to the order of operations this could be written in standard mathematical notation as

 

.

Thus the area represents the quantity

 

 

This makes perfect sense.  Recall that 40 miles/hour is the speed and 3 hours the time interval.  Moving at 40 miles/hour for 3 hours, an object would move 120 miles.

 

Any 'graph rectangle' which represents speed vs. time interval will have an area which represents the product of the speed and the time interval.  This product is the distance an object will travel at that speed, during that time interval.

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique rating:

Question:  `q002

On a graph of income stream in dollars per month vs. clock time in months, we find a graph rectangle with base 36 and altitude 1000. 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

Given Solution: 

A graph of y vs. x represents y on the vertical axis and x on the horizontal axis.  So a graph of income stream in dollars / month vs. clock time in months represents income stream as the 'vertical' quantity and clock time as the 'horizontal' coordinate. 

On the present graph, the income stream is said to be in dollars / month and the clock time is said to be in months. 

 

Thus the altitude of the graph represents the income stream in dollars / month, and the base represents the change in clock time in months.

 

In the process of solving this problem you should have sketched a graph, with the vertical axis labeled 'income stream in dollars / month' and the horizontal axis labeled 'clock time in months'. 

When you multiply the base by the altitude to get the area, your result will represent the product of 1000 dollars / month and 36 months.

 

area = 1000 * 36 = 36 000, representing 1000 dollars / month * 36 months = 36 000 dollars / month * months.

 

The units of the product are dollars / month * months.

 

dollars / month * months  = dollars / month * months / 1 = dollars * months / (month * 1) = dollars * month / month = mile * (months / month) = dollars

 

Thus the area represents the quantity

This makes perfect sense.  Recall that 1000 dollars/month is the income stream and 36 months the time interval.  Earning money at a rate of 1000 dollars/month for 36 months, we would earn 36 000 dollars.

 

Any 'graph rectangle' which represents income stream vs. time interval will have an area which represents the product of the income stream and the time interval. 

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique rating:

Question:  `q003On a graph of force in pounds vs. position in feet, we find a graph rectangle with base 200 and altitude 30. 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

Given Solution: 

A graph of y vs. x represents y on the vertical axis and x on the horizontal axis.  So a graph of force in lb vs. position in ft represents force as the 'vertical' quantity and position as the 'horizontal' coordinate. 

On the present graph, the force is said to be in lb and the position is said to be in ft. 

 

Thus the altitude of the graph represents the force in lb, and the base represents the change in position in ft.

 

In the process of solving this problem you should have sketched a graph, with the vertical axis labeled 'force in lb' and the horizontal axis labeled 'position in ft'. 

When you multiply the base of the rectangle by the altitude to get the area, your result will represent the product of 30 lb and 200 ft.

The units of the product are lb * ft.  These units do not simplify.

 

Thus the area represents the quantity

This quantity might not make sense in terms of what you know right now.  However it does make physical sense, and actually does so with a couple of different interpretations.  You aren't expected to know these interpretations at this point, but it would be your advantage to 'get the wheels turning' by thinking a little bit about them now:

 

In one interpretation you can think of pushing a box down a long hall. 

In another interpretation you can think of two people trying to raise a heavy rock with levers. 

A graph of force vs. position will usually have the 'work' interpretation.  In this case any 'graph rectangle' will have an area which represents the product of the force and the displacement, which represents work.  The topic of work and energy is developed elsewhere in the course.

 

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique rating:

Question:  `q004On a graph of density in grams / centimeter vs. position in centimeters, we find a graph rectangle with base 16 and altitude 50.

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

Given Solution: 

A graph of y vs. x represents y on the vertical axis and x on the horizontal axis.  So a graph of density in g / cm vs. position in cm represents density as the 'vertical' quantity and position as the 'horizontal' coordinate. 

On the present graph, the density is said to be in g / cm and the position is said to be in cm. 

 

Thus the altitude of the graph represents the density in g / cm, and the base represents the change in position in cm.

 

In the process of solving this problem you should have sketched a graph, with the vertical axis labeled 'density in g / cm' and the horizontal axis labeled 'position in cm'. 

When you multiply the base of the rectangle by the altitude to get the area, your result will represent the product of 50 g / cm and 16 cm.

The units of the product are g / cm * cm.  These units simplify to g * cm / cm = g * (cm/cm) = g * 1 = g.

 

Thus the area represents the quantity

This quantity has a fairly straightforward meaning. 

 

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique rating:

 

Question:  `q005A 'graph trapezoid' is defined by two points on a graph, as follows:

The 'graph slope' between two points is the slope of the 'slope segment' of the graph trapezoid defined by the two points.

On a graph of speed in miles / hour vs. clock time in hours, we find graph points (2, 50) and (7, 60)

 

A graph of y vs. x represents y on the vertical axis and x on the horizontal axis.  So a graph of speed in miles / hour vs. clock time in hours represents speed as the 'vertical' quantity and clock time as the 'horizontal' coordinate. 

On the present graph, the speed is said to be in miles/hour and the clock time is said to be in hours. 

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

Given Solution: 

A graph of y vs. x represents y on the vertical axis and x on the horizontal axis.  So a graph of speed in miles / hour vs. clock time in hours represents speed as the 'vertical' quantity and clock time as the 'horizontal' coordinate. 

On the present graph, the speed is said to be in miles/hour and the clock time is said to be in hours. 

 

Thus an altitude of the graph represents a velocity in miles / hour, and the base represents the change in clock time in hours.

 

In the process of solving this problem you should have sketched a graph, with the vertical axis labeled 'speed in miles / hour' and the horizontal axis labeled 'clock time in hours'. 

The 'slope segment' of this trapezoid rises from its left altitude 50 to its right altitude 60, representing the velocities 50 miles/hr and 60 miles/hr

 

The rise of this trapezoid is the rise of its 'slope segment', which is the change in its 'graph altitude'. 

The run of this trapezoid is the run of its 'slope segment', which is the change in the horizontal coordinate on the corresponding interval

The slope of the trapezoid is 'rise / run', the rise of the slope segment divided by its run. 

 

Details of the calculation of units: 

 

The order of operations can be used to represent this series of calculations as follows:

 

To interpret the meaning of this slope:

The base of the equal-area rectangle represents the change in the 'horizontal' quantity.  The 'horizontal quantity' in this case the clock time, so the base represents the change in clock time.

The equal-area rectangle is formed by cutting the trapezoid along a horizontal line originating from the midpoint of the slope segment.  This cuts the trapezoid into two pieces, one being a triangle which can be rotated 180 deg and 'pasted' to the other piece to form a rectangle.  Since the rectangle is formed by the two pieces of the original trapezoid, it has the same area as that trapezoid.

 

The 'graph altitude' of this rectangle is halfway between the 'graph altitudes' of the original trapezoid, and can therefore be calculated by averaging the two 'altitudes':

In this case the 'graph altitudes' are 50 and 60, representing 50 miles/hr and 60 miles/hr, so the altitude of the 'equal-area rectangle' is

and represents

which is the average of the initial and final velocities on the interval.

 

This quantity makes perfect sense as the approximate average velocity for this interval, which based only on the given information we expect (but are not assured) will be between the initial and final velocity and will lie somewhere around halfway between the two.

 

When you multiply the base of the rectangle by the altitude to get the area, your result will represent the product of the average altitude and the base of the original trapezoid.

 

In this case we obtain

The units of the product are miles / hour * hours = miles

 

Thus the area represents the quantity

This makes perfect sense

We can generalize this:

 

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique rating:

Question:  `q006On a graph of income stream in dollars per month vs. clock time in months, we find the two points (2, 50) and (7, 60). 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

Given Solution: 

A graph of y vs. x represents y on the vertical axis and x on the horizontal axis.  So a graph of income stream in dollars/month vs. clock time in month(s) represents income stream as the 'vertical' quantity and clock time as the 'horizontal' coordinate. 

On the present graph, the income stream is said to be in dollars/month and the clock time is said to be in month(s). 

 

Thus an altitude of the graph represents a income stream in dollars/month, and the base represents the change in clock time in month(s).

 

In the process of solving this problem you should have sketched a graph, with the vertical axis labeled 'income stream in dollars/month' and the horizontal axis labeled 'clock time in month(s)'. 

The 'slope segment' of this trapezoid rises from its left altitude 50 to its right altitude 60, representing the income streams 50 dollars/month and 60 dollars/month

 

The rise of this trapezoid is the rise of its 'slope segment', which is the change in its 'graph altitude'. 

The run of this trapezoid is the run of its 'slope segment', which is the change in the horizontal coordinate on the corresponding interval

The slope of the trapezoid is 'rise / run', the rise of the slope segment divided by its run. 

 

Details of the calculation of units: 

 

The order of operations can be used to represent this series of calculations as follows:

 

To interpret the meaning of this slope:

The base of the equal-area rectangle represents the change in the 'horizontal' quantity.  The 'horizontal quantity' in this case the clock time, so the base represents the change in clock time.

The equal-area rectangle is formed by cutting the trapezoid along a horizontal line originating from the midpoint of the slope segment.  This cuts the trapezoid into two pieces, one being a triangle which can be rotated 180 deg and 'pasted' to the other piece to form a rectangle.  Since the rectangle is formed by the two pieces of the original trapezoid, it has the same area as that trapezoid.

 

The 'graph altitude' of this rectangle is halfway between the 'graph altitudes' of the original trapezoid, and can therefore be calculated by averaging the two 'altitudes':

In this case the 'graph altitudes' are 50 and 60, representing 50 dollars/month and 60 dollars/month, so the altitude of the 'equal-area rectangle' is

and represents

which is the average of the initial and final velocities on the interval.

 

This quantity makes perfect sense as the approximate average income stream for this interval, which based only on the given information we expect (but are not assured) will be between the initial and final income stream and will lie somewhere around halfway between the two.

 

When you multiply the base of the rectangle by the altitude to get the area, your result will represent the product of the average altitude and the base of the original trapezoid.

 

In this case we obtain

The units of the product are dollars/month * month(s) = dollars

 

Thus the area represents the quantity

This makes perfect sense

We can generalize this:

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique rating:

Question:  `q007On a graph of force in pounds vs. position in feet, we find a graph rectangle with base 200 and altitude 30. 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

Given Solution: 

On a graph of force in pounds vs. position in feet, we find a graph rectangle with base 200 and altitude 30. 

A 'graph trapezoid' is defined by two points on a graph, as follows:

The 'graph slope' between two points is the slope of the 'slope segment' of the graph trapezoid defined by the two points.

On a graph of force in lb vs. position time in ft, we find graph points (70, 20) and (190, 50)

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

Given Solution: 

A graph of y vs. x represents y on the vertical axis and x on the horizontal axis.  So a graph of force in lb vs. position in ft represents force as the 'vertical' quantity and position as the 'horizontal' coordinate. 

On the present graph, the force is said to be in lb and the position is said to be in ft. 

 

Thus an altitude of the graph represents a force in lb, and the base represents the change in position in ft.

 

In the process of solving this problem you should have sketched a graph, with the vertical axis labeled 'force in lb' and the horizontal axis labeled 'position in ft'. 

The 'slope segment' of this trapezoid rises from its left altitude 20 to its right altitude 50, representing the forces 20 lb and 50 lb.  

 

The rise of this trapezoid is the rise of its 'slope segment', which is the change in its 'graph altitude'. 

The run of this trapezoid is the run of its 'slope segment', which is the change in the horizontal coordinate on the corresponding interval

The slope of the trapezoid is 'rise / run', the rise of the slope segment divided by its run. 

Details of the calculation of units: 

To interpret the meaning of this slope:

The base of the equal-area rectangle represents the change in the 'horizontal' quantity.  The 'horizontal quantity' in this case the position, so the base represents the change in position.

The equal-area rectangle is formed by cutting the trapezoid along a horizontal line originating from the midpoint of the slope segment.  This cuts the trapezoid into two pieces, one being a triangle which can be rotated 180 deg and 'pasted' to the other piece to form a rectangle.  Since the rectangle is formed by the two pieces of the original trapezoid, it has the same area as that trapezoid.

 

The 'graph altitude' of this rectangle is halfway between the 'graph altitudes' of the original trapezoid, and can therefore be calculated by averaging the two 'altitudes':

In this case the 'graph altitudes' are 20 and 50, representing 20 lb and 50 lb, so the altitude of the 'equal-area rectangle' is

and represents

which is the average of the initial and final forces on the interval.

 

This quantity makes perfect sense as the approximate average force for this interval, which based only on the given information we expect (but are not assured) will be between the initial and final force and will lie somewhere around halfway between the two.

 

When you multiply the base of the rectangle by the altitude to get the area, your result will represent the product of the average altitude and the base of the original trapezoid.

 

In this case we obtain

The units of the product are lb * ft = lb * ft

 

Thus the area represents the quantity

This makes perfect sense

We can generalize this:

 

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique rating:

Question:  `q008On a graph of density in grams / centimeter vs. position in centimeters, we find the points (5, 12) and (20, 10).

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

Given Solution: 

A 'graph trapezoid' is defined by two points on a graph, as follows:

The 'graph slope' between two points is the slope of the 'slope segment' of the graph trapezoid defined by the two points.

On a graph of density in grams / cm vs. position time in cm, we find graph points (5, 12) and (20, 10)

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

Given Solution: 

A graph of y vs. x represents y on the vertical axis and x on the horizontal axis.  So a graph of density in grams / cm vs. position in cm represents density as the 'vertical' quantity and position as the 'horizontal' coordinate. 

On the present graph, the density is said to be in grams / cm and the position is said to be in cm. 

 

Thus an altitude of the graph represents a density in grams / cm, and the base represents the change in position in cm.

 

In the process of solving this problem you should have sketched a graph, with the vertical axis labeled 'density in grams / cm' and the horizontal axis labeled 'position in cm'. 

The 'slope segment' of this trapezoid rises from its left altitude 12 to its right altitude 10, representing the densityPlural 12 grams / cm and 10 grams / cm

 

The rise of this trapezoid is the rise of its 'slope segment', which is the change in its 'graph altitude'. 

The run of this trapezoid is the run of its 'slope segment', which is the change in the horizontal coordinate on the corresponding interval

The slope of the trapezoid is 'rise / run', the rise of the slope segment divided by its run. 

Details of the calculation of units: 

The order of operations can be used to represent this series of calculations as follows:

 

To interpret the meaning of this slope:

The base of the equal-area rectangle represents the change in the 'horizontal' quantity.  The 'horizontal quantity' in this case the position, so the base represents the change in position.

The equal-area rectangle is formed by cutting the trapezoid along a horizontal line originating from the midpoint of the slope segment.  This cuts the trapezoid into two pieces, one being a triangle which can be rotated 180 deg and 'pasted' to the other piece to form a rectangle.  Since the rectangle is formed by the two pieces of the original trapezoid, it has the same area as that trapezoid.

 

The 'graph altitude' of this rectangle is halfway between the 'graph altitudes' of the original trapezoid, and can therefore be calculated by averaging the two 'altitudes':

In this case the 'graph altitudes' are 12 and 10, representing 12 grams / cm and 10 grams / cm, so the altitude of the 'equal-area rectangle' is

and represents

which is the average of the initial and final densities on the interval.

 

This quantity makes perfect sense as the approximate average density for this interval, which based only on the given information we expect (but are not assured) will be between the initial and final density and will lie somewhere around halfway between the two.

 

When you multiply the base of the rectangle by the altitude to get the area, your result will represent the product of the average altitude and the base of the original trapezoid.

 

In this case we obtain

The units of the product are grams / cm * cm = grams

 

Thus the area represents the quantity

This makes perfect sense

We can generalize this:

 

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique rating:

Question:  `q00. 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

Given Solution: 

 

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique rating:

Question:  `q00. 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

Given Solution: 

 

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique rating:

Question:  `q00. 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

Given Solution: 

 

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique rating:

Question:  `q00. 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

Given Solution: 

 

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique rating: